A Morse-Bott approach to monopole Floer homology and the Triangulation conjecture
aa r X i v : . [ m a t h . G T ] F e b A Morse-Bott approach to monopole Floerhomology and the Triangulation conjecture
Francesco Lin
Abstract.
In the present work we generalize the construction of monopole Floer homologydue to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singulari-ties. Focusing then on the special case of a three-manifold equipped equipped with a spin c structure which is isomorphic to its conjugate, we define the counterpart in this contextof Manolescu’s recent Pin(2)-equivariant Seiberg-Witten-Floer homology. In particular, weprovide an alternative approach to his disproof of the celebrated Triangulation conjecture. Department of Mathematics, Massachusetts Institute of Technology
E-mail address : [email protected] ontents Introduction 5Chapter 1. Basic setup 111. The monopole equations 112. Blowing up the configuration spaces 153. Completion and slices 174. Perturbations 21Chapter 2. The analysis of Morse-Bott singularities 291. Hessians and Morse-Bott singularities 292. Moduli spaces of trajectories 353. Transversality 484. Compactness and finiteness 595. Gluing 666. The moduli space on a cobordism 83Chapter 3. Floer homology for Morse-Bott singularities 931. Homology of smooth manifolds via stratified spaces 932. Floer homology 1003. Invariance and functoriality 111Chapter 4. Pin(2)-monopole Floer homology 1231. An involution in the theory 1232. Equivariant perturbations and Morse-Bott transversality 1273. Invariant chains and Floer homology 1394. Some computations 1465. Manolescu’s β invariant and the Triangulation conjecture 153Bibliography 159 ntroduction The present work has two purposes. The first one is to generalize Kronheimer andMrowka’s construction of monopole Floer homology ([
KM07 ]) to the case in which the gradi-ent flow has Morse-Bott singularities. The second one is to define in this framework the coun-terpart of Manolescu’s recent Pin(2)-equivariant Seiberg-Witten-Floer homology ([
Man13a ]),and use it to give an alternative approach to his recent breakthrough, the disproof of the long-standing Triangulation conjecture.Monopole Floer homology is a set of invariants of three manifolds equipped with a spin c structure. They are obtained by studying the gradient flow equation of the Chern-Simons-Dirac functional, the Seiberg-Witten monopole equations. These equations have a natural S -symmetry. The construction of [ KM07 ] deals with case in which the singularities ofthis flow are non degenerate, meaning that the Hessian is invertible at those points. Theanalogue in the finite dimensional case are the usual Morse singularities. However, in manysituations it is convenient to be able to deal with a more general kind of singularity, which isanalogous to the Morse-Bott case in a finite dimensional setting. These critical points formsmooth submanifolds, and the Hessian is invertible in the normal directions. For example,these type of objects naturally arise when dealing with the Chern-Simons-Dirac functional onSeifert-fibered spaces, see [
MOY97 ].Another context in which this kind of singularities arises is the case in which the spin c structure is actually induced by a genuine spin structure. In this case, it has been known fora long time (see for example [ Mor96 ]) that the problem has more symmetry due to the thequaternionic structure of the spinor bundle. By exploiting this additional data, Manolescuwas able to construct his new invariants of three manifolds based on the Lie groupPin(2) = S ∪ j · S ⊂ H , and disproof the longstanding Triangulation conjecture. The Triangulation conjecture.
The present historical discussion is taken from the niceexposition [
Man13b ], to which we refer for more details. The Triangulation conjecture, firstformulated by Kneser in [
Kne26 ], asserts that every topological manifold is homeomorphicto a simplicial complex. This was known to be true for manifolds of dimension at most three([
Rad25 ], [
Moi52 ]) and false in dimension four ([
AM90 ]), while the answer was unknownin dimension at least five.One could also ask the stronger question of whether every topological manifold M admitsa pl -structure, i.e it is homeomorphic to a simplicial complex such that the link of each vertexis pl -homeomorphic to a sphere. This was answered by Kirby and Siebenmann in [ KS77 ]. In particular, they construct a class∆( M ) ∈ H ( M ; Z / Z )which vanishes if an only if M admits a pl -structure, and show that in each dimension atleast five this obstruction is effective.The case of general simplicial complexes is more subtle, and has very deep connectionswith low dimensional topology. Define the homology cobordism group Θ H to be the groupwhose elements are oriented pl -homology three spheres up to homology cobordism and forwhich the sum is given by connected sum. The definition makes sense in every dimension,but by a result of Kervaire [ Ker69 ] it is trivial in all these other cases. On the other hand,one can show that Θ H is not trivial because of the existence of a surjective map µ : Θ H → Z / Z , the Rokhlin homomorphism. This is obtained by sending a homology three sphere to sign( W ) / W is any spin four manifold bounding it.Given a triangulation K on M (which we suppose to be closed and oriented), one canform the so called Sullivan-Cohen-Sato class c ( K ) = X σ ∈ K ( n − [link K ( σ )] · σ ∈ H n − ( M ; Θ ) ∼ = H ( M ; Θ ) . The short exact sequence of groups(0.1) 0 → ker µ → Θ H µ −→ Z / Z → H ( M ; Θ H ) µ ∗ −→ H ( M ; Z / Z ) δ −→ H ( M ; ker µ )and it can be shown that the image of c ( K ) is the class ∆( M ) discussed above. In particularthis implies that if M admits a triangulation then δ (∆( M )) is zero. Work of Galewski-Stern[ GS80 ] and Matumoto [
Mat78 ] shows that the vanishing of this element is also a sufficientcondition for the existence of a triangulation. Of course, if the exact sequence (0.1) splitsthen δ (∆( M )) is always zero. In [ GS80 ] and [
Mat78 ] it is also shown that this condition issufficient, hence in order to disproof the Triangulation conjecture it is sufficient to show thatthe sequence does not split. In fact, Manolescu proved the following result.
Theorem
Man13a ]) . There are no elements of order two and Rokhlin invariantone in the homology cobordism group. Hence the Triangulation conjecture is false in alldimensions at least five.
In general, the structure of the homology cobordism group remains a mystery. It is knownthat it is not finitely generated ([
Fur90 ], [
FS90 ]), but for example it is not known whetherit has torsion or not. In order to prove this theorem, Manolescu defines for each homologysphere Y a Z -valued invariant β ( Y ) satisfying the following properties:(1) it is invariant under homology cobordism;(2) it reduces modulo two to the Rokhlin invariant µ ;(3) β ( − Y ) = − β ( Y ).The theorem follows because if 2[ Y ] = 0 in Θ H , then [ Y ] = [ − Y ] so β ([ Y ]) = β ( − [ Y ]) = − β ([ Y ]) NTRODUCTION 7 hence β ([ Y ]) is zero, and so is the Rokhlin invariant. The invariant β arises as an analoguein the Pin(2)-equivariant context of Frøyshov’s invariant in Seiberg-Witten Floer homol-ogy ([ Frø10 ], [
KM07 ]) and Ozsvath-Szabo’s correction term in Heegaard Floer homology([
OS03 ]). We now describe its construction in our theory.
An overview of
Pin(2) -monopole Floer homology.
Manolescu’s construction of hisnew invariants follows the framework of his previous work [
Man03 ], and relies on the theoryof Conley index and finite dimensional approximations of the monopole equations. In thepresent work we define the counterpart of these objects in Kronheimer and Mrowka’s theory.We expect the two definitions to agree for rational homology spheres. Even though theconstruction is quite involved, the final result has some desirable features that are missing inthe case of Manolescu’s invariants. To each closed oriented three manifold Y equipped witha self-conjugate spin c structure s (i.e. s is isomorphic to its conjugate ¯ s ) we associate threegraded topological abelian groups called Pin(2) -monopole Floer homology groups . These aredenoted by c HS • ( Y, s ) c HS • ( Y, s ) HS • ( Y, s )where the S stands for spin , and they are pronounced “ H-S-to ”, “
H-S-from ” and “
H-S-bar ”respectively. These groups are also graded topological modules over the graded topologicalring R = F [[ V ]][ Q ] / ( Q )where V and Q have degree respectively − −
1, and F is the field with two elements.The three groups are related by a long exact sequence of R -modules(0.2) · · · i ∗ −→ d HM k ( Y, s ) j ∗ −→ d HM k ( Y, s ) p ∗ −→ HM k − ( Y, s ) i ∗ −→ d HM k − ( Y, s ) j ∗ −→ . . . . When compared to Manolescu’s invariants, the groups resulting from this alternative approachhave the following two important additional features: • they are defined for every three manifold, not only for rational homology spheres; • they are functorial in the sense that a spin c cobordism ( X, s X ) from ( Y , s ) to ( Y , s )determines a group homomorphism c HS • ( X, s X ) : c HS • ( Y , s ) → c HS • ( Y , s ) , and similarly for the other versions. In the case the spin c structure s X is actually agenuine spin structure, the map is in fact a homomorphism of graded R -modules.Furthermore, in forthcoming work (see for example [ Lin15 ]) we will develop in this set-ting some computational tools that are available in monopole Floer homology but not inManolescu’s theory. For many of these developments it is useful to also take into accountalso not self-conjugate spin c structures. Denote by the action by conjugation on the setSpin c ( Y ) of spin c structures of Y , and by [ s ] the equivalence class of s . We then define for s = ¯ s the group c HS • ( Y, [ s ]) as d HM • ( Y, s ) = d HM • ( Y, ¯ s )where these are canonically identified via conjugation, and the total group c HS • ( Y ) = M [ s ] ∈ Spin c ( Y ) / c HS • ( Y, [ s ]) . INTRODUCTION
This defines a functor from the category cob of compact connected oriented three manifoldsand isomorphism classes of cobordism between them to the category of topological F [[ V ]]-modules.In the case of S , the Pin(2)-monopole Floer homology groups can be identified as thegraded R -modules HM • ( S ) = F [ V − , V ]][ Q ] / ( Q ) {− } d HM • ( S ) = (cid:0) F [ V − , V ]][ Q ] / ( Q ) {− } (cid:1) / R · d HM • ( S ) = R{− } where the braces indicate grading shifts. In particular, the minimum degree of an element isthe to group is zero. For a general homology three sphere the bar group is isomorphic (up tograding shift) to HM • ( S ), the to group is bounded below and the from group is boundedabove. The exact sequence (0.2) implies that the R module given by the image i ∗ (cid:0) HM • ( S ) (cid:1) ⊂ d HM • ( S )decomposes as the direct sum of three copies of the F [[ V ]]-module F [[ V − , V ]] / F [[ V ]], whichare related to each other by the action Q . The invariant β ( Y ) used by Manolescu to proveTheorem 0.1 is then defined so that 2 β ( Y ) + 1 is the minimum grading of an element in themiddle tower, i.e. the one on which Q acts non trivially and Q acts trivially. For example,it is zero in the case of S .The key point of Kronheimer and Mrowka’s construction of monopole Floer homology isthe introduction of the blow-up of the configuration space on which the functional is defined.This is done in order to deal properly with the reducible configurations. In fact, the natural S -action on the configuration space is not free and this caused serious invariance and func-toriality issues in the earlier approaches to the Floer homology of the monopole equations.These have been tackled in various ways in the last twenty years (see for example [ MW01 ],[
Man03 ] and [
Frø10 ]). The approach of [
KM07 ] naturally leads to consider Morse flows onmanifolds where the boundary is invariant for the gradient flow (and is therefore not
Morse-Smale). This requires a definition of Morse homology for a manifold with boundary whichdiffers from the more classical one as it has to deal with some new phenomena, in particularthe existence of boundary obstructed trajectories (see Chapter 2 of [
KM07 ] for an introduc-tion to the finite dimensional case).There are many approaches to Morse-Bott homology in literature, and for many reasonsthe one that fits our problem of defining the invariants the best is the one introduced byFukaya ([
Fuk96 ]) in the context of instanton Floer homology. The chain complex associatedto a Morse-Bott function f on a smooth manifold X that he defines has underlying vectorspace the direct sum of (some modified version of) the singular chain complexes of the criticalsubmanifolds C ∗ ( X, f ) = M C∈ Crit( f ) ˜ C ∗ ( C ) NTRODUCTION 9 and the differential of a chain σ is given by ∂σ = ˜ ∂σ + X C ′ ∈ Crit( f ) σ × ˘ M + ( C , C ′ ) . Here ˜ ∂ denotes the differential in the modified chain complex ˜ C ∗ ( C ) and each element in thesum is the fibered product of σ and the compactified moduli space of trajectories ˘ M + ( C , C ′ )connecting C and C ′ (using the evaluation map on the negative end), which is a chain in C ′ via the evaluation map on the positive end. Of course σ belongs to some appropriateclass of geometric objects which is closed under fibered products with the moduli spaces (in[ Fuk96 ] the author considers polyhedra with singularities in codimension at least two). Fur-thermore, we require some transversality conditions in order for the fibered product to bein this class. The nice feature of this construction is that it works for generic (in a suit-able sense) Morse-Bott perturbations, and it can be adapted to work in our case where alsoboundary obstructedness phenomena come into play. A manifestation of the latter is thatthe compactified moduli spaces of trajectories are not in general manifolds with corners, asthey have a more complicated structure both combinatorially and topologically (see Section19.5 in [
KM07 ]). Nevertheless we will construct a version of the singular chain complex ofa smooth manifold (inspired by the work [
Lip14 ]), and prove that it computes the usualsingular homology. With this in hand, we will define the three Floer chain complexes byadapting the construction sketched above to the case where the manifold X (which in ourcase is the moduli space of configurations) has boundary and the gradient of f is tangent to it.When the spin c structure is self-conjugate, the configuration space admits a naturalPin(2)-action, and the idea is to exploit this additional symmetry to construct a chain com-plexes computing monopole Floer homology groups with an additional chain involution in-duced by the action of the element j in Pin(2). The Pin(2)-monopole Floer homology groupswill be then defined as the homologies of the invariant subcomplexes. In order to do this wewill perturb the problem while respecting the additional symmetry. Unfortunately this classof perturbations will not allow us to achieve genericity in the sense of [ KM07 ], but never-theless they will be generic enough so that we can arrange the singularities to be Morse-Bottand apply our approach.Many of the proofs and results in the present work (and especially in the first part) aregeneralizations of the ones contained in [
KM07 ]. In order to keep the length of the presentwork somehow contained, we will often rely on the work already done there, and we will onlyexpand the details which are significantly different or which are notably interesting. We willassume that the reader has a reasonable understanding of the content of [
KM07 ], and inorder to help her/him we will always give precise references of the omitted passages. We willrefer to [
KM07 ] as the book . Plan of the work.
In Chapter 1, we quickly review the main protagonists of the presentwork, namely the Chern-Simons-Dirac functional and its gradient flow equation, the Seiberg-Witten monopole equations. In particular, we briefly discuss the content of the Chapter4 −
11 in the book regarding the monopole equations, the blow-up of the configurationsspaces, their completions and finally the theory of tame perturbations, which will be essentialin the construction of Pin(2)-monopole Floer homology.
Chapter 2 is dedicated to the local and global analysis of Morse-Bott singularities, fol-lowing Chapters 12 −
19 in the book. After defining them, we will discuss the properties ofthe space of solutions to the monopole equations connecting two such critical submanifolds,and prove the fundamental transversality, compactness and gluing results.In Chapter 3, following Chapters 22 −
25 in the book, we show how to define monopoleFloer homology when the singularities of the flow are Morse-Bott. We construct a modifiedversion of the singular chain complex of a smooth manifold and adapt Fukaya’s approach toour context. We show that the functoriality and invariance properties of the Floer groupshold in this setting, proving among the other things that the result of the new constructioncanonically agrees with the one of Kronheimer and Mrowka.Finally, in Chapter 4 we focus on the case of a three manifold equipped with self-conjugatespin c structure. We show how to perturb the equations in a way which is compatible withthe additional symmetry of the equations. This naturally leads to Morse-Bott singularities.The symmetry of the equations will give rise a symmetry of the chain complex defining ourinvariants, and exploiting this we will be able to define the Pin(2)-monopole Floer homologyand prove all the properties we briefly discussed above. After providing some simple calcula-tions (in particular for S × S and the three-torus), we will give an alternative disproof ofthe Triangulation conjecture. Acknowledgements.
The author would sincerely like to thank his advisor Tom Mrowkafor introducing him to the subject, for suggesting the present problem, and for his patient helpand support throughout the development of the project. Without his guidance and expertisethis work would not have been possible at all. The author would also like to express hisgratitude to Jonathan Bloom, especially for the interesting discussions related to the content ofChapter 3. Finally, he would like to thank Michael Andrews, Lucas Culler, Michael Hutchings,Ciprian Manolescu, Roberto Svaldi and Umut Varolgunes for the useful conversations. Thiswork was partially funded by NSF grants DMS-0805841 and DMS-1005288.HAPTER 1
Basic setup
This chapter contains a quick discussion of the background required for the constructionof Floer homology in the Seiberg-Witten setting. We start by describing the differentialgeometry needed to write down the monopole equations and then introduce the fundamentalconstruction of Kronheimer and Mrowka’s approach, the blow up of the moduli spaces. Wethen set up the functional spaces on which we will study the analytical problems, and recallthe class of perturbation we allow. The material is treated in the same way as in the bookand the aim of our discussion is to review the content and the notation of Chapters 4 to 11.In particular, the result will be cited without proofs (for which we refer the reader to thebook).
1. The monopole equations
In order to introduce the Seiberg-Witten equations we first have to discuss spin c structures.This can be done in many ways and we expose a very concrete one which will be the mostuseful in the rest of the work. Let Y be an oriented closed connected riemannian 3-manifold.A spin c structure s on Y is given by a rank-2 hermitian vector bundle S on Y together witha map ρ : T X → End( S ) , called Clifford multiplication , satisfying the following properties. It is a bundle map thatidentifies
T X isometrically with the subbundle su ( S ) of traceless skew-adjoint endomorphismsof S (equipped with the metric tr( a ∗ b )) and it respects orientations, i.e. if e , e , e is anoriented orthonormal frame then ρ ( e ) ρ ( e ) ρ ( e ) = 1 S . This means that at any point we can always find a basis of the fiber such that the Cliffordmultiplication by ρ ( e i ) are given by the Pauli matrix σ i : σ = (cid:18) i − i (cid:19) σ = (cid:18) −
11 0 (cid:19) σ = (cid:18) ii (cid:19) . The action of ρ is extended to cotangent vectors using the metric, and then to (complexvalued) forms using the rule ρ ( α ∧ β ) = 12 (cid:16) ρ ( α ) ρ ( β ) + ( − deg( α ) deg( β ) ρ ( β ) ρ ( α ) (cid:17) . On a 3-manifold Y spin c structures always exist. In fact one can take a trivialization of thetangent bundle T Y and define on C × Y a Clifford multiplication given globally using the
112 1. BASIC SETUP
Pauli matrices. Furthermore, given a spin c structure ( S , ρ ) and an hermitian line bundle L → Y we can define a new spin c structure given by S = S ⊗ Lρ ( e ) = ρ ( e ) ⊗ L . This construction gives the space of spin c structures the structure of an affine space over thegroup of isomorphism classes of complex line bundles over Y or, equivalently, H ( Y ; Z ). Remark . Notice that our definition depends on the prior choice of a Riemannianmetric. However, we identify two spin c structures s and s associated to two metrics g and g such that there is a path of metrics g t joining g and g and a corresponding continousfamily s t over [0 , c structures asbeing associated to a smooth closed connected oriented manifold Y .On a 4-manifold the story is analogous. Let X be a closed oriented riemannian 4-manifold.A spin c structure s X is given by a rank 4 hermitian vector bundle S X on X together with aClifford multiplication ρ X , i.e. a bundle map ρ X : T X → End( S X )such that at each x ∈ X we can find an oriented orthonormal frame e , e , e , e with ρ X ( e ) = (cid:18) − I I (cid:19) ρ X ( e i ) = (cid:18) − σ ∗ i σ i (cid:19) i = 1 , , , in some orthonormal basis of the fiber. Here I is the 2 × σ i arethe Pauli matrices as above. Extending the Clifford multiplication to (complex) forms as inthe 3-dimensional case, we have that in the same basis ρ (vol x ) = (cid:18) − I I (cid:19) where vol = e ∧ e ∧ e ∧ e is the oriented volume form. Hence we get a orthogonaldecomposition S X = S + ⊕ S − respectively as the − ρ (vol). Then the Clifford multiplication by atangent vector interchanges the two factors, and we have the bundle isometry ρ : Λ + X → su ( S + )where Λ + X is the space of self-dual 2-forms on X . The existence and classification resultsfor spin c structures on a 4-manifold are identical to the 3-dimensional case, even though theexistence of at least one such structure is more subtle.Finally we discuss the relation between spin c structures on 3 and 4-manifolds. Suppose Y = ∂X , and consider a spin c structure on X . Then using the outward normal vector field ν one obtains an identification ρ ( ν ) : S + | Y → S − | Y . Hence we recover a spin c structure on Y with S = S + | Y ∼ = S − | Y ρ ( v ) = ρ X ( ν ) − ρ X ( v ) . . THE MONOPOLE EQUATIONS 13 Fix now a spin c structure s on a riemannian 3-manifold Y . This data allows us to considerthe following two objects. First, one has spinors , i.e. sections Ψ of S . Then one has spin c connections , which are unitary connections B on S such that ρ is parallel, namely for everyvector field ξ and every spinor Ψ one has ∇ B ( ρ ( ξ )Ψ) = ρ ( ∇ ξ )Ψ + ρ ( ξ ) ∇ B (Ψ) , where ∇ is the Levi-Civita connection. Any such connection determines to a connection B t on the line bundle det( S ), and given a base spin c connection B , all the other ones can bewritten as B = B + b for some b ∈ i Ω ( Y ; R ). Notice that this also implies B t = B t + 2 b . We denote the space ofspin c connections by A ( Y, s ). We define the configuration space C ( Y, s ) = { ( B, Ψ) } = A ( Y, s ) × Γ( S ) . The group of automorphisms of the spin c structure G = Map( Y, S ) , also called the gauge group , acts on C ( Y, s ) via the map u · ( B, Ψ) = ( B − u − du, u Ψ) . We denote the quotient space of this action by B ( Y, s ), the moduli space of configurations.We are interested in paths in C ( Y, s ) satisfying the following flow equations:(1.1) ddt B = − (cid:18) ∗ F B t + ρ − (ΨΨ ∗ ) (cid:19) ⊗ S ddt Ψ = − D B Ψ . Here (ΨΨ ∗ ) denotes the traceless part of the endomorphism ΨΨ ∗ ., and D B is the Diracoperator associated to B , i.e. the compositionΓ( S ) ∇ B −−→ Γ( T ∗ X ⊗ S ) ρ −→ Γ( S ) . The Dirac operator D B is a first order elliptic self-adjoint operator, hence it has index zero.These equations can be thought as describing the trajectories for the flow of the formal L gradient of the Chern-Simons-Dirac functional(1.2) L ( B, Ψ) = − Z Y ( B t − B t ) ∧ ( F B t + F B t ) + 12 Z Y h D B Ψ , Ψ i d vol , where B is some fixed spin c connection. The equations for the critical points(1.3) ρ ( F B t ) − (ΨΨ ∗ ) = 0 D B Ψ = 0are called the
Seiberg-Witten equations or monopole equations . The Chern-Simons-Diracfunctional is not invariant under G , but one has the relation L ( u · ( B, Ψ)) − L ( B, Ψ) = 2 π ([ u ] ∪ c ( S ))[ Y ]where [ u ] ∈ H ( Y ; Z ) is the cohomology class corresponding to the map u : Y → S . Inparticular, L is invariant under the identity component G e of the gauge group, and descends to a well defined function with values in R / (2 π Z ).The monopole equations have a special class of solutions. We say that a configuration( B, Ψ) (not necessarily a solution) is reducible if Ψ = 0, and irreducible otherwise. The subsetof irreducible configurations is denoted by C ∗ ( Y, s ). If there is a reducible solution ( B,
0) tothe monopole equations (1.3), then the connection B t on det ( S ) is flat, hence c ( S ) is torsion.On the other hand, if the first Chern class of S is torsion, there are always reducible solutions,and the space of reducible solutions up to gauge transformations can be identified with a torus H ( Y ; i R ) / πiH ( Y ; Z ) ⊂ B ( Y, s ) . The distinction between irreducible and reducible plays an important role because an ir-reducible configuration has trivial stabilizer under the action of the gauge group, while areducible configuration has stabilizer S consisting of constant gauge transformations.The flow equations (1.1) can also be interpreted purely in 4-dimensional terms. We candefine the configuration space on a 4-manifold X as C ( X, s ) = A ( X, s ) × Γ( S + )where the first factor is the space of spin c connections on S X . The gauge group G =Map( X, S ) acts on it with quotient B ( X, s ). A spin c structure on Y naturally induces aspin c structure on the infinite cylinder Z = R × Y as follows. The bundle S Z will be (thepull-back of) S ⊕ S , and the Clifford multiplication ρ Z is defined as ρ Z ( ∂/∂t ) = (cid:18) − I S I S (cid:19) ρ Z ( v ) = (cid:18) − ρ ( v ) ∗ ρ ( v ) 0 (cid:19) for v ∈ T Y.
A time-dependent spinor naturally defines a section of S + , while a time dependent connection B on Y gives rise to a connection A on Z defined as(1.4) ∇ A = ddt + ∇ B . In this setting, one can write the flow equations (1.1) as a gauge invariant equations for apair ( A, Φ) of the form ρ Z ( F + A t ) = (ΦΦ ∗ ) D + A Φ = 0where D + A is the Dirac operator arising as the compositionΓ( S + ) ∇ A −−→ Γ( T ∗ X ⊗ S + ) ρ −→ Γ( S − ) . We will also think the equations as the differential operator F : C ( Z, s ) → C ∞ ( X ; i su ( S + ) ⊕ S − )( A, Φ) (cid:18) ρ ( F + A t ) − (ΦΦ ∗ ) , D + A Φ (cid:19) . It is important to notice that the last set of equations makes sense for a general configuration( A, Φ) on a general 4-manifold X , and these are indeed the original Seiberg-Witten equations([ Mor96 ]). . BLOWING UP THE CONFIGURATION SPACES 15
Remark . In the case X is compact without boundary it follows from the Atiyah-Singer index theorem that D + A has complex indexInd D + A = 18 (cid:0) c ( S + ) [ X ] − σ ( X ) (cid:1) , where σ ( X ) is the signature of the intersection form of X .On the infinite cylinder Z the 4-dimensional equations make sense for every spin c connec-tion A , not necessary coming from a time dependent connection B on Y as in equation (1.4)(which we call in temporal gauge ). In general we can always write such a connection as A = B + ( cdt ) ⊗ S , where c is a time dependent imaginary valued function on Y , and writing down the Seiberg-Witten equations in a 3-dimensional fashion we obtain the gauge invariant equations ddt B − dc = − (cid:18) ∗ F B t + ρ − (ΨΨ ∗ ) (cid:19) ⊗ S ddt Ψ + c Ψ = − D B Ψ . Given a configuration ( A, Φ) on Z we will denote by ( ˇ A ( t ) , ˇΦ( t )) the path of configurationson Y it determines.
2. Blowing up the configuration spaces
The key construction in monopole Floer homology is the blow-up of the moduli spacealong the reducible locus. This is done in order to be able to deal properly with reduciblesolutions, which are the fixed points for the action of the constant gauge transformations.In particular, our goal will be to define in an infinite dimensional setting an analogue of S -equivariant homology.Consider the configuration space C ( X, s ) on a 4-manifold (possibly with boundary) witha fixed spin c structure. Define the blown-up moduli space as C σ ( X, s ) = (cid:8) ( A, s, φ ) | s ≥ k φ k L ( X ) = 1 (cid:9) ⊂ A ( X, s ) × R × Γ( S + ) . This comes with the natural blow-down map π : C σ ( X, s ) → C ( X, s )( A, s, φ ) ( A, sφ ) . Such a map is a bijection over the irreducible locus C ∗ ( X, s ), while the fiber over a reducibleconfiguration ( A,
0) is a sphere S ( C ∞ ( X ; S + )).The Seiberg-Witten equations are defined on the blow up as follows. One can see theoperator F as a section on the trivial vector bundle V ( X, s ) → C ( X, s ) with fiber C ∞ ( X ; i su ( S + ) ⊕ S − ). This vector bundle can be pulled back to C σ ( X, s ) alongthe projection π to get a vector bundle V σ ( X, s ). Then one defines the section F σ : C σ ( X, s ) → V σ ( X, s )( A, s, φ ) → (cid:0) ρ ( F + A t ) − s ( φφ ∗ ) , D + A φ (cid:1) . We call these the
Seiberg-Witten equations on the blown up configuration space . Notice thatthe gauge group G ( X ) naturally acts on both C σ ( X, s ) and V σ ( X, s ) and the section F σ ( X, s )is equivariant for this action.The blow-up of the configuration space on a 3-manifold is defined in an analogous way as C σ ( Y, s ) = (cid:8) ( B, r, ψ ) | r ≥ k ψ k L ( Y ) = 1 (cid:9) ⊂ A ( Y, s ) × R × Γ( S ) . On a finite cylinder Z = I × Y , a configuration γ σ = ( A, s, φ ) ∈ C σ ( Z, s )gives rise to a path of configurations in C σ ( Y, s )ˇ γ σ ( t ) = ( ˇ A ( t ) , s k ˇ φ ( t ) k L ( Y ) , ˇ φ ( t ) / k ˇ φ ( t ) k L ( Y ) )provided k ˇ φ ( t ) k L ( Y ) = 0 for every t ∈ I. Because of a unique continuation result for solutions of the equation F σ ( γ σ ) = 0(see Chapter 7 in the book), this is true if we restrict to this class of paths. If A is also in tem-poral gauge, the 4-dimensional Seiberg-Witten equations on the blow-up can be interpretedas the equations for the path ( B ( t ) , r ( t ) , ψ ( t )) given by(1.4) ddt B = − (cid:18) ∗ F B t + r ρ − ( ψψ ∗ ) (cid:19) ⊗ S ddt r = − Λ( B, r, ψ ) rddt ψ = − D B ψ + Λ( B, r, ψ ) ψ where we have defined the functionΛ( B, r, ψ ) = h ψ, D B ψ i L ( Y ) . The right hand side of the equations defines a vector field of C σ ( Y, s ), the blown-up gradientof the Chern-Simons-Dirac functional , which we denote by (grad L ) σ . This vector field is thepull-back of grad L on the irreducible locus C ∗ ( Y, s ), and is tangent to the locus r = 0, i.e.the boundary of C σ ( Y, s ). Notice that L also pulls back to a well defined function on C σ ( Y, s ),but it is important to notice that (grad L ) σ is not the gradient of this function with respectto any natural metric. It is easy to identify the critical points of (grad L ) σ . In particular theyare: • configurations with r = 0 such that ( B, rψ ) ∈ C is a critical point of grad L . • configurations with r = 0, such that ( B, ∈ C is a critical point of grad L and ψ is aneigenvector of D B (with eigenvalue Λ( B, , ψ )). . COMPLETION AND SLICES 17 When dealing with configurations on a cylinder Z = I × Y , there is another version ofthe blow-up which will be very useful when studying flow lines. This is the τ -model C τ ( Z, s ) ⊂ A ( Z, s ) × C ∞ ( I, R ) × Γ( S + )which consists of the set of triples ( A, s, φ ) satisfying • s ( t ) ≥ • k φ ( t ) k L ( Y ) = 1.An element γ = ( A, s, φ ) in C τ ( Y, s ) determines a path ˇ γ = ( ˇ A, s, ˇ φ ) in C σ ( Y, s ), and thecorrespondence is bijective if we restrict to configurations in temporal gauge. In general, theflow equations (2) can be rewritten for our path in the following gauge-invariant form ddt ˇ A = − (cid:18) ∗ F ˇ A t + dc + r ρ − ( ˇ ψ ˇ ψ ∗ ) (cid:19) ⊗ S ddt s = − Λ( ˇ
A, s, ˇ ψ ) sddt ˇ ψ = − D ˇ A ˇ ψ − c ˇ φ + Λ( ˇ A, s, ˇ ψ ) ˇ ψ where A = ˇ A + cdt . It is useful to interpret these equations in the τ -model in a 4-dimensionalfashion (which also makes gauge-invariance more manifest). The equations can be written as12 ρ ( F + A t ) − s ( φφ ∗ ) = 0 ddt s + Re h D + A φ, ρ ( dt ) − φ i L ( Y ) s = 0 D + A φ − Re h D + A φ, ρ ( dt ) − φ i L ( Y ) φ = 0 . The left hand side on this equations determine a section F τ of the vector bundle V τ ( Z, s ) → C τ ( Z, s )with fiber over ( A, s, φ ) the vector space (cid:8) ( η, r, ψ ) | h Re ˇ φ ( t ) , ˇ ψ ( t ) i L ( Y ) = 0 for all t (cid:9) ⊂ C ∞ ( Z, i su ( S + )) ⊕ C ∞ ( I, R ) ⊕ C ∞ ( Z, S − ) . A configuration in C σ ( Z, s ) such that the restriction to each slice { t } × Y is non zero gives riseto a well defined element of C τ ( Z, s ) by rescaling. A unique continuation result (see Chapter7 in the book) tells us that if solution of the equations is such that the spinor vanishes ona slice, then the spinor vanishes everywhere. Hence we get a bijection between the set ofsolutions of F σ and F τ .
3. Completion and slices
We now define suitable completions of our configuration spaces in order to be able towork in a Hilbert (or Banach) manifold setting which will be essential for the kind of analysiswe will do. From now on, given a vector bundle E over M with an inner product and a connection ∇ , and given integer k , we will denote by L pk ( M ; E ) the Sobolev space obtainedby completing the space of sections of such a bundle with respect to the Sobolev norm k f k pL pk = Z M | f | p + |∇ f | p + · · · + |∇ k f | p d vol . If M is closed, we can also define the fractional Sobolev spaces L k ( M ; E ) as the completionof the space of sections with respect to the norm k f k L k = k (1 + ∆) k/ f k L . This definition is equivalent to the previous one in the case k is an integer.We then focus in the case when M is a 3-manifold or a 4-manifold (possibly with bound-ary). We will write W for the bundle S in the former case and for S + in the latter. Fix asmooth connection A on M . We define C k ( M, s ) = ( A ,
0) + L k ( M ; iT ∗ M ⊕ W ) = A k ( M, s ) × L k ( M ; W )where A k ( M, s ) = A + L k ( M ; iT ∗ M )is the space of L k connections on M . Similarly the completion of the blown-up configurationspace is defined as C σk ( M, s ) = (cid:8) ( A, s, φ ) | s ≥ , k φ k L ( X ) = 1 (cid:9) ⊂ A k ( M, s ) × R ≥ × S ( L k ( M ; W ))where the sphere is still taken with respect to the L norm. This is a smooth Hilbert manifoldwith boundary given by the locus s = 0. For 2( k + 1) > dim M , the gauge group G k +1 ( M ) ⊂ L k ( M ; C )is defined as the subset of functions with pointwise norm 1. With the topology inherited from L k +1 ( M ; C ), this is a Hilbert Lie group which acts smoothly and freely on C σk ( M, s ).The tangent bundle of C σk ( M ; s ) ⊂ A k × R × L k ( M ; W ) can be regarded as the bundlewith fiber at γ = ( A , s , φ ) given by { ( a, s, φ ) | Re h φ , φ i L = 0 } ⊂ L k ( M ; iT ∗ M ) × R × L k ( M ; W ) . We also define the bundles T σj → C σk ( M, s ) obtained from the tangent bundle by fiberwisecompletion in the L j norm for j ≤ k . The fiber at a point γ = ( A, s, φ ) ∈ C σk ( M, s ) is givenby { ( a, s, φ ) | Re h φ , φ i L = 0 } ⊂ L j ( M ; iT ∗ M ) × R × L j ( M ; W ) . In the case of non blown-up configuration spaces, the same construction yields the productbundles T j = L j ( M ; iT ∗ M ⊕ W ) × C k ( M, s ) . In both cases, for j = k we obtain the usual tangent bundle.For 2( k + 1) > dim M , we introduce the quotient spaces B k ( M, s ) = C k ( M, s ) / G k +1 ( M ) B σk ( M, s ) = C σk ( M, s ) / G k +1 ( M ) . COMPLETION AND SLICES 19 In the latter case, the action of the gauge group is free and the quotient space is Hausdorff(see Proposition 9 . . j ≤ k a smooth bundledecomposition T σj = J σj ⊕ K σj defined as follows. The fiber J σγ,k at a point γ = ( A , s , φ ) is the image of the derivative ofthe gauge group action d σγ : T e G k +1 ( M ) = L k +1 ( M, i R ) → T γ C σk ( M )(1.5) ξ ( − dξ, , ξφ ) , and similarly we define J σγ,j for j ≤ k by extending the map in Sobolev spaces of lowerregularity. The fiber of K σj over a point γ is defined as the subspace of T σγ,j consisting of thespace of triples ( a, s, φ ) satisfying h a, ν i = 0 at ∂M − d ∗ a + is Re h iφ , φ i = 0Re h iφ , φ i L ( M ) = 0 . This decomposition can be thought as the extension to the boundary of the analogous de-composition on C ∗ k ( M, s ) ⊂ C k ( M, s ) given by T j = J j ⊕ K j . Here as before J γ,j at the point γ = ( A , Φ ) is the completion of the image of the derivativeof the gauge group action d γ : T e G k +1 ( M ) = L k +1 ( M, i R ) → T γ C k ( M ) (1.6) ξ ( − dξ, ξ Φ )and K γ,k is its orthogonal complement with respect to the standard L inner product, namely(1.7) { ( a, φ ) | − d ∗ a + i Re h i Φ , φ i = 0 and h a, ν i = 0 at ∂M } . It is important to notice that the decomposition T σj = J σj ⊕ K σj is not orthogonal with respectto any natural metric on the blown-up configuration space.Given a configuration γ ∈ C σk ( M, s ) we are now ready to define the Coulomb-Neumannslice S σk,γ through it. This will be a closed Hilbert submanifold containing γ whose tangentspace at γ is K σk . A small open neighborhood of U ⊂ S σk,γ of γ will provide a local charts forthe Hilbert manifold B σk ( M, s ) near [ γ ], via the composition(1.8) S σk,γ ֒ → C σk ( M, s ) → B σ ( M, s )where the first map is the inclusion and the second is the quotient map, see Corollary 9 . . S σk,γ with γ = ( A , s , φ ) to be the closed Hilbert submanifold of C σk ( M, s ) consisting of triples ( A + a, s, φ ) satisfying h a, ν i = 0 at ∂M − d ∗ a + iss Re h iφ , φ i = 0Re h iφ , φ i L ( M ) = 0 . In the case γ is irreducible, one can define the affine subspace S k,γ ⊂ C k ( M, s ) given by S k,γ = { ( A + a, Φ) | | − d ∗ a + i Re h i Φ , Φ i = 0 and h a, ν i = 0 at ∂M } , and S σk,γ is simply the proper transform of this under blow-up.In the case γ = ( A ,
0) is a reducible configuration the equations defining S k,γ are simply ( d ∗ a = 0 h a, ν i = 0 at ∂M. These equations define a global slice for the gauge group action. Looking for a gauge trans-formation of the form u = e ξ to put a given configuration in the slice S k,γ is equivalent tosolve the problem h dξ, ν i = h a, ν i at ∂M ∆ ξ = d ∗ a which is known to have a unique solution ξ ∈ L k +1 such that Z M ξ = 0 . If we introduce the subgroup G ⊥ k +1 = (cid:26) e ξ | Z M ξ = 0 (cid:27) , the action of the gauge group gives us a diffeomorphism G ⊥ k +1 × K k,γ → C k ( e ξ , ( a, φ )) ( A + ( a − dξ ) ⊗ , e ξ φ ) . The gauge group can be decomposed as G k +1 = G h × G ⊥ k +1 , where G h is the group of theharmonic maps M → S sitting in the short exact sequence(1.9) 1 → S → G h → H ( M ; Z ) → . Such a sequence splits (but not canonically). In particular, up to homotopy one can identify B σ = K ∗ k,γ / G h . Such a space is a fiber bundle (by projecting the spinor away)( L k ( M ; W ) \ { } ) /S ֒ → K ∗ k,γ / G h → H ( M ; i R ) /H ( M ; i Z )which is trivial by Kuiper’s theorem on the contractibility of the unitary group of a Hilbertspace. Hence it has the homotopy type of a product CP ∞ × H ( M ; i R ) /H ( M ; i Z ) , and its cohomology ring (over Z ) is isomorphic to Z [ U ] ⊗ Λ ∗ ( H ( M ; Z ) / torsion) , where deg U = 2. Similarly, one shows that B k ( M, s ) has the homotopy type of a torus H ( M ; i R ) /H ( M ; i Z ). . PERTURBATIONS 21 We define the Seiberg-Witten equations on the completions of the configuration spaces.In the case of a compact 4-manifold (perhaps with boundary), we define for each j ≤ k thetrivial vector bundle V j ( X, s ) over C k ( X, s ) with fiber L j ( X ; i su ( S + ) ⊕ S − ), and the bundle V σj ( X, s ) → C σk ( X, s )as its pull-back under the blow-down map. The section F σ then extends to these Sobolevcompletions as a section F σ : C σk ( X, s ) → V σk − ⊂ V σj . The gauge group acts smoothly on such bundles for j ≤ k + 1, and the section F σ is G k +1 ( X )-equivariant for j = k −
1. The case of a 3-dimensional manifold is analogous. In this case weobtain the sections grad L : C k ( Y, s ) → T k − (grad L ) σ : C k ( Y, s ) → T σk − , which are both smooth.We can also easily adapt this story in the case of the τ model on a cylinder I × Y . Wedefine C τk ( Z, s ) ⊂ A k ( Z, s ) × L k ( I, R ) × L k ( Z ; S + )consisting of the triples ( A, s, φ ) with • s ( t ) ≥ • k φ ( t ) k L ( Y ) = 1 for all t ∈ I .Notice that this space is not a Hilbert manifold (nor even a manifold with boundary) becauseof the condition s ≥
0, but is a closed subspace of the Hilbert manifold˜ C τk ( Z, s ) ⊂ A k ( Z, s ) × L k ( I, R ) × L k ( Z ; S + )consisting of the triples ( A, s, φ ) with k φ ( t ) k L ( Y ) = 1 for all t ∈ I . There is a naturalinvolution of this space(1.10) i : ˜ C τk ( Z, s ) → ˜ C τk ( Z, s )( A, s, φ ) ( A, − s, φ )and the gauge group G k +1 ( Z ) acts smoothly and freely on it. The completion and slices storyin this framework is essentially unchanged, and we refer the reader to Sections 9.2 and 9.4 inthe book for the details.
4. Perturbations
As it is usual in Floer theory, we need to introduce suitable perturbations of the Seiberg-Witten equations in order to construct monopole Floer homology. The space of perturbationshas to be a large enough in order to achieve transversality for the moduli spaces of solutions.On the other hand, these perturbations have to be mild enough so that the perturbed equa-tions still have the nice properties (smoothness, unique continuation and compactness amongthe others) of the original ones. In this section we review the construction of the perturba-tions we will use, the ones arising as formal gradients of cylinder functions. We do this in quite detail as we will perform a variant of such a construction in Chapter 4, and we start bydiscussing the abstract theory of perturbations.. The perturbations we will deal with will arise as the formal gradient of a given gauge-invariant function f : C ( Y ) → R , i.e. a section q : C ( Y ) → T such that for every path γ : [0 , → C ( Y ) we have f ◦ γ (1) − f ◦ γ (0) = Z h ˙ γ, q i L ( Y ) dt. We will write −L = L + f for the perturbed Chern-Simons-Dirac functional, and writegrad −L = grad L + q , which is a G ( Y )-invariant section of T → C ( Y ).We can then pull-back such a perturbation to the cylinders I × Y in order to obtain asection ˆ q : C ( Z ) → V ( Z )as follows. From a configuration ( A, Φ) ∈ C ( Z ) we obtain by restricting to slices a continuouspath ( ˇ A ( t ) , ˇΦ( t )) in C ( Y ), hence a continuous path q ( ˇ A ( t ) , ˇΦ( t )) in L ( Y ; iT ∗ Y ⊕ S ). Byidentifying iT ∗ Y ⊕ S with i su ( S + ) ⊕ S − via the Clifford multiplication on the first factor weobtain an element of V ( Z ).In the following definition we recollect the analytic requirements we make on such aperturbation. Definition . Let k ≥ q : C ( Y ) → T is k - tame if it is the formal gradient of a G ( Y )-equivariant function on C ( Y ) and it satisfiesthe following properties:(1) the associated 4-dimensional perturbation ˆ q defines a smooth sectionˆ q : V k ( Z ) → C k ( Z );(2) for every integer j ∈ [1 , k ] the section ˆ q extends to a continuous sectionˆ q : V j ( Z ) → C j ( Z );(3) for every integer j = [ − k, k ] the first derivative D ˆ q ∈ C ∞ ( C k ( Z ) , Hom( T C k ( Z ) , V k ( Z )))extends to a map D ˆ q ∈ C ∞ ( C k ( Z ) , Hom( T C j ( Z ) , V j ( Z ))) ;(4) there is a constant m such that k q ( B, Ψ) k L ≤ m ( k Ψ k L + 1)for every ( B, Ψ) ∈ C k ( Y ); . PERTURBATIONS 23 (5) for any fixed smooth connection A , there is a real function µ such that the inequality k ˆ q ( A, Φ) k L ,A ≤ µ ( k ( A − A , Φ) k L ,A )holds for all configurations ( A, Φ) ∈ C k ( Z );(6) the 3-dimensional perturbation q defines a C section q : C ( Y ) → T . We say that q is tame if it is k -tame for every k ≥ V k ⊂ V k − , we candefine the section F q = F + ˆ q and the perturbed Seiberg-Witten equations on Z as F q ( A, Φ) = 0 . In a more expanded version, we write the perturbation as q = ( q , q ) with q ∈ L ( Y ; iT ∗ Y ) q ∈ L ( Y ; S )and the induced 4-dimensional perturbation as ˆ q = (ˆ q , ˆ q ), whereˆ q ∈ L ( Z ; i su ( S + ))ˆ q ∈ L ( Z ; S − ) . The equation F q ( A, Φ) = 0 can then be written as ρ Z ( F + A t ) − ∗ ) = − q ( A, Φ) D + A Φ = − ˆ q ( A, Φ) ) or, when interpreted as a gradient-flow equation, in the form ddt B t = − ∗ F B t − ρ − (ΨΨ ∗ ) − q ( B, Ψ) ddt Ψ = − D B Ψ − q ( B, Ψ) . To write the perturbed equations on the blow-up, one notices that the section ˆ q of V k givesrise to a section ˆ q σ : C σk ( Z ) → V σk as follows. Gauge invariance implies thatˆ q ( A,
0) = 0 , hence one can define the section ˆ q ,σ : C σk ( Z ) → V ,σk ( A, s, φ ) Z ( D ( A,rsφ ) ˆ q )( φ ) dr and finally ˆ q σ = (ˆ q , ˆ q ,σ ) . It is straightforward that ˆ q σ is a smooth section as ˆ q is. In any case, F σ q = F σ + ˆ q σ : C σk → V σk − is a smooth section and F σ ˆ q = 0 are the perturbed Seiberg-Witten equations on the blow-up .There is also a natural 3-dimensional counterpart of the last construction, giving riseto the perturbed 3-dimensional gradient in the blow-up. The perturbation q gives rise to aperturbation q σ on the blow-up C σk ( Y ), and we can write(grad −L ) σ = (grad L ) σ + q σ , which is a smooth section of the vector bundle T σk − → C σk . The equations of the gradientflow of a path ( B ( t ) , r ( t ) , ψ ( t )) are ddt B t = − ∗ F B t − r ρ − ( ψψ ∗ ) − q ( B, rψ ) ddt r = − Λ q ( B, r, ψ ) rddt ψ = − D B Ψ − ˜ q ( B, Ψ) + Λ q ( B, r, ψ ) r where ˜ q is defined similarly to ˆ q ,σ to be˜ q ( B, r, ψ ) = Z D ( B,srψ ) q (0 , ψ ) ds, and we have defined the functionΛ q ( B, r, ψ ) = Re h ψ, D B ψ + ˜ q ( B, r, ψ ) i L . Writing this in coordinates we have that q σ ( B, r, ψ ) = (cid:16) q ( B, rψ ) , h ˜ q ( B, r, ψ ) , ψ i L ( Y ) r, ˜ q ( B, r, ψ ) ⊥ (cid:17) where ⊥ denotes the orthogonal projection to the real orthogonal complement of ψ . Wedenote by D B, q the operator on sections of S defined as D B, q ψ = D B ψ + D ( B, q (0 , ψ ) . Then a critical point (
B, r, ψ ) of the gradient is of the form: • r = 0 and ( B, rψ ) is a critical point of grad −L ; • r = 0, ( B,
0) is a critical point of grad −L and ψ is an eigenvector of D B, q .The story for the τ model is essentially identical, see Section 10.4 in the book.There is a very large class of tame perturbations arising as formal gradients of cylinderfunctions f : C ( Y ) → R . We recall their construction (as it will be very useful later in Chapter4) and their most important properties.Given a coclosed 1-form c ∈ Ω ( Y ; i R ) we can define the function r c : C ( Y ) → R ( B + b ⊗ , Ψ) Z Y b ∧ ∗ ¯ c = h b, c i Y . . PERTURBATIONS 25 This is generally invariant only under the identity component G e of the gauge group, but itis fully gauge invariant if c is coexact. One also has the G -invariant map C ( Y ) → T = H ( Y ; i R ) / (2 πiH ( Y ; Z ))( B + b ⊗ , Ψ) [ b harm ]where b harm denotes the harmonic part of the 1-form b and the brackets denote the equivalenceclass. Picking an integral basis ω , . . . ω t for H ( Y ; R ) we can identify the torus T with R t / (2 π Z t ) and the map can be written as( B, Ψ) ( r ω ( B, Ψ) , . . . , r ω t ( B, Ψ)) (mod 2 π Z t ) . Consider now a splitting v of the exact sequence in equation (1.9), S → G h v ⇆ H ( Y ; Z ) . We can choose it so that G h,o := v ( H ( Y ; Z )) = { u | u ( y ) = 1 } ⊂ G h for some chosen basepoint y ∈ Y . We then define the subgroup G o ( Y ) = G h, × G ⊥ ( Y ) ⊂ G ( Y )which acts freely on C ( Y ), and consider the based configuration space as the quotient of therespective completions B ok ( Y ) = C k ( Y ) / G ok +1 ( Y ) . This space is a Hilbert manifold and the space B k ( Y ) is obtained as the quotient by theremaining circle action ( B, Ψ) ( B, e iϑ Ψ) . Consider now the smooth bundle S over T × Y obtained as the quotient of H ( Y ; i R ) × S → H ( Y ; i R ) × Y by the group G h,o . Any smooth section Υ of S can be lifted to a section˜Υ : H ( Y ; i R ) × Y → H ( Y ; i R ) × S respecting the following quasi-periodicity condition. For every κ ∈ H ( Y ; Z ) we have˜Υ α + κ ( y ) = u ( y ) ˜Υ α ( y ) , where u = v ( κ ) ∈ G h,o and we write ˜Υ b ( y ) instead of ˜Υ( b, y ). Hence a section Ψ of S givesrise to the G o ( Y )-equivariant map Υ † : C ( Y ) → C ∞ ( S )( B + b ⊗ , Ψ) e − Gd ∗ b ˜Υ b harm , where G : L k − ( Y ) → L k +1 ( Y )is the Green’s operator of ∆ = d ∗ d . In turn we define the G o ( Y )-invariant map q Υ : C ( Y ) → C ( B, Ψ) Z Y h Ψ , Υ † ( B, Ψ) i = h Ψ , ˜Υ † i Y , which is also equivariant for the remaining S action if we make S act on C by complexmultiplication. Choose a finite collection of coclosed 1-forms c , . . . , c n + t with the first n being coexactand the remaining t representing an integral basis of H ( Y ; R ), and a collection of smoothsections Υ , . . . , Υ m of S . This gives rise to the map p : C ( Y ) → R n × T × C m ( B, Ψ) (cid:0) r c ( B, Ψ) , . . . , r c n + t ( B, Ψ) , q Υ ( B, Ψ) , . . . , q Υ m ( B, Ψ) (cid:1) (mod2 π Z t )which is G o ( Y )-invariant and equivariant with respect to the remaining S action. Definition . We say that a gauge invariant function f : C ( Y ) → R is a cylinderfunction if it is of the form f = g ◦ p where: • the map p : C ( Y ) → R n × T × C m is defined as above with any choice of n coclosed1-forms and m sections of S , n, m ≥ • the function g is an S -invariant smooth function on R n × T × C m with compact support.We summarize the two main features of cylinder functions in the following proposition(see Chapter 11 in the book). Proposition . If f is a cylinder function, then its formal gradient grad f : C ( Y ) → T is a tame perturbation. Furthermore, for every compact subset K of a finite dimensionalsubmanifold M ⊂ B ok ( Y ) , which are both S -invariant, one can find a collection of coclosedforms c µ , sections Υ ν and a neighborhood U of K inside M such that p : B ok ( Y ) → R n × T × C m gives an embedding of U . Finally, in order to apply the Sard-Smale theorem we will need to be in a Banach manifoldsetup. To do this, we first specify a countable collection of cylinder functions as follows. Forevery pair ( n, m ), choose a countable collection of ( n + m )-tuples ( c , . . . , c n , Υ , . . . , Υ m )which are dense in the C ∞ topology in the space of all such ( n + m )-tuples. Choose acountable collection of compact subsets K ⊂ R n × T × C m which is dense in the Hausdorfftopology and, for each K , a countable collection of S -invariant functions g α with support in K which are dense (in the C ∞ topology) in the space of S -invariant functions with supportin K . We also require that the subset of g α that vanish on K ∩ ( R n × T × { } )is dense (in the C ∞ topology) in the space of S -invariant functions with support on K andvanishing on K ∩ ( R n × T × { } ). Combining all these choices, we obtain a countable collectionof cylinder functions, and denote the corresponding set of tame perturbations by { q i } i ∈ N . Wecan then construct a Banach space of perturbations containing all the ones in this family, asstated in the next result. Proposition . There exists a separable Banach space P and a linear map D : P → C ( C ( Y ) , T ) λ q λ such that every q λ is a tame perturbation and the image contains the family { q i } i ∈ N . Fur-thermore we have that: . PERTURBATIONS 27 • for a cylinder Z = I × Y and all k ≥ , the map P × C k ( Z ) → V k ( Z ) given by ( λ, γ ) ˆ q λ ( γ ) is smooth; • the map P × C ( Y ) → T ( Y ) given by ( λ, β ) q λ ( β ) is continuous and satifies bounds k q λ ( B, Ψ) k L ≤ k λ k m ( k Ψ k L + 1) k q λ ( B, Ψ) k L ,A ≤ k λ k µ (cid:16) k B − B , Ψ k L ,A (cid:17) for some constants m and a real function µ . By a large Banach space of tame perturbations we will mean a pair ( P , D ) arising fromthe proposition. In the rest of the work we will always assume that a large Banach space ofperturbations is fixed, and we will identify it with its image (and denote it by P ), even if itis clear that the Banach space topology is not the topology as a subspace of the set of tameperturbation, for any any natural choice of the topology on the latter.HAPTER 2 The analysis of Morse-Bott singularities
In this chapter we study the analytical and structural properties of the space of solu-tions to the Seiberg-Witten equations when the singularities of the (blown-up) gradient are
Morse-Bott . This kind of singularities is not generic, but arises quite naturally when deal-ing with many problems, e.g. when dealing with the equation on a Seifert-fibered space (see[
MOY97 ]). The case in which we will be interested in the most is the case of a spin c structureinduced by a spin structure, which will be discussed in the last chapter of the present work.The content of this chapter can be thought as a generalization of the material appearing inChapters 12 to 19 in the book, to which we refer as the Morse or classical case. We will oftenmake use of results proven there providing precise references.
1. Hessians and Morse-Bott singularities
We define the class of differential operators we will be dealing with, namely k - almost self-adjoint first-order elliptic operators (abbreviated k - asafoe ), see Section 12 . L is called k - asafoe if it can be written as L = L + h where: • L is a first order self-adjoint elliptic differential operator with smooth coefficients,acting on sections of a vector bundle E → Y ; • h is an operator on sections of Eh : C ∞ ( Y ; E ) → L ( Y ; E )which extends to a bounded map on the spaces L j ( Y ; E ) for all | j | ≤ k .We also say that L is asafoe if it is k - asafoe for every k ≥
0. So k - asafoe operators are aclass of (non necessarily symmetric) compact perturbations of first order elliptic self-adjointoperators. The main example to keep in mind is that of the perturbed Dirac operators D q ,B ,which indeed motivates the definition. This class of operators satifies very nice regularity andspectral properties, see Section 12 . k - asafoe operator hyperbolic if its spectrum does not intersect the imaginary line.We first define Morse-Bott singularities in the non blown-up configuration space. Recallthat the irreducible part C ∗ k ( Y ) ⊂ C k ( Y ) we have the decomposition(2.1) T j | C ∗ k = J j ⊕ K j ,
290 2. THE ANALYSIS OF MORSE-BOTT SINGULARITIES see Section 3 of Chapter 1. As this decomposition is orthogonal, the vector field grad −L is asection of K k − ⊂ T k − , so we can define the Hessian operator at a configuration α ∈ C ∗ k ( Y )Hess q ,α : K k,α → K k − ,α as the restriction to K k,α ⊂ T k,α of the linear mapΠ K k − ◦ D α (grad −L ) : T α C ∗ k ( Y ) = T k,α → K k − ,α where Π K k − : T k − → K k − is the L orthogonal projection with kernel J k − . As α varies,one obtains a G k +1 -equivariant smooth bundle mapHess q : K k → K k − . The operators Hess q ,α are symmetric, as they are the pull back of the covariant Hessian of thecircled valued function −L on B ∗ ( Y ). Here we identify K k with the pull-back of the tangentbundle of B ∗ ( Y ), and K k − with the pull back of [ T k − ] = ( T k − / J k − ) / G k +1 . It also satisfiesnice spectral properties which are manifest from the following construction (see Proposition12 . . extended Hessian at a configuration α as the map d Hess q ,α : T k,α ⊕ L k ( Y ; i R ) → T k − ,α ⊕ L k − ( Y ; i R )given by the matrix d Hess q ,α = (cid:20) D α grad L d α d ∗ α (cid:21) . Here d α is the linearization of the gauge group action at α , see equation (1.6), and d ∗ α is thelinear operator (1.7) defining K k,α (which coincides with the adjoint of d α ). Such an operatoris symmetric k - asafoe ,. This can be seen using the decomposition T j,α = L j ( Y ; S ) ⊕ L j ( Y ; iT ∗ Y ) . The standard Hessian appears using the decomposition T j,α = J j,α ⊕ K j,α as a block in the matrix d Hess q ,α = d α q ,α d ∗ α + ˜ h where ˜ h is symmetric, bounded between the spaces of lower regularity and vanishes at acritical point. Hence we obtain that Hess q ,α has a complete orthornormal system { e n } ofsmooth eigenvectors with real eigenvalues which is dense in each K j,α , such that there is afinite number of eigenvalues in every bounded interval. Finally Hess q ,α is Fredholm of indexzero, hence it is surjective if and only if it is injective.Suppose now that we have a smooth submanifold [ C ] ⊂ B ∗ k ( Y ) (which we will alwayssuppose to be closed, connected and finite dimensional) consisting of gauge equivalence classescritical points of grad −L . A standard regularity argument implies that every [ a ] ∈ [ C ] alwaysadmits a smooth representative (see Corollary 12 . . . HESSIANS AND MORSE-BOTT SINGULARITIES 31 gauge-equivalence classes of critical points is independent of the choice of k . Call C ⊂ C ∗ k ( Y )the corresponding gauge-invariant critical set. Given α ∈ C , define T α C ⊂ K k,α to be the inverse image of T [ α ] [ C ] via the identification K k,α → [ T k ] arising from the localchart provided by the slice ¯ı : U ⊂ S k,α → B k ( Y ) . This gives rise to a finite dimensional subbundle T C ⊂ K j | C over C which is invariant underthe action of the gauge group. We can then define the normal bundle to C as N j = K j / T C → C. As C consists of critical points for grad −L , the space T α C is contained in the kernel of Hess q ,α ,and hence there is an induced mapHess n q ,α : N k,α → N k − ,α called the normal Hessian . This is symmetric and inherits all the nice spectral propertiesof the Hessian discussed above. It is Fredholm of index 0 and it defines a smooth gauge-equivariant map(2.2) Hess ν q : N k → N k − of bundles over C . Even though the decomposition (2.1) does not extend smoothly to thewhole configuration space, all the definitions we have provided continue to make sense withoutany change also for critical submanifolds [ C ] consisting entirely of reducible configurations .For such critical submanifolds one can define the normal Hessian as in the irreducible case2.2. We are now ready to define the notion of Morse-Bott singularity. Definition . We say that a (closed, connected and finite dimensional) submanifold ofgauge equivalence classes of critical points [ C ] ⊂ B k ( Y ) with the property that if [ C ] containsa reducible configuration then in consists entirely of reducible configurations is a Morse-Bottsingularity if for every α ∈ C the normal HessianHess n q ,α : N k,α → N k − ,α is an isomorphism. We say that grad −L and the perturbation q are Morse-Bott if the set ofgauge equivalence classes of critical points is an union of Morse-Bott singularities.For brevity, we will call a submanifold of gauge equivalence classes of critical points simplya critical submanifold . It is an immediate consequence of the definition that for a Morse-Bottgrad −L all the critical submanifolds are isolated, as in that case T α C is exactly the kernel ofHess q ,α . Furthermore the requirement that the submanifold is finite dimensional is redundantas it follows from the spectral properties of the Hessian.The compactness properties of the space of solutions of the (perturbed) Seiberg-Wittenequations (see Section 10 . . −L ) σ is not the gradient of a function in any natural way. The Hessian Hess σ q : K σk → K σk − is obtained by restricting to K σk ⊂ T σk the smooth bundle map T σk → K σk − x Π K σk − ◦ D (grad −L ) σ ( x ) . Here D (grad −L ) σ is the vector field obtained by differentiating (grad −L ) σ as a vector field alongthe submanifold C σk ( Y ) ⊂ A k × R × L k ( Y ; S ) . As in the previous case, it is useful to work with the extended Hessian d Hess q ,α : T k,α ⊕ L k ( Y ; i R ) → T k − ,α ⊕ L k − ( Y ; i R ) , which will also be a protagonist when studying moduli spaces on infinite cylinders and isdefined as follows. Consider the operator d σ, † a : T σk, a → L k − ( Y ; i R )given at configuration a = ( B , s , ψ ) by the map( b, s, ψ )
7→ − d ∗ b + is Re h iψ , ψ i + i | ψ | Re( µ Y h iψ , ψ i ) . This is the operator defining the subspace K σk, a (see Section 3 of Chapter 1), and it is importantto notice that this is not the adjoint of the differential of the gauge group action d σ a : L k − ( Y ; i R ) → T σk,α , defined in equation (1.5). Then we define the extended Hessian by the matrix(2.3) d Hess σ q ,α = (cid:20) D a (grad −L ) σ d σα d σ, † a (cid:21) . Using the further decomposition T σk, a = J σk, a ⊕ K σk, a we see that the operator has the form d Hess σ q ,α = x d σα y Hess σ q ,α d σ, † a , and at a critical point a the terms x and y vanish, so that the Hess σ q , a is a direct summand ofthe extended Hessian. Notice though that the extended Hessian is not a k - asafoe operator,for the simple reason that it does not act on the space of sections of a fixed vector bundleover Y . This problem can be fixed by means of the following construction (which will be usedagain in the rest of the present work): we can convert an element( b, r, ψ ) ∈ T σj, a where a = ( B , r , ψ ) to a section of the vector bundle iT ∗ Y ⊕ R ⊕ S simply by setting(2.4) ( b, r, ψ ) ( b, r, ψ ) where ψ = ψ + rψ . From this perspective we see that the extended Hessian is a k - asafoe operator. (See Section12 . . HESSIANS AND MORSE-BOTT SINGULARITIES 33 As before, suppose we are given a smooth submanifold [ C ] ⊂ B σk ( Y ) consisting of criticalpoints of (grad −L ) σ . We will always suppose that such a submanifold is closed, connectedand finite dimensional. Furthermore, we suppose that if it contains a reducible configuration,then all configurations in it are reducible. Call C ⊂ C σk ( Y ) the corresponding gauge-invariantcritical set. As before, given a ∈ C , we can define T a C ⊂ K k, a to be the inverse image of T a [ C ] via the identification K σk,α → [ T k ] provided by the local chartgiven by the slice ¯ ι : U ⊂ S σk, a → B k ( Y ) . This gives rise to a finite dimensional vector bundle T C → C which is a subbundle of K σj | C for each j = 1 , . . . k , and we can define the normal bundle N σj = K σj / T C . Now, C consists of critical points of (grad −L ) σ hence T a C is contained in the kernel of Hess σ q , a .We can then define the normal Hessian in the blow up as the operatorHess σ,n q , a : N σ a ,k → N σ a ,k − which also induces a smooth gauge-equivariant mapHess σ,n q : N σk → N σk − between bundles over C . Definition . We say that a (closed, connected and finite dimensional) submanifoldof gauge equivalence classes of critical points [ C ] ⊂ B σk ( Y ) is a Morse-Bott singularity if thefollowing conditions hold: • [ C ] consists entirely of reducible or irreducible configurations; • for every a ∈ C the normal HessianHess σ,n q , a : N σ a ,k → N σ a ,k − is an isomorphism; • in the case [ C ] consists of reducible configurations, we also require the condition thatits blow-down [ C ] is also a reducible Morse-Bott critical submanifold (in the sense ofDefinition 1.1), and the restriction of the blowdown map is a fibration.We say that (grad −L ) σ and the perturbation q are Morse-Bott if all submanifolds of gaugeequivalence classes of critical points are Morse-Bott singularities.As before, for brevity we will call [ C ] a critical submanifold. Furthermore, Morse-Bottsubmanifolds are isolated, as T a C is exactly the kernel of Hess σ q , a . The blow down inducesa bijective correspondence between irreducible critical points, and the condition of being aMorse-Bott singularity is preserved under blow down. The reducible case is slightly morecomplicated. We impose the condition that the blow down map is a fibration in order toensure that the moduli spaces are regular in the sense of Section 3 (see Definition 3.14 andthe end of the proof in Theorem 3.17). This condition is of course satisfied when the criticalsubmanifold downstairs consists of a single point. It also implies that the correspondingeigenspaces of the Dirac operators all have the same dimension. Example . Consider the unperturbed equations on a three manifold Y with positivescalar curvature for some torsion spin c structure. Then the torus of flat connections is aMorse-Bott singularity (downstairs), because the corresponding Dirac operators do not havekernel (see Chapter 36 in the book). For the unperturbed equation on the flat torus with thetorsion spin c structure, the torus of flat connections is not a Morse-Bott singularity. Thisis because the Dirac operator D B for the flat connection with trivial holonomy B has twodimensional kernel consisting of constant section (see also Chapter 37 in the book). Example . An example of Morse-Bott singularities (in the blown down configurationspace) arises when studying Seifert fibered space homology spheres, as shown in [
MOY97 ].Notice that in the paper a non standard reducible connection is used instead of the Levi-Civitaone, but all the definitions still apply.
Example . The key example we are interested in is the case of a 3-manifold equippedwith a self-conjugate spin c structure which will be extensively studied in Chapter 4. In thatcase three kinds of singularities will arise: • irreducible non-degenerate critical points; • reducible non-degenerate critical points; • reducible critical submanifolds diffeomorphic to S blowing down to a single reducibleconfiguration.The present work is developed in order to be able to deal properly with the last kind ofsingularity.We have the following easy lemmas. Lemma . If [ C ] is a Morse-Bott singularity consisting of reducible configurations, thereis no point [ b ] ∈ [ C ] such that Λ q ([ b ]) = 0 . Proof.
This readily follows from the fact that one of the summands of the Hessian at areducible critical point [ b ] is the multiplication map on R given by t Λ q ([ b ]) t, see Lemma 12 . . R is not in the tangent space of [ C ] because it only consistsof reducibles. (cid:3) Lemma . If a ∈ C is a Morse-Bott singularity, then the extended Hessian d Hess σ q ,α isFredholm of index , has real spectrum and kernel consisting exactly of T a C . Proof.
The first two properties are already proven in the book, see Section 12 .
4. Forthe last one, the operator at a critical point a has the form d Hess σ q ,α = d σα σ q ,α d σ, † a and the block (cid:20) d σα d σ, † a (cid:21) is invertible (see Section 12 . (cid:3) . MODULI SPACES OF TRAJECTORIES 35 It is clear that Morse-Bott singularies are not generic in any sense. If a perturbation q issuch that (grad −L ) σ has Morse-Bott singularities, a nearby perturbation will generally have adifferent set of critical points. In what follows we will suppose that a Morse-Bott perturbation q is fixed, and we need to perturb the vector field away from the singularities in order toachieve transversality. This can be done as follows.In the based configuration space B ok ( Y ) = C k ( Y ) / G ok +1 ( Y ) the image of the critical setis a finite collection of S -invariant submanifolds. This is because the set of critical pointsin the non blown-up setting is compact (see Corollary 10 . . p : B ok ( Y ) → R n × T × C m defined by a collection of coclosed 1-forms and sections of S such that p is an embedding ofimage of the critical set. For each critical submanifold [ C ] ⊂ B k ( Y ), let O [ C ] ⊂ B k ( Y ) be anopen S -invariant neighborhood of the corresponding S -orbit, chosen so that their imagesunder p have disjoint closures, and write O = [ [ C ] O [ C ] ⊂ B ok ( Y ) . We also require that each O [ C ] is path connected and that the relative fundamental groups π (cid:16) p ( O [ C ] ) , p ([ C ]) (cid:17) are trivial. Consider the subset P O ⊂ P of perturbations(2.5) P O = { q ∈ P | q | O = q | O } which is a closed linear subspace of P , hence a Banach space itself. Then there is an openneighborhood q in P O such that for all q in this neighborhood, the vector fields grad L q and(grad L q ) σ have no zeroes outside O , hence they have exactly the same zeroes as grad L q and(grad L q ) σ (see Proposition 11 . . Definition . Suppose we are given a tame Morse-Bott perturbation q . We say thata tame perturbation q is adapted to q if q agrees with q in a neighborhood O as above andthe perturbed vector field grad L + q does not have any critical point outside of O .The discussion above tells us that there is an open set of tame perturbations in P O adaptedto a given q inside the Banach space of perturbation satisfying the property (2 . sa in Section 12 .
2. Moduli spaces of trajectories
In this section, which is analogous of Chapter 13 in the book, we provide two descriptionsof the space of solutions connecting two given critical submanifolds. Each of them will be useful for different aspects of the theory in the following sections. In order to prove that thesedescriptions are equivalent, we need to prove some estimates for solutions of the perturbedSeiberg-Witten equations on a finite cylinder Z = I × Y which always stay in a small neighbor-hood of a given critical point. We study this problem in the first part of the section, and thenfocus in the second part on the basic properties of the solutions on the infinite cylinder. Wewill always assume that a perturbation q is chosen so that all the singularities are Morse-Bott.Let b be a critical point of the perturbed equations on C σk ( Y ) and let α be its image in C k ( Y ). Denote the corresponding translation-invariant solutions on I × Y as γ a = ( A b , s b , φ b ) ∈ C τk ( I × Y ) γ α = ( A b , Φ b ) ∈ C k ( I × Y ) . Given an element γ τ = ( A, s, φ ) ∈ C τk ( I × Y )covering a configuration γ = ( A, sφ ) = ( A, Φ) ∈ C k ( I × Y )we can write A − A b = b ⊗ c ⊗ dtφ = φ b + ψ where b , c are time dependent 1 and 0-forms respectively, and ψ is a time dependent sectionof S → Y such that φ a + ψ ( t ) is of unit norm for every t ∈ I . In what follows Sobolev normof a difference of two configurations in the τ -model such as k γ τ − γ b k L k,A b ( I × Y ) is intended as the norm in the bigger affine space L k ( I × Y ; iT ∗ Z ) × L k ( I ; R ) × L k,A a ( I × Y ; S ) . The result and proofs we are going to discuss are very close to the case of Morse sin-gularities in the book, and they can be thought as a parametric version of them. Beforestating them, we need to define a nice parametrization of a neighborhood of a critical point b ∈ C σk ( Y ). Definition . Fix k ≥
1. Given a point b in a critical submanifold C , an L k -compatibleproduct chart around b is a pair ( U , ϕ ) where U ⊂ T b C ⊕ N σ b ,k ⊕ J σ b ,k is an open neighborhood of { } and there exists a small neighborhood U ⊂ C σk ( Y ) of b and amap ϕ : U → U such that the following hold: • ϕ is an L k -compatible diffeomorphism, i.e. there exists C > k v k L k /C ≤ k dϕ ( v ) k L k ≤ C k v k L k for every v ∈ T U , and D ϕ = Id; . MODULI SPACES OF TRAJECTORIES 37 • the restriction of ϕ to U ∩ ( T b C ⊕ N σ b ,k ⊕ { } ) is a local chart for the Coulomb slice S σk, b through b such that the restriction to U ∩ ( T b C ⊕ { } ⊕ { } ) is a local chart for C ∩ S σk, b with ϕ (0) = b ; Lemma . Every critical point b ∈ C σk ( Y ) admits a L k -compatible chart around it. Proof.
Consider the map Ψ : T b C ⊕ N σ b ,k → S σk, a obtained by composing the local chart from Section 3 in Chapter 1 ι : K σk, b → S σk, b with a linear isomorphism of Hilbert spaces ψ : T b C ⊕ N σ b ,k → K σk, b where ψ is the inclusion on the first summand and a left inverse of the quotient map K σk, b → N σ b ,k on the second summand. The map Ψ is a diffeomorphism in a neighborhood U ′ of { } ⊕ { } ,and by the implicit function theorem we can describe Ψ − ( C ) ∩ U ′ in a smaller neighborhoodas the graph of a smooth function f : T b C → N σ b ,k . The origin is a critical point of f because T b C is by definition tangent to C . To obtain aparametrization of a neighborhood in C σk ( Y ) we use the action of the gauge group as follows.The differential of the gauge group action gives the isomorphism d σ b : T e G k +1 ( Y ) → J σ b ,k and we define the map ϕ : U ′ ⊕ J σ b ,k → C σk ( Y )( v t , v n , v j ) exp (cid:0) ( d σ b ) − ( v j ) (cid:1) · Ψ( v t , v n + f ( v t )) . Here we are exponentiating the element in the Lie algebra to obtain a gauge transformation.The differential of this map at the origin is the identity by definition, and this is in fact L k -compatible product chart by applying the inverse function theorem. (cid:3) Remark . If 1 ≤ j ≤ k the construction above implies that if b is an L k configurationwe can construct an L k -compatible product chart which is the restriction of an L j -compatibleproduct chart. This follows from the fact that the differential at the origin is the identity andthe uniqueness statement in the inverse function theorem. We will use this fact in Section5. Indeed, one can slightly modify the construction to obtain a L k -compatible product chartwhich is defined in an L j neighborhood of the origin, see Lemma 5.13. The chart we haveconstructed does not have this property, but it has the desirable property of being gaugeinvariant is some sense that will be made precise in Remark 2.4.Given such an L k -compatible product chart ( U , ϕ ), for a configuration ( B, r, ψ ) ∈ U = ϕ ( U ) one can define its normal component ( B, r, ψ ) ν = ϕ (cid:0) Π ν ◦ ϕ − ( B, r, ψ ) (cid:1) ∈ U where Π ν : T a C ⊕ N σ a ,k ⊕ J σk, a → { } ⊕ N σ a ,k ⊕ J σk, a is the projection with kernel T a C ⊕ { } ⊕ { } . Furthermore, given any L pathˇ γ : I → C σk ( Y )such that ˇ γ ( t ) is in U for every t ∈ I , we can define its normal part γ ν to be the path definedas ˇ γ ν ( t ) = ˇ γ ( t ) ν for t ∈ I. This is still an L path. The most interesting case is that of a solution γ ∈ C τk ( I × Y ) of theSeiberg-Witten equations on the cylinder. After a suitable gauge transformation, this givesrise to a L path (cid:0) ˇ γ ( t ) , c ( t ) (cid:1) ∈ C σk ( Y ) × L k ( Y, i R ) , see Corollary 10 . . normal part , denotedby γ ν , to be the L path (ˇ γ ν ( t ) , c ( t ) (cid:1) .Similarly, there is a notion of tangent component ( B, r, ψ ) t = ϕ (cid:0) Π t ◦ ϕ − ( B, r, ψ ) (cid:1) ∈ S σk, b where Π t : T a C ⊕ N σ a ,k ⊕ J σk, a → T a C ⊕ { } ⊕ { } is the projection with kernel { } ⊕ N σ a ,k ⊕ J σk, a , and one can define the tangent part of a L path in the identical fashion as above. Remark . It follows from the construction of Lemma 2.2 that we can choose the L k -compatible product chart so that the tangent part of any two gauge equivalent configurationsin the image of the chart is the same.Furthermore, the images of the trivial bundles over U with fiber T a C ⊕ { } ⊕ { } and { } ⊕ N σ a ,k ⊕ { } , which we denote by T tk and T nk determine via fiberwise completion a smoothbundle decomposition T l | ˜ U = T tj ⊕ T nj ⊕ J σj for every j ≤ k (notice that the first summand is not affected by the completion) on someneighborhood ˜ U ⊂ U . We will restrict our chart to this neighborhood. In particular, we candecompose the gradient of the Chern-Simons-Dirac in its tangent and normal part (grad −L ) σ,t and (grad −L ) σ,n . There is an analogous notion of L k -compatible product chart for a Morse-Bott critical point in the non blown-up configuration space, which will be useful in whatfollows.We start by studying the situation downstairs. From now on we suppose that a L -compatible product chart is fixed. We have the following key estimates for near constantsolutions (see Proposition 13 . . k is an integer not less than 2. . MODULI SPACES OF TRAJECTORIES 39 Proposition . For any Morse-Bott critical point β there exists a gauge invariantneighborhood U of γ β in C k ( I × Y ) and a constant C such that any solution γ in U in theCoulomb-Neumann slice S k,γ β satisfies the estimate k γ ν k L ( I × Y ) ≤ C (cid:0) −L ( s ) − −L ( s ) (cid:1) . Here we are choosing the gauge invariant neighborhood to be small enough so that thenormal part is well defined for a configuration in Coulomb-Neumann slice. The proof ofthis result proceeds as the one in the book, the only difference being the fact that the nondegeneracy of the normal Hessian gives us control only on the normal part of the configurationand not on the whole configuration. Furthermore, we cannot estimate the L k +1 norm (upto gauge and on a smaller interval I ′ ⊂ I as in the statement in Proposition 13 . .
1) becausethe normal part of a solution is generally not a solution, hence the bootstrapping argumentdoes not apply. On the other hand, the rest of the proof works few modifications. The keyobservation is the following.
Lemma . Given an irreducible Morse-Bott critical point β there is a constant C > and a neighborhood ( β, ∈ U Y ⊂ C ( Y ) × L ( Y, i R ) such that for every ( β + v, c ) ∈ U Y we have k (( β + v ) ν , c ) k L ( Y ) ≤ C (cid:0) k d ∗ β v k + k d β + v c k + k grad −L ( β + v ) k (cid:1) . In the reducible case the same results holds with the additional hypothesis R Y c = 0 . Notice that we need to choose first a neighborhood small enough of β in order to have anotion of normal component. Proof.
Consider the map C ( Y ) × L ( Y ; i R ) → J ⊕ T n ⊕ L ( Y ; i R )( β + v, c ) (cid:0) d β + v c, (grad −L ) n ( β + v ) , d ∗ β v (cid:1) where we use the compatible product chart to define the normal part of the gradient. Usingthe identification T ,β ≡ J ,β ⊕ T β C ⊕ N ,β , the linearization of this map at ( β,
0) can be written as( v J , v τ , v n , c ) ( d β c, Hess nβ v n , d ∗ β v J ) . Here we use that the perturbation q is a C section, as stated in condition (6) in the Definition4.1 in Chapter 1. As Hess nβ : N ,β → N ,β is an isomorphism and the map d β : L ( Y ; i R ) → J ,β is invertible in the irreducible case, we obtain that the linearization is an isomorphism whenrestricted to the subspace( J ,β ⊕ { } ⊕ N ,β ) ⊕ L ( Y ; i R ) ⊂ T ,β ⊕ L ( Y ; i R ) . This, together with the L -compatibility of the chart, implies that the estimate k (( β + v ) ν , c ) k L ( Y ) ≤ C ′ (cid:0) k d ∗ β v k + k d β + v c k + k (grad −L ) n ( β + v ) k (cid:1) holds in some L neighborhood of ( β, β and a costant C ′′ such that for β + w in this neighborhood one has(2.6) k (grad −L ) ν ( β + v ) k ≤ C ′′ k grad −L ( β + v ) k , hence we obtain the required estimate. (cid:3) Proof of Proposition 2.5.
Following the proof of Lemma 13 . .
4, we obtain an identityof the form2 (cid:0) −L ( s ) − −L ( s ) (cid:1) == Z I (cid:0) k dds ˇ γ ( s ) k + k dds c ( s ) k (cid:1) ds + Z I (cid:0) k d ∗ β (ˇ γ ( s ) − β ) k + k d ˇ γ ( s ) c ( s ) k + k grad −L (ˇ γ ( s )) k (cid:1) ds + Z I (cid:10) (ˇ γ − β ) ♯c, dds ˇ γ (cid:11) ds where ♯ is a bilinear operator involving only pointwise multiplication. By the previous lemma(which can be applied to the restriction because k ≥
2) the second term on the right handside bounds from above Z I k (cid:0) ˇ γ ν ( s ) , c ( s ) (cid:1) k L ( Y ) ds. We claim that, after possibly rescaling the neighborhood of the critical point β , we can obtainan estimate of the form(2.7) k dds ˇ γ t ( s ) k L ≤ K k ˇ γ ν ( s ) k L ( Y ) for some K >
0. From this, we see that the inequality can be rearranged (using that indimension 4 the L norm is controlled by the L norm) to be k ( γ − γ β ) ν k L ( I × Y ) ≤ C ′ (cid:0) −L ( s ) − −L ( s ) (cid:1) + K k γ − γ β k L ( I × Y ) k ( γ − γ β ) ν k L ( I × Y ) , hence we obtain the result by restricting to a suitably small neighborhood of γ β .To prove the estimate (2.7), we just notice that for a suitably small neighborhood of β we have for every small ε > k ˇ γ t ( s + ε ) − ˇ γ t ( s ) k L ( Y ) ≤ Z s + εs k (grad −L ) t (ˇ γ ( s )) k L ( Y ) ds ≤ C ′′ Z s + εs k grad −L (ˇ γ ( s )) k L ( Y ) ds ≤ K Z s + εs k ˇ γ ν ( s ) k L ( Y ) ds. where the last inequality comes from the fact that −L is constant along the critical submanifold,and the fact that the gradient of −L is C in the L topology. Then our desired inequalityfollows by dividing both sides by ε and taking the limit for ε going to zero. (cid:3) So far we have treated the case of configurations spaces on a finite cylinder Z = I × Y .In the case of an infinite cylinder, the situation is a little different (see Section 13 . . MODULI SPACES OF TRAJECTORIES 41 book). For any interval I ⊂ R , one introduces the configuration space˜ C τk, loc ( I × Y ) = { ( A, s, φ ) | k ˇ φ ( t ) k L ( Y ) = 1 for every t ∈ I }⊂ A k, loc ( I × Y ) × L k, loc ( I ; R ) × L k, loc ( I × Y ; S + ) , where A k, loc ( I × Y ) = A + L k, loc ( I × Y ; iT ∗ Z )for any smooth spin c connection A . We consider on this space the topology of L k convergenceon compact subsets (it is clear that for I compact this definition coincides with the usual one).Furthermore, we define the closed subspace C τk, loc ( I × Y ) ⊂ ˜ C τk, loc ( I × Y )consisting of configurations ( A, s, φ ) with s ≥
0. The appropriate gauge group to consider is G k +1 , loc ( I × Y ), the group of L k +1 , loc maps with values in the circle S ⊂ C . The quotientspaces will be denoted by B τk, loc ( I × Y ) = C τk, loc ( I × Y ) / G k +1 , loc ( I × Y )˜ B τk, loc ( I × Y ) = ˜ C τk, loc ( I × Y ) / G k +1 , loc ( I × Y ) . Suppose a perturbation q ∈ P is fixed so that all critical points of (grad −L ) σ are Morse-Bott singularities. The choice of q determines the perturbed 4-dimensional Seiberg-Wittenequations, which is a section F τ q of the map V τk − , loc ( I × Y ) → ˜ C τk, loc ( I × Y ) . Here the fiber at γ = ( A , s , φ ) is the subspace V τk − , loc ,γ ⊂ L k − , loc ( I × Y ; i su ( S + )) ⊕ L k − , loc ⊕ L k − , loc ( I × Y ; S − )consisting of triples ( a, s, φ ) with Re h ˇ φ ( t ) , ˇ φ ( t ) i L ( Y ) = 0for every t ∈ I . It is important to remark that here we do not use the language of vectorbundles, as V τk − , loc is not locally trivial in any straightforward way.If b ∈ C σk ( Y ) is a critical point, then the corresponding translation invariant configuration γ b is a solution of the equations, i.e. F τ q ( γ b ) = 0, and we write [ γ b ] for its gauge-equivalenceclass. We say that a configuration [ γ ] ∈ ˜ B τk, loc is asymptotic to [ b ] as t → ±∞ if[ τ ∗ t γ ] → [ γ b ] in B τk, loc ( Z ) as t → ±∞ where τ t : Z → Z ( s, y ) ( s + t, y )is the translation map. We will respectively writelim → [ γ ] = [ b ] and lim ← [ γ ] = [ b ] . Definition . Suppose [ C − ] and [ C + ] are critical submanifolds for (grad −L ) σ . We write M ([ C − ] , [ C + ]) for the space of all configurations [ γ ] in B τk, loc ( Z ) which are asymptotic to apoint in [ C − ] for t → −∞ , asymptotic to a point in [ C + ] for t → + ∞ and solve the perturbedSeiberg-Witten equations: M ([ C − ] , [ C + ]) = (cid:8) [ γ ] ∈ B τk, loc ( Z ) | F τ q ( γ ) = 0 , lim ← [ γ ] ∈ [ C − ] , lim → [ γ ] ∈ [ C + ] (cid:9) . We refer to this as a moduli space of trajectories on the cylinder Z = R × Y . We can similarlydefine the subset ˜ M ([ C − ] , [ C + ]) of the large space ˜ B τk, loc ( Z ). Given open subsets [ U ± ] ⊂ [ C ± ]we write M ([ U − ] , [ U + ]) ⊂ M ([ C − ] , [ C + ])for the subspace of configurations which are asymptotic at ±∞ to the critical points in theopen set [ U ± ].Notice that we suppressed the value of k from our notation. Because of the bootstrappingproperties of the Seiberg-Witten equations, the space is essentially independent of the choiceof k . This is stated precisely in the next lemma (see Proposition 13 . . Lemma . Let M ([ C − ] , [ C + ]) k temporarily denote the moduli space M ([ C − ] , [ C + ]) ⊂B τk, loc ( Z ) . (1) If [ γ ] is in M ([ C − ] , [ C + ]) k , then there is a gauge representative γ ∈ C τk, loc ( Z ) which is C ∞ on Z ; (2) The naturally induced bijections M ([ C − ] , [ C + ]) k → M ([ C − ] , [ C + ]) k for all k , k ≥ are homeomorphisms. If [ γ ] ∈ M ([ C − ] , [ C + ]), then there is a corresponding (smooth) path [ˇ γ ] in B σk ( Y ) ap-proaching [ C − ] and [ C + ] at the two ends. Hence we can decompose the space according tothe relative homotopy class of the path z ∈ π ( B σk ( Y ) , [ C − ] , [ C + ])and write M ([ C − ] , [ C + ]) = [ z M z ([ C − ] , [ C + ]) . The set of homotopy classes is an affine space on H ( Y ; Z ), the component group of the gaugegroup. Furthermore, we have the continuous evaluation maps ev + : M ([ C − ] , [ C + ]) → [ C + ]ev − : M ([ C − ] , [ C + ]) → [ C − ]obtained by sending a solution to its limit points at ±∞ ,ev ± [ γ ] = lim t →±∞ [ˇ γ ( t )] . While the definition of the moduli space we have just given will be useful when studyingcompactness issues, it will not fit when dealing with transversality problems, because thespaces involved are not Banach manifolds in any natural way. Furthermore, unlike the casetreated in the book, our operators are not well behaved on the L k spaces on the cylinderbecause we are dealing with Morse-Bott singularities. In particular, their linearization isnot Fredholm on such spaces. We tackle the problem in the following way, see for example . MODULI SPACES OF TRAJECTORIES 43 [ Don02 ] and [
MMR94 ]. Recall that for a vector bundle E → Z which is the pullback ofa bundle on Y the definition of the weighted Sobolev space L k,δ ( Z ; E ) with weight δ ∈ R > .Pick a smooth function f : R → R > such that f ( t ) = e δ | t | for | t | >> . Then L k,δ ( Y ; E ) is the space f − L k ( Z ; E ):(2.8) s ∈ L k,δ ( Z ; E ) if and only if f · s ∈ L k ( Z ; E ) . In this case, the norm on L k,δ ( Y ; E ) is defined so that the multiplication by f is an isometry L k,δ → L k . Furthermore, different functions define equivalent norms. It may be usefulsometimes to use the equivalent norm(2.9) k s k k,δ = Z Z f (cid:16) | s | + |∇ s | + · · · + |∇ k s | (cid:17) d vol . With this in mind, the following embedding and multiplication results are straightfor-wardly adapted from the unweighted case (see Theorems 13 . . . . Proposition . There is a continuous inclusion L pk,δ ( Z ) ֒ → L ql,δ ′ ( Z ) for k ≥ l , δ ′ ≤ δ , p ≤ q and ( k − n/p ) ≥ ( l − n/q ) , with the further assumption that if the lastinequality is an equality < p ≤ q < ∞ . This embedding is compact if and only if δ δ ′ . Proposition . Suppose δ + δ ′ ≥ δ ′′ , k, l ≥ m and /p + 1 /q ≥ /r , with p, q, r ∈ (1 , ∞ ) . Then the multiplication L pk,δ ( Z ) × L ql,δ ′ ( Z ) → L rm,δ ′′ ( Z ) is continuous in any of the following three cases: (1) (a) ( k − n/p ) + ( l − n/q ) ≥ m − n/r , and (b) k − n/p < , and (c) l − n/q < ;or (2) (a) min { ( k − n/p ) , ( l − n/q ) } ≥ m − n/r , and (b) either k − n/p > or l − n/q > ;or (3) (a) min { ( k − n/p ) , ( l − n/q ) } ≥ r − n/m , and (b) either k − n/p = 0 or l − n/q = 0 .When the map is continuous, it is a compact operator as a function of g for fixed f provided l > m and l − n/q > m − n/r . Suppose we are given critical submanifolds [ C − ] and [ C + ]. Our aim is to define the space oftrajectories connecting these two submanifolds that converge exponentially fast up to gaugetransformation. In order to do so, first choose two contractible open sets [ U − ] ⊂ [ C − ] and[ U + ] ⊂ [ C + ] that admit smooth lifts U − and U + . We can choose these open sets and their liftsso that the latter are contained in the slice passing through configurations b ± ∈ C σk ( Y ) with[ b ± ] ∈ [ U ± ]. Choose then a smooth family of smooth base configurations { γ } parametrizedby ( a − , a + ) ∈ U − × U +4 2. THE ANALYSIS OF MORSE-BOTT SINGULARITIES such that γ ( a − , a + ) agrees near ±∞ with the translation invariant configuration γ a − and γ a + respectively. Choosing the lifts in the appropriate component of the gauge group or-bit, we can arrange that the family [ γ ] defines any given relative homotopy class z ∈ π ( B σk ( Y ) , [ C − ] , [ C + ]). As C τk, loc ( Z ) is a subset of an affine space˜ C τk, loc ⊂ A k, loc ( Z ) × L k, loc ( R , R ) × L k, loc ( Z ; S + ) , we can interpret the difference of two configurations as elements of the vector space L k, loc ( Z ; iT ∗ Z ) × L k, loc ( R , R ) × L k, loc ( Z ; S + ) , and it makes sense to ask for the L k,δ norm to be finite. Hence we can introduce the config-uration space C τk,δ ( U − , U + ) defined as { γ ∈ C τk, loc ( Z ) | γ − γ ( a − , a + ) ∈ L k,δ ( Z ; iT ∗ Z ) × L k,δ ( R , R ) × L k,δ ( Z ; S + )for some ( a − , a + ) ∈ U − × U + } . It is clear that such a definition is independent of the choice of the family of base configurations { γ } . Similarly we can introduce the larger space ˜ C τk,δ ( U − , U + ) as a subspace of ˜ C τ k, loc ( Z ). Wealso introduce the gauge group G k +1 ,δ ( Z ) which is the subgroup of G k +1 , loc ( Z ) that preserves C τk,δ ( U − , U + ). This group has the following simple characterization, whose proof is readilyadapted from the one of Lemma 13 . . Lemma . The group G k +1 ,δ ( Z ) is independent of U − and U + and can be described as G k +1 ,δ ( Z ) = { u : Z → S | − u ∈ L k +1 ,δ ( Z ; C ) } Furthermore, its component group is Z , the identification given by the winding number of themap t u ( t, y ) for any fixed basepoint y . We can then define the quotient spaces B τk,δ,z ([ U − ] , [ U + ]) = C τ,δ k ( U − , U + ) / G k +1 ,δ ( Z )˜ B τk,δ,z ([ U − ] , [ U + ]) = ˜ C τk,δ ( U − , U + ) / G k +1 ,δ ( Z )where we have chosen the lifts U − and U + so that a path connecting them projects to a pathin the class z . To within a canonical identification, B τk,δ,z ([ U − ] , [ U + ]) is independent of thechoice of these smooth lifts. Also, if [ U ′− ] ⊂ [ U − ] and [ U ′ + ] ⊂ [ U + ] we have a natural inclusion B τk,δ,z ([ U − ] ′ , [ U ′ + ]) ֒ → B τk,δ,z ([ U − ] , [ U + ]) , and an analogous one for the larger spaces with tildes. Using these identifications we candefine the spaces of configurations B τk,δ,z ([ C − ] , [ C + ]) = a B τk,δ,z ([ U − ] , [ U + ]) / ∼ ˜ B τk,δ,z ([ C − ] , [ C + ]) = a ˜ B τk,δ,z ([ U − ] , [ U + ]) / ∼ and their union over all relative homotopy classes B τk,δ ([ C − ] , [ C + ]) = [ z B τk,δ,z ([ C − ] , [ C + ])˜ B τk,δ ([ C − ] , [ C + ]) = [ z ˜ B τk,δ,z ([ C − ] , [ C + ]) . . MODULI SPACES OF TRAJECTORIES 45 The key result of the section is the following.
Theorem . For a fixed tame perturbation q such that all singularities are Morse-Bott there exists a δ > with this property. For any two critical submanifolds [ C − ] and [ C + ] and any γ ∈ C τk, loc ( Z ) representing an element [ γ ] ∈ M z ([ C − ] , [ C + ]) , and let b ± besuitable lifts of ev ± ([ γ ]) as above. Then there is a gauge transformation u ∈ G k +1 , loc suchthat u · γ ∈ C τk,δ ( b − , b + ) , and if u ′ is another such a gauge transformation, then u − u ′ ∈ G k +1 .The resulting bijection is a homeomorphism M z ([ C − ] , [ C + ]) → (cid:8) [ γ ] ∈ B τk,z,δ ([ C − ] , [ C + ]) | F τ q ( γ ) = 0 (cid:9) . Finally, if the statement holds for δ , then it holds for every < δ ′ < δ . Before proving the theorem, we discuss a key auxiliary result (see Proposition 13 . . Lemma . For every solution γ τ ∈ C τk, loc of the perturbed Seiberg-Witten equations on [0 , ∞ ) × Y asymptotic to a Morse-Bott critical point [ b ] ∈ C σk ( Y ) there exists a t such thatfor all t ≥ t −L ( γ τ ( t )) − −L ( b ) ≤ Ce − δt where C = −L ( γ τ ( t )) − −L ( b ) . Furthermore, if a tame perturbation q such that all singularitiesare Morse-Bott is fixed, then such a δ > can be chosen uniformly for all critical points. Proof.
This is analogous to the Morse case. Let C be the critical submanifold β itbelongs to, and fix a L -compatible product chart around β . As −L is C on C ( Y ) withvanishing derivative at β = ( B, Φ), we have for some C > |−L ( β + w ) − −L ( β ) | = |−L ( β + w ) − −L (( β + w ) τ )) | ≤ C k ( β + w ) ν k L ,B for all w ∈ T β C ( Y ) with [ β + w ] in some L neighborhood U of [ β ] in B ( Y ) (here we use the L compatibility of the local chart used to define the normal part). Then the non-degeneracyof the normal Hessian tells us that k (grad −L ) n ( β + w ) k L ≥ C k ( β + w ) ν k L for β + w in the Coulomb slice S ,β (so that ( β + w ) ν is also in the Coulomb slice) and [ β + w ]in some L neighborhood of [ β ], and, after possibly restricting to a smaller neighborhood U , k grad −L ( β + w ) k L ≥ C ′ k ( β + w ) ν k L As the path [ˇ γ ( t )] converges to [ b ], we can assume that ˇ γ ( t ) lies in the Coulomb slice for t ≥ t , so it lies in U ∩ U and we have that −L satisfies the differential inequality dds −L (ˇ γ ( s )) = −k grad −L (ˇ γ ( t )) k L ≤ − δ ( −L (ˇ γ ( s )) − −L ( β ))where δ = C ′ /C . This implies the exponential decay in the statement. The fact that δ > . . (cid:3) Proof of Theorem 2.12.
We just need to prove that for every solution γ τ of the per-turbed equations on [0 , ∞ ) × Y converging to a Morse-Bott critical point [ b ] there exists a δ depending only in b and a gauge transformation u such that u · γ τ − γ b ∈ L k,A b ,δ ([0 , ∞ ) × Y ) . We first focus on the case of the solutions in the blow down. To prove this, consider thesequence of cylinders [ i − , i + 1] × Y . From Proposition 2.5, we see that there is an i suchthat for every i ≥ i we can find a sequence of gauge transformations u i ∈ G k +1 ([ i − , i + 1] × Y )such that if we call γ i = u i · γ τ we have k γ νi k L ,Aβ ([ i − ,i +1] × Y ) ≤ N i where N i = C (cid:0) −L ( i − − −L ( i + 1)) / (cid:1) . Indeed, u i is simply the gauge transformation that puts γ τ in Coulomb-Neumann slice on thegiven interval. On the other hand, the norm of the tangent part γ ti (which is gauge invariantby definition) can be estimated as follows. As in the proof of the inequality (2.7), after passingto a smaller neighborhood U ′ ⊂ U , we have that for any ε ∈ [0 ,
1] and i ≥ j one has that thetangent part of the path satisfies the inequality k ˇ γ ti ( i − / ε ) − ˇ γ ti ( i − / k L ( Y ) ≤ C Z i − / εi − / k ˇ γ νi ( s ) k L ,B ( Y ) ds. As before, this implies the inequality for s big enough k dds ˇ γ ti ( s ) k L ( Y ) ≤ C k ˇ γ νi ( s ) k L ,B ( Y ) . Define the quantity M i = ∞ X j = i N j Suppose we have chosen an L -compatible product chart so that the tangent part of a con-figuration is gauge invariant (see Remark 2.4). Using the triangular inequality and the factthat ˜ γ t converges to zero by hypothesis, the same inequality implies a bound of the form k ˇ γ ti ( i − / ε ) k L ( Y ) ≤≤ k ˇ γ ti ( i − / ε ) − ˇ γ ti ( i + 1 / k L ( Y ) + ∞ X j = i +1 k ˇ γ tj ( j − / − ˇ γ tj ( j + 1 / k L ( Y ) ≤ CM i for every ε ∈ [0 , k ˇ γ ti ( i − / ε ) k L ,B ( Y ) ≤ C ′ M i . By integrating we obtain k ˇ γ ti k L ,B ([ i − ,i +1] × Y ) ≤ K Z (cid:0) k dds ˇ γ ti k L ( Y ) + k ˇ γ ti k L ,B ( Y ) (cid:1) ≤ K ′ M i , . MODULI SPACES OF TRAJECTORIES 47 hence k ˇ γ i − γ β k L ,Aβ ([ i − ,i +1] × Y ) ≤ K ′′ M i and finally by a bootstrapping argument(2.10) k ˇ γ i − γ β k L k,Aβ ([ i − / ,i +3 / × Y ) ≤ K ′′′ M i . The exponential decay of N i (which follows from Lemma 2.13) implies the exponential decayfor M i . We can then choose a ˜ δ > ∞ X i =0 e ˜ δi M i < ∞ . Now we just need to glue carefully all the gauge transformations u i in order to obtain onethat preserves the summability property. To do so, one just applies the construction in theproof of Proposition 13 . . q , which can be used to give bounds analogous to Lemma13 . . (cid:3) Lemma . Let γ τ ∈ C τk, loc a solution of the perturbed Seiberg-Witten equations on [0 , ∞ ) × Y with lim → [ γ τ ] = [ b ] , a Morse-Bott critical point. Then there is a δ > such thatthe function f ( t ) = Λ q ( t ) − Λ q ( b ) satisfies the bound | f ( t ) | ≤ Ce − δt . Proof.
As mentioned above, we can assume that b is reducible. The result follows if wecan prove a differential inequality of the form ddt f ≤ − δ | f | + Ke − δt for some K >
0. By Lemma 13 . . ddt Λ q ( γ τ ( t )) = − k φ ′ k + h φ, L ′ φ i . Notice that all three terms are gauge invariant. Working in temporal gauge, we have fromthe proof of Corollary 13 . . O ( e − δt ). We also have the lower boundon the first term k φ ′ k = k (grad −L ) σ, ( γ τ ( t )) k ≥ C k φ ν k L ,B b + O ( e − δt ) . To see this, we work in the Coulomb slice S σk,γ b on [ t − , t + 1]. As in Lemma 13 . . b is Morse-Bott and the differentiability of (grad −L ) σ, thatfor all ( B, r, φ b + ψ ) in a neighborhood of b in C σ ( Y ) we have k ψ ν k L ,B b ≤ C k (grad −L ) σ, ( B, r, φ b + ψ ) k + K ( k B − B b k L ( Y ) + r ) . The bound follows because of the exponential decay of the last term as proved in the propo-sition above, see equation (2.10). Finally using again the exponential decay of the connectioncomponent we have for a configuration (
B, r, φ ) in neighborhood in C σ ( Y ) of b the estimate | Λ q ( t ) − Λ q ( b ) | = | Re h φ, D q ,B φ i − Re h φ b , D q ,B b φ b i| ≤≤ | Re h φ, D q ,B b φ i − Re h φ b , D q ,B b φ b i| + O ( e − δt ) ≤ C ′ k φ ν k L ,B b + O ( e − δt )where the last estimate follows from the fact that the map on the unit sphere in the L norm S ( L ( Y ; S )) → R ψ Re h ψ, D q ,B b ψ i is C and has the unit sphere of the eigenspace to which φ b belongs to as a Morse-Bott criticalsubmanifold. The claimed differential inequality follows from these two inequalities. (cid:3)
3. Transversality
In this section we deal with transversality for the moduli spaces of solutions on an infinitecylinder. These are treated in the same way as in the Morse case (see Chapter 14 in thebook), with the only difference that we need to work in the weighted Sobolev space settingas the extended Hessian now has kernel at the limit configurations. We start by defining theanalytical setup for the problem. From now on we suppose that a tame perturbation q suchthat all critical points are Morse-Bott is fixed.We recall a useful trick to study differential operators on weighted Sobolev spaces, see forexample [ Don02 ]). We focus on the first order case as it will be useful later. Suppose wehave a vector bundle E → Z pulled back from E → Y together with a family of k - asafoe operators { L ( t ) } on E . Then the action of the differential operator d/dt + L ( t ) : L k,δ ( Z ; E ) → L k − ,δ ( Z ; E )is conjugated via the multiplication by f (the function defining the weighted Sobolev norm,see equation (2.8)) to the action of the differential operator(2.11) d/dt + L ( t ) + σ ( t ) : L k ( Z ; E ) → L k − ( Z ; E ) , where again σ is defined as − f ′ /f . In particular, we reduced to the more familiar study ofthe family of k - asafoe operators { L ( t ) + σ ( t ) } acting on the unweighted spaces, see Chapter14 in the book. The key point of the introduction of the weighted spaces is that for suitablechoice of δ this family of operators will be hyperbolic at the ends even though the one westarted with was not.Consider as in the previous section two contractible open sets [ U − ] , [ U + ] of two criticalsubmanifolds [ C − ] and [ C + ]. Choose smooth lifts U − , U + ⊂ C σk ( Y ) contained in the Coulombslices S σk, b ± for some b ± . Fix some δ > . TRANSVERSALITY 49 Banach manifold of paths ˜ C τk,δ ( U − , U + ). We write T τj,δ for the L j,δ completion of its tangentbundle. In particular the fiber T τj,γ at γ = ( A , s , φ ) withev ± ( γ ) = b ± ∈ U ± is given by the subset (cid:8) ( a, s, φ ) | Re h φ ( t ) , φ ( t ) i L ( Y ) = 0 (cid:9) ⊂ L j, loc ( Z ; iT ∗ Z ) ⊕ L j, loc ( R , R ) ⊕ L j,loc,A ( Z ; S + )consisting of the configurations v such that there exist v ± ∈ T b ± C ± with v − γ v − ,v + ∈ L j,δ ( Z ; iT ∗ Z ) ⊕ L j,δ ( R , R ) ⊕ L j,,δ,A ( Z ; S + )where we define γ v − ,v + = β ( − t ) v − + β ( t ) v + for any fixed smooth cut-off function β such that β ( s ) = ( x ≤ , x ≥ . It is clear that such v ± are unique and we call them the evaluations ev ± ( v ). We also definethe Hilbert norm of v ∈ T τj,γ as k v k L j,δ ( Z ) := k v − γ ev + ( v ) , ev − ( v ) k L j,δ ( Z ) + k ev − ( v ) k L ( Y ) + k ev + ( v ) k L ( Y ) . Different choices of the function β clearly define equivalent norms. One can also define T τj,δ, = { v | ev − ( v ) = ev + ( v ) = 0 } ⊂ T τj,δ , which is a closed subspace with finite codimension dim[ C − ] + dim[ C + ].The derivative of the gauge group action, regarded as a bundle map, is given by d τ : Lie( G j +1 ,δ ) × ˜ C τk,δ ( b − , b + ) → T τj,δ ( ξ, γ ) ( − dξ, , ξφ ) , where γ = ( A , s , φ ), and the image is contained in the subspace T τj,δ, . For a fixed configu-ration γ one can define the Coulomb gauge fixing conditionCoul τγ : C τk,δ ( U − , U + ) → L k − ,δ ( Z ; i R )given by the equation( A + a, s, φ )
7→ − d ∗ a − σ ( t ) c + iss Re h iφ , φ i + i | φ | Re ( µ Y ( h iφ , φ i ))Here we are considering a = cdt + b. for a time dependent imaginary valued function c and a family b of imaginary valued one-formson Y , and σ is the function depending only on time appearing in equation (2.11). Notice thatthe image is contained in L k − ,δ ( Z ; i R ) because we have chosen the lifts U − , U + to be in aCoulomb slice and the family of functions c is in L j . For all 1 ≤ j ≤ k , the linearization ofCoul τγ extends to the operator d τ, † γ : T τj,δ → L j − ,δ ( Z ; i R )given by the map( A + a, s, φ )
7→ − d ∗ a − σ ( t ) c + is Re h iφ , φ i + i | φ | Re ( µ Y ( h iφ , φ i )) The definition of the Coulomb slice we have chosen differs from that in the book (in whichthe term with the function σ is not present) because we are working in the weighted setting.In fact, the operator a
7→ − d ∗ a − σ ( t ) c is the adjoint of the operator − d in the weighted norm, as it can be easily seen by applyingthe trick in equation 2.11.Define for a configuration γ in ˜ C τk ( U − , U + ) the subspaces K τj,δ,γ ⊂ T τj,δ,γ J τj,δ,γ ⊂ T τj,δ,γ . to be respectively the kernel of d τ, † γ and the image of d τγ . One then has the following result,which is the analogue of Proposition 14 . . Proposition . For each δ > the subspaces J τj,δ,γ and K τj,δ,γ are complementary at eachconfiguration γ , so they define a smooth bundle decomposition in closed subbundles T τj,δ = J τj,δ ⊕ K τj,δ . Furthermore, J τj,δ = J τ ,δ ∩ T τj,δ . Proof.
We can restrictict our attention to the finite codimension subbundle T τj,δ, ,γ Weshow that J τj,δ is a closed subspace at each point. Recall that it is defined as the image of theoperator d τγ : L j +1 ,δ ( Z ; i R ) → T τj,δ, ( ξ, γ ) ( − dξ, , ξφ ) . We have the estimate k d τ ξ k L j,δ = k dξ k L j,δ + k ξφ k L j,δ ≥ k dξ k L j,δ + 12 Z R f · (cid:16) k d Y ξ ( t ) k L j ( Y ) + k ξ ( t ) φ ( t ) k L j ( Y ) (cid:17) dt ≥ k dξ k L j,δ + C k ξ k L δ ≥ C ′ k ξ k L j +1 ,δ , which shows that the image is closed. Here we used the alternative definition of the weightednorms (see equation (2.9)).The fact that the subspaces are in direct sum follows if we show that d τ, † γ d τγ : L j +1 ,δ ( Z ; i R ) → L j − ,δ ( Z ; i R )is an isomorphism. This follows because we have chosen the operator d τ, † γ so that its partacting on one-forms is the adjoint of − d on the weighted spaces. In particular this allows toproof injectivity by the usual integration by parts argument. Finally also the surjectivity andthe closedness of the image follow as the proof in the book. (cid:3) Corollary . The moduli space of configuration B τk,δ ([ U − ] , [ U + ]) is a Hilbert manifold,and the evaluation maps ev ± are smooth. . TRANSVERSALITY 51 Proof.
The fact that the quotient is Hausdorff follows as Proposition 13 . . S τk,δ,γ = (Coul τγ ) − (0)is a slice for the gauge group action. In particular, there is a neighborhood U γ ⊂ S τk,δ,γ of γ such that the restriction of the map induced by the quotient map¯ ι : S τk,δ,γ → ˜ B τk,δ ([ U − ] , [ U + ])is a diffeomorphism onto its image, which is a neighborhood of [ γ ] ∈ ˜ B τk,δ ([ U − ] , [ U + ]). Finally,the smoothness of the evaluation maps follows from their smoothness when restricted to alocal slice. (cid:3) We now describe the linearization of the Seiberg-Witten equations when restricted to aslice. One can define the vector bundle V τj,δ → C τk,δ ( U − , U + )whose fiber over γ is the vector space V τj,δ,γ = { ( η, r, ψ ) | Re h φ ( t ) , φ ( t ) i L ( Y ) = 0 for all t }⊂ L j,δ ( Z ; i su ( S + )) ⊕ L j,δ ( R ; R ) ⊕ L j,δ,A ( Z ; S − ) . As in Lemma 14 . . F τ q of the vector bundle V τk − ,δ → C τk,δ ( U − , U + ) . As we have chosen δ > M z ([ U − ] , [ U + ]) arises as the quotient of the locus (cid:8) F τ q = 0 (cid:9) by the action of the gauge group. In order to understand its local structure we need to studythe derivative of F τ q . As V τk,δ is not a trivial vector bundle, the definition of such a derivativeinvolves a projection. In our case, we define the projectionΠ τγ : L j,δ ( Z ; i su ( S + )) ⊕ L j,δ ( R ; R ) ⊕ L j,δ,A → V τj,δ,γ obtained by applying the L projection on each slice { t } × Y , that isΠ τγ ( η, r, ψ ) = ( η, r, Π ⊥ φ ( t ) ψ )where Π ⊥ φ ( t ) ψ = ψ − Re h ˇ φ ( t ) , ψ ( t ) i L ( Y ) φ . Then the derivative D F τ q is defined as the derivative in the ambient Hilbert space followedby the projection Π τγ . Because of condition (iii) in the definition of a tame perturbation(Definition 4.1 in Chapter 1), the derivative extends to smooth bundle maps D F τ q : T τj,δ → V τj − ,δ for any j ∈ [ − k, k ]. The restriction of the operator D F τ q to the finite codimension subspace T τj,δ,γ , can beinterpreted in the following way (we will temporarily ignore the question of regularity). Letˇ γ ( t ) = ( B ( t ) , r ( t ) , φ ( t )) : R → C σ ( Y )be the path defined by γ . The codomain of the derivative can be then interpreted as sectionsalong ˇ γ of the tangent bundle T σ , by identifying an endomorphism η of the spin bundle S and with an imaginary valued one form b on Y via Clifford multiplication. The domain ofthe operator can be similarly interpreted as the space of sections ( V, c ) along ˇ γ of the bundle T σ ⊕ L ( Y ; i R ) → C σ ( Y ) . Indeed a 1-form a on R × Y can be written as b + cdt with b in temporal gauge and c animaginary valued function, and hence interpreted as a pair ( b, c ) consisting of a path b of1-forms on Y and a path c of functions on Y .Also in this case the derivative along the path involves a projection, as the vector bundle T σ ( Y ) is not trivial along the path. We set Ddt V = ( dbdt , drdt , Π ⊥ φ ( t ) dψdt )where as before Π ⊥ φ ( t ) is the orthogonal projection to the orthogonal complement of φ ( t ).With these identifications, one can write the operator D F τ q as( V, c ) Ddt V + D (grad −L ) σ ( V ) + d σγ ( t ) c where D (grad −L ) σ is the derivative of the vector field (grad −L ) σ and d σ is the derivative of thegauge group action on C σ ( Y ), see Equation (1.5) in Chapter 1.We then impose a Coulomb type gauge-fixing condition, namely the linearization of theCoulomb slice condition discussed above d τ, † γ ( V, c ) = 0which can be rephrased in our new language by the condition (cid:18) ddt − σ ( t ) c (cid:19) + d σ, † γ ( t ) ( V ) = 0where d σ, † is the linearized gauge-fixing operator on C σ ( Y ) defined in Section 1. The appro-priate operator to study for our problem is then Q γ = D γ F τ q ⊕ d τ, † γ : T τj,δ,γ , → V τj − ,δ,γ ⊕ L j − ,δ ( Z ; i R )which can be written in path notation if γ is in path notation(2.12) ( V, c ) Ddt ( V, c ) + L γ ( t ) ( V, c ) . Here if γ ( t ) is b ∈ C σ ( Y ) we have that(2.13) L γ ( t ) = (cid:20) D b (grad −L ) σ d σ b d σ, † b − σ ( t ) (cid:21) which is obtained from the extended Hessian we have encountered before in Section 1 (equation(2.3) by adding the term − σ ( t ) : L j ( Y ; i R ) → L j − ( Y ; i R ) . TRANSVERSALITY 53 the lower right entry. We call this operator the weighted extended Hessian at time t , anddenote it by L b ,t . The proof of Lemma 1.7 carries over without modifications to show thefollowing. Lemma . If b ∈ C is a Morse-Bott singularity, then for any t the weighted extendedHessian L b ,t is Fredholm of index , has real spectrum and kernel consisting exactly of T b C . We are finally ready to state the basic Fredholm property for the linearized equations onthe infinite cylinder. Indeed the choice of the weighted Sobolev spaces is made so that thenext result holds.
Proposition . Suppose we are given two critical submanifolds U − , U + in Coulombslice. Then for each γ ∈ C τk,δ ( U − , U + ) the linear operator Q γ : T τj,δ,γ → V τj − ,δ,γ ⊕ L j − ,δ ( Z ; i R ) is Fredholm for every ≤ j ≤ k and satisfies the G˚arding inequality k u k L j,δ ≤ C ( k Q γ u k L j − ,δ + C k u k L j − ,δ ) . The index of Q γ is independent of j and δ > sufficiently small. Here by sufficiently small we mean that δ > | Λ q ( b ) | as b varies among the critical submanifolds (see also Lemma1.6). The latter condition is included to assure invariance of the index, as it will be clear fromthe spectral flow interpretation. Proof.
Here again we restrict to the finite codimension subspace T τj,δ,γ , , for which wecan apply Atiyah-Patodi-Singer techniques. The family { L ( t ) } of weighted extended Hessiansin equation 2.13 which is not necessarily hyperbolic at the limit points. As discussed in theintroduction of the section idea is to study the operators on the weighted Sobolev spaces, anduse the observation of equation (2 .
11) to obtain the operator ddt + L ( t ) + σ ( t )acting on the unweighted spaces. But σ ( t ) = − δ for t >> σ ( t ) = δ for t <<
0, so for δ > { L ( t ) + σ ( t ) } is hyperbolic. Hence, we can apply theAtiyah-Patodi-Singer techniques to this family of operators and the proof of the propositionfollows in the exact same way as in Theorem 14 . . (cid:3) Corollary . The restriction of the bundle map D F τ q D F τ q : K τj,δ,γ → V τj − ,δ,γ is Fredholm and has the same index as Q γ . By Atiyah-Patodi-Singer techniques the index of Q γ on the subspace T τj,δ,γ , , which wedenote by Ind Q γ , is equal to the spectral flow of the family of operators (cid:8) L ˇ γ ( t ) + σ ( t ) (cid:9) , and it should be clear from this description that it depends only on the critical submanifolds C ± . We then introduce the following definition. Definition . Given critical points b − , b + ∈ C σk ( Y ) which have images respectively in[ C − ] and [ C + ], we define gr( b − , b + ) = Ind Q γ + dim[ C + ] . Given [ b − ] , [ b + ] ∈ B σk ( Y ) and a relative homotopy class z ∈ π ( B σk ( Y ) , [ C − ] , [ C + ]) connectingthem we also define gr z ([ b − ] , [ b − ]) = gr( b − , b − )for any pair of lifts b − , b + ∈ C σk ( Y ) such that a path connecting them defines in the quotientthe given homotopy class. We call these quantities the relative gradings between the criticalpoints. In the similar way we define relative gradings between critical submanifolds.The interesting point of this definition is that even though the index of Q γ is not additivein the setting of weighted Sobolev spaces, the relative grading is. Lemma . If a , b and c are three critical points, then gr( a , c ) = gr( a , b ) + gr( b , c ) . Proof.
This is very well understood with the interpretation of the index as spectral flow.Call the submanifolds these critical points belong to [ C a ] , [ C b ] and [ C c ]. Then the indices of Q γ and Q γ ′ for some paths γ, γ ′ connecting a to b and b to c on the right hand side can beinterpreted as the spectral flow of two families of k - asafoe operators L t + σ ( t ) and L ′ t + σ ( t )where L ( t ) and L ′ ( t ) are the weighted extended Hessians along a path connecting the criticalpoints, and lim t → + ∞ L t = lim t →−∞ L ′ t = L. On the other hand, one can concatenate the two paths to obtain a third path γ ′′ with asso-ciated family of operators L ′′ ( t ). Thengr( a , c ) = sf( L ′′ t + σ ( t )) + dim[ C a ]= (cid:0) sf( L t + σ ( t )) + sf( L ′ t + σ ( t )) + dim(ker L ) (cid:1) − dim[ C a ] = gr( a , b ) + gr( b , c ) . The result follows as ker L is identified with T b [ C b ]. (cid:3) The ideas in the proof above can be exploited to prove the following adaptation of Lemma14 . . Lemma . For the closed loop z u based at [ b ] ∈ B σk ( Y ) , we have gr z u ([ b ] , [ b ]) = ([ u ] ∪ c ( S )) [ Y ] where [ u ] denotes the homotopy class of u : Y → S , identified with an element of H ( Y ; Z ) . In general we cannot expect the operator Q γ to be surjective because of the boundaryobstructedness phenomenon that arises already in the finite dimensional case (see Section 2 . . TRANSVERSALITY 55 Definition . A reducible configuration a ∈ C σk ( Y ) (not necessarily a critical point) is boundary stable if Λ q ( a ) < boundary unstable if Λ q ( a ) >
0. We say that a pair of criticalsubmanifolds ([ C − ] , [ C + ]) is boundary obstructed if the points in [ C − ] are boundary-stable andthe points in [ C − ] are boundary-unstable, and so we call the moduli space of trajectories M ([ C − ] , [ C + ]).To clarify the second part of the definition, notice that because of the Lemma 1.6 if areducible critical point is boundary stable (unstable), then all the points in the critical man-ifold it belongs to are stable (unstable). Also, if a moduli space M ([ C − ] , [ C + ]) contains anirreducible trajectory, then [ C − ] consists of irreducible or boundary unstable points and [ C + ]consists of irreducible or boundary stable points (see Lemma 14 . . γ is a reducible trajectory. Then the operator Q γ decomposes as the sumof two operators Q ∂γ and Q νγ , reflecting the decomposition of the involution i : ˜ C τk,δ ( U − , U + ) → ˜ C τk,δ ( U − , U + )( A, s, φ ) ( A, − s, φ ) . The first operator is Q ∂γ = ( D γ F τ q ) ∂ ⊕ d τ, † γ where ( D γ F τ q ) ∂ : ( T τk,δ,γ ) ∂ → ( V τk,δ,γ ) ∂ is the part invariant under the involution i , while the second operator is Q νγ : L k,δ ( R ; i R ) → L k − ,δ ( R ; i R ) s dsdt + Λ q (ˇ γ ) s. The following elementary result (which is Lemma 14 . . Q νγ . Lemma . The dimensions of the kernel and the cokernel of Q νγ are: • and if a and b are boundary stable and unstable respectively; • and if a and b are boundary unstable and stable respectively; • and in the other cases. We introduce a first notion of transversality, which corresponds in the finite dimensionalcase to the condition that stable and unstable manifolds intersect transversely.
Definition . Let γ be a solution in M z ([ C − ] , [ C + ]). If the pair ([ C − ] , [ C + ]) is notboundary obstructed, we say that γ is Smale-regular if Q γ is surjective. In the boundaryobstructed case, γ must be reducible and we say that it is regular if Q ∂γ is surjective. We saythat M z ([ C − ] , [ C + ]) is Smale-regular if it is
Smale-regular at any point.The point of the definition is that Smale-regular moduli spaces are transversely cut outsmooth manifolds by the inverse function theorem, as stated in the following proposition (seeProposition 14 . . Proposition . Consider critical manifold [ C ± ] such that M z ([ C − ] , [ C + ]) is Smale-regular, and set d = gr z ([ C − ] , [ C + ]) + dim[ C − ] . Then the moduli space M z ([ C − ] , [ C + ]) is: • a smooth d -manifold consisting entirely of irreducible solutions if either [ C − ] or [ C + ] isirreducible; • a smooth d -manifold with boundary [ C − ] and [ C + ] are boundary-unstable and stablerespectively; • a smooth d -manifold consisting entirely of reducibles if [ C − ] and [ C + ] are either bothboundary stable or unstable; • a smooth d + 1 -manifold consisting entirely of reducibles in the boundary obstructedcase.In the second case, the boundary of the moduli space consists of the reducible elements. Remark . It is important to notice that the zero locus of the section F τ q modulogauge action is not the moduli space we are interested in, but a bigger space ˜ M z ([ C − ] , [ C + ]).The actual space is the quotient of this space by the involution i : ˜ M z ([ C − ] , [ C + ]) → ˜ M z ([ C − ] , [ C + ])[ A, s, φ ] [ A, − s, φ ] , see also equation (1.10). Here we use the fact that for a solution the function s is alwayspositive, always negative or constantly zero.In what follows, we will need a slightly stronger notion of transversality. When M z ([ C − ] , [ C + ])is Smale-regular, the evaluation mapsev ± : M z ([ C − ] , [ C + ]) → [ C ± ]are smooth maps. Given a sequence of critical submanifolds C = ([ C i ]) i =0 ,...,n and relative homotopy classes z = ( z i ) i =0 ,...,n − with z i ∈ π ( B σk ( Y ) , [ C i ] , [ C i +1 ]) we can define the space M z ( C ) consisting of n -uples [ γ i ] ∈ M z i ([ C i ] , [ C i +1 ]) such that ev + [ γ i ] = ev − [ γ i +1 ]for every 0 ≤ i ≤ n −
2. This space comes with natural continuous evaluation maps ev ± tothe critical submanifolds [ C ] and [ C n ] respectively. Definition . Suppose we are given a Morse-Bott tame perturbation q such that allthe moduli spaces are Smale-regular as in Definition 3.11. We then say that q is regular ifthe following holds. For every sequence of critical submanifolds C = ([ C i ]) i =0 ,...,n and corre-sponding relative homotopy classes z , and any other critical submanifold [ C + ] and homotopyclass z + ∈ π ( B σk ( Y ) , [ C n ] , [ C + ]), the mapsev + : M z ( C ) → [ C n ]ev − : M z + ([ C n ] , [ C + ]) → [ C n ]are transverse smooth maps. . TRANSVERSALITY 57 Notice that unless M z ( C ) has a natural smooth structure, the condition of being transverseis not defined. The definition needs to be interpreted in the following inductive way. Supposethat M z ( C ) has a canonical smooth structure for which the evaluation maps are smooth (thisis true in the case the sequence has length one by the Smale-regularity assumption). Thenthe space M ( z ,z + ) ([ C ] , . . . , [ C n ] , [ C + ])has a natural smooth structure as the fibered product of the two evaluation maps, as thesetwo are smooth transverse map by the regularity assumption. Furthermore the evaluationmaps to [ C ] and [ C − ] are smooth. Remark . Some approaches (see for example [
AB95 ]) impose stronger transversalitycondition on the moduli spaces, as the fact that the evaluation maps are submersions. Onthe other hand, it is not hard to construct examples in our problem such that this propertydoes not hold (even after small prturbation).Before proving the main transversality result we characterize the moduli spaces be-tween two reducible critical submanifolds ( C , C ) lying over the same reducible configuration( B, ∈ C ⊂ C k ( Y ), corresponding to eigenvalues λ and λ of D q ,B . Let i = ( |{ µ ∈ Spec( D q ,B ) | λ ≤ µ < λ }| , if λ ≥ λ , −|{ µ ∈ Spec( D q ,B ) | λ > µ ≥ λ }| , if λ ≤ λ . Of course we are counting eigenvalues with multiplicity. Notice that this definition is slightlydifferent from the one given in Section 14 . z joining [ C ] and [ C ] in B σk ( Y ) arising from a pathconnecting C and C in C σk ( Y ). The is a natural map(2.14) π : M z ([ C ][ C ]) → [ C ]sending a trajectory to the reducible critical point it lies over. We then have the followingresult, see Section 14 . Lemma . We have the following: gr( C , C ) = i, if λ and λ have the same sign, i − , if λ is positive and λ is negative, i + 1 if λ is negative and λ is positive.In the last case the relative grading is a negative integer. The moduli spaces are empty whenthe relative grading is negative. When the relative grading is a non negative number N , themap in equation (2.14) is a fibration and each fiber is diffeomorphic to the complement inthe projective space C P N +dim[ C ] of two hyperspaces of codimension respectively dim[ C ] and dim[ C ] . Furthermore both evaluation maps from M z ([ C ][ C ]) are fibrations. Proof.
This is proved as in Proposition 14 . . B, C ∗ of the space of solutions of thetranslation invariant Dirac equation ddt Φ( t ) = − D q ,B Φ( t ) with asymptotics Φ( t ) ∼ c e − λ t as t → + ∞ Φ( t ) ∼ c e − λ t as t → −∞ . In particular, we can write such a solution asΦ = X µ c µ e − µt φ µ where µ is an eigenvalue in the interval [ λ , λ ], with the coefficients c λ and c λ both nonzero. In particular the evaluation maps for these spaces are submersions, and the resultfollows because by definition also the blow down map from [ C ] and [ C ] is a fibration. (cid:3) We are now ready to discuss the main transversality result of the section. Recall we haveintroduced in Definition 1.8 the notion of adapted perturbation.
Theorem . For a fixed Morse-Bott perturbation q and for δ > small enoughthere is a residual subset of the space of adapted perturbations P O which consists of regularperturbations. Proof.
The proof, which is analogous to the one of Theorem 15 . . . . U − ] and [ U + ], with smooth lifts inCoulomb gauge U − and U + . Define the smooth map of Banach manifolds N : C τk,δ ( U − , U + ) × P O → V τk − ,δ ( Z )( γ, q ) F τ q ( γ ) . The derivative of N at a point ( γ, q ) is a map D ( γ, q ) N : T τk,δ,γ × T q P O → V τk − ,δ,γ ( Z ) . We first focus in the case in which γ is an irreducible configuration, and need to show that N is surjective. After we use the same trick already employed in the proof of Proposition 3.4in order to reduce to a problem on the unweighted spaces, the proof of Proposition 15 . . D ( γ, q ) N ) : T τk,δ, ,γ × T q P O → V τk − ,δ,γ ( Z )is surjective. Using then the familiar strategy, we can define the universal moduli space M z ([ U − ] , [ U + ]) = N − (0 , s ) / G k +1 ,δ ⊂ B τk,δ,z ([ U − ] , [ U + ]) × P O , which is a smooth Banach manifold and has the property that the projection to P O is Fred-holm. Hence the Sard-Smale theorem provides a residual subset of regular values in P O , andwe restrict our attention to the subset of adapted ones.The case of γ reducible not projecting to a reducible critical point in the blow down isanalogous to the classical case, as in this case the derivative of N has the summand( D ( γ, q ) N ) ∂ : ( T τk,δ,γ ) ∂ × T q P O → ( V τk − ,δ,γ ( Z )) ∂ , . COMPACTNESS AND FINITENESS 59 and the result follows in an identical way using the surjectivity of the restriction( D ( γ, q ) N ) ∂ : ( T τk,δ, ,γ ) ∂ × T q P O → ( V τk − ,δ,γ ( Z )) ∂ , whose proof is also contained in the proof of Proposition 15 . . γ lies over a single reducible ( B, ψ
7→ − ddt ψ + D q ,B ψ + σ ( t ) ψ. Notice that this is not the adjoint in the equivalent norm in equation (2.9). The operator ψ
7→ − ddt ψ + D q ,B ψ is an isomomorphism on the unweighted spaces as zero is not in the spectrum of D q ,B (seeLemma 1.6). Hence for δ > f defin-ing the weighted Sobolev norm) the operator in equation (3) will also be an isomorphism,and indeed we can find a δ > C − ] , [ C ] , [ C + ] at a point in which both trajectories ([ γ − ] , [ γ + ]) areirreducible. The regularity property at this point is equivalent to the fact that the map N ′ : C τk,δ ( U − , U ) × C τk,δ ( U , U + ) × P O → V τk − ,δ ( Z ) ⊕ V τk − ,δ ( Z ) × U × U ( γ − , γ + , q ) (cid:0) F τ q ( γ − ) , F τ q ( γ + ) , ev + ( γ − ) , ev − ( γ + ) (cid:1) . is transverse to ( { } ⊕ { } ) × ∆, where ∆ ⊂ U × U is the diagonal. This will follow if we can prove that the restrictions of the linearizations D ( γ − ,γ + , q ) N ′ : T τk,δ, ,γ − × T τk,δ, ,γ + × T q P O → V τk − ,δ,γ − ( Z ) × V τk − ,δ,γ + ( Z ) × { } × { } are surjective. Even though this does not follow from the surjectivity of the maps ( D ( γ, q ) N ) above, the proof of Proposition 15 . . C ] and [ C ]are critical submanifolds lying over the same reducible [ C ] the evaluation maps on the modulispaces lying over constant trajectories are submersions because of Lemma 3.16. (cid:3)
4. Compactness and finiteness
In this section, which closely follows Chapter 16 in the book, we discuss the compactessproperties for moduli spaces of trajectories, and construct the space of unparametrized brokentrajectories.
We say that a trajectory in the moduli space M z ([ C − ] , [ C + ]) is non trivial if it is notinvariant under the action of R by translations on the infinite cylinder. This is equivalent tosay that either it has distinct endpoints or if they coincide then the relative homotopy class z is non trivial. Definition . An unparametrized trajectory connecting [ C − ] to [ C + ] is an equivalenceclass of non trivial trajectories in M z ([ C − ] , [ C + ]) under the action of translations. We write˘ M z ([ C − ] , [ C + ])for the space of unparametrized trajectories. Definition . An unparametrized broken trajectory joining two critical submanifolds[ C − ] to [ C + ] consists of the following data: • an integer n ≥
0, the number of components ; • an ( n + 1)-tuple of critical submanifolds [ C ] , . . . , [ C n ] with [ C ] = [ C − ] and [ C n ] = [ C + ],the resting submanifolds ; • for each 1 ≤ i ≤ n , an unparametrized trajectory[˘ γ i ] ∈ ˘ M z i ([ C i − ] , [ C i ]) , the i-th component of the broken trajectory, with the property thatev + [˘ γ i ] = ev − [˘ γ i +1 ]and we call this critical point the i th restpoint .The homotopy class of the broken trajectory is the relative homotopy class of the path ob-tained by concatenating representatives of the classes z i . We write ˘ M + z ([ C − ] , [ C + ]) for thespace of unparametrized trajectories in the homotopy class z , and write the typical elementas [ ˘ γ ] = ([˘ γ ] , . . . , [˘ γ n ]) . Finally, there are naturally defined evaluation mapsev ± : ˘ M + z ([ C − ] , [ C + ]) → [ C ± ] . Remark . We consider also broken trajectories with n = 0 components for bookkepingpurposes. If z is the class of the constant path for the submanifold [ C ], then ˘ M + z ([ C ] , [ C ])consists of a single point, the broken trajectory with no components.The space of unparametrized broken trajectories is topologized as follows. Consider anelement [ ˘ γ ] = ([˘ γ ] , . . . , [˘ γ n ]) ∈ ˘ M + z ([ C − ] , [ C + ]) , with [˘ γ i ] ∈ ˘ M z i ([ C i − ] , [ C i ]) being represented by a parametrized trajectory[ γ i ] ∈ M z i ([ C i − ] , [ C i ]) . Let U i ⊂ B τk, loc ( Z ) an open neighborhood of [ γ i ], and let T ∈ R + . We defineΩ = Ω( U , . . . U n , T )to be the subset of ˘ M + z ([ C − ] , [ C + ]) consisting of broken unparametrized trajectories[ ˘ δ ] = ([˘ δ ] , . . . , [˘ δ m ]) . COMPACTNESS AND FINITENESS 61 satisfying the following condition. There exists a map( , s ) : { , . . . n } → { , . . . , m } × R such that • [ τ ∗ s ( i ) δ ( i ) ] ∈ U i ; • if 1 ≤ i ≤ i ≤ n , then either ( i ) ≤ ( i ), or ( i ) = ( i ) and s ( i ) + T ≤ s ( i ).We take the sets of the form Ω = Ω( U , . . . U n , T ) to be a neighborhood base for [˘ γ ] in˘ M + z ([ C − ] , [ C + ]).The first goal of the section is to prove the following result (see Proposition 16 . . Theorem . For any
C > and critical submanifolds [ C ± ] , there are only finitelymany z with energy E q ( z ) ≤ C for which ˘ M + z ([ C − ] , [ C + ]) is non-empty. Furthermore eachspace ˘ M + z ([ C − ] , [ C + ]) is compact. The proof of this proposition is essentially identical to the one in the Morse case (Propo-sition 16 . . downstairs , i.e. for blown down trajectories. This willfollow from the compactness properties for the Seiberg-Witten equations on a finite cylinder(see Chapter 5 in the book). We will then deduce the case we are actually interested in.Suppose our critical submanifolds [ C ± ] blow down to critical submanifolds [ C ± ] in B k ( Y ),we introduce moduli spaces N z ([ C − ] , [ C + ]) ⊂ B k, loc ( Z ) N ([ C − ] , [ C + ]) = [ z N z ([ C − ] , [ C + ])of solutions to the perturbed equations asymptotic to [ C − ] and [ C + ] at ±∞ . This comes witha blow down map π : M z ([ C − ] , [ C + ]) → N z ([ C − ] , [ C + ]) . We define a trajectory in N z ([ C − ] , [ C + ]) to be non-trivial if it is not invariant under transla-tion. We can introduce for these moduli spaces the analogues of the unparametrized trajec-tories ˘ N z ([ C − ] , [ C + ]) and broken trajectories ˘ N + z ([ C − ] , [ C + ]), and the disjoint union˘ N + = [ [ C − ] , [ C + ] ˘ N + ([ C − ] , [ C + ]) . For this space, we have the following compactness result, which differs from Proposition4.4 as it deals with the union over all critical submanifolds, see Proposition 16 . . Proposition . For any
C > there are only finitely many [ C − ] , [ C + ] and z such that E q ( z ) ≤ C and the space ˘ N + z ([ C − ] , [ C + ]) is non empty. Furthermore, each ˘ N + z ([ C − ] , [ C + ]) is compact. In other words, the space of broken trajectories in ˘ N + with energy E q ≤ C iscompact. The key ingredient in the proof is the following basic lemma. As this is the only point inwhich the proof of Proposition 4.5 differs from the one of Proposition 16 . . Lemma . Let [ γ ] ∈ B k, loc be a solution of the equations with finite energy. Then [ γ ] ∈ N ([ C − ] , [ C + ]) for some critical submanifolds [ C − ] , [ C + ] . Before proving this lemma, we recall a basic lemma following from the compactness prop-erties of the moduli space on a finite cylinder (see Lemma 16 . . . . Lemma . Fix a collection A of critical points [ α ] in B k ( Y ) and for each of them agauge invariant open neighborhood U α ⊂ C k ( I × Y ) of the translation invariant configuration γ α such that their union contains all translation invariant solutions. Let C be any constant,and I ′ ⊂ I any other interval of non zero length. Then there exists ǫ > such that if γ is atrajectory satisfying E q ( γ ) ≤ C and E I ′ q ( γ ) ≤ ǫ, then γ | I × Y ∈ U α for some critical point [ α ] in A . Here by E I ′ q ( γ ) we mean the (perturbed) energy of the trajectory when restricted to theinterval I ′ . As the critical submanifolds are compact, we can suppose that the family A contains only a finite number of manifolds from each of them. Definition . Fix an interval I . We say that a collection U of gauge invariant neighbor-hoods U α ⊂ C k ( I × Y ) with α ∈ A as in the previous lemma has the separating property if thefollowing holds. There should exist neighborhoods V [ α ] ⊂ B k − ( Y ) of the critical submanifoldssuch that • V [ α ] and V [ α ′ ] are disjoint if α and α ′ do not belong to the same critical submanifold; • each V [ α ] is path-connected and simply connected; • if γ ∈ U α , then [˘ γ ( t )] ∈ V [ α ] for every t ∈ I .For a critical submanifold [ C ] we define V [ C ] to be neighborhood of [ C ] obtained as the unionof the V [ α ] with [ α ] in [ C ]. Proof of lemma 4.6.
Fix an interval I and a collection of neighborhoods U with theseparating property. The finite energy condition implies that the translates τ ∗ t ( γ ) are suchthat E I q ( τ ∗ t γ ) → t → + ∞ so from Lemma 4.7 above the translate ( τ ∗ t γ ) | I belongs to U α t for some critical point [ α t ] in A . Because of the separating property we have that [ˇ γ ( t )] ∈ V [ C ] for some critical submanifold[ C ] for all t ≥ t . By choosing big intervals I and small neighborhoods V [ C ] this shows thatthe function −L (ˇ γ ( t )) converges for t → + ∞ . We then need to prove that the solution hasexactly one limit point on such a critical submanifold, i.e. it does not “spiral around”. Thisphenomenon might happen already in finite dimensions when considering the gradient flow ofa smooth function, but does not happen for analytic functions (see [ MMR94 ] and [
Don02 ]).We define the L metric on B k − ( Y ) given by d ([ α ] , [ α ′ ]) = inf (cid:8) k α − u · α ′ k L ( Y ) | u ∈ G k ( Y ) (cid:9) . To check that this is a metric we need to show that two configurations α, α ′ ∈ C k − ( Y ) suchthat there is a sequence { u n } n ∈ N ⊂ G k ( Y ) with u n · α ′ converging to α in the L norm are . COMPACTNESS AND FINITENESS 63 actually gauge equivalent. This is proved in a similar way as Proposition 9 . . u n is in the identity component of the gauge group, so it canbe written as e ξ n + ξ ⊥ n . Calling B, B ′ the connection component of the configurations, we havethat dξ ⊥ n − ( B − B ′ ) → L . In particular the ξ ⊥ n form a Cauchy sequence in L , so they converge in the L topologyto a configuration ξ ⊥ which is in L k by elliptic regularity. This implies that the gaugetransformations u n converge in the L topology to a gauge transformation u ∈ G k . On theother hand, u n · α ′ converges in the L norm to u · α ′ (for the spinor part we use that indimension three L embeds in L hence in L ), so the latter coincides with α .The result will follow by taking the initial interval I as large as we wish and the separatingneighborhood as small as we want if we show that the path [ˇ γ ( t )] has finite length in thismetric. Let I = [ s − , s + 1], and suppose that γ | I × Y ∈ U α for some α in C . We will supposethe configuration is in Coulomb-Neumann gauge with respect to γ α . After possibly restrictingboth family neighborhoods, we can carry over the estimates of Lemma 2.13 to the function −L (ˇ γ ( s )) − −L (ˇ γ ([ C ]). Furthermore, because there are only finitely many neighborhoods U α involved, we can choose a constant C so that this in equality holds for all of them. As theChern-Simons-Dirac operator −L converges along the trajectory to −L ([ C ]), this implies thatits value along the trajectory converges exponentially fast. By applying the Cauchy-Schwarzinequality we have that Z s +1 s − k grad −L (ˇ γ ( t )) k L ( Y ) dt ≤ C (cid:18)Z s +1 s − k grad −L (ˇ γ ( t )) k L ( Y ) dt (cid:19) / == C ( −L (ˇ γ ( s − − −L (ˇ γ ( s + 1))) / . The rightmost term is a well defined real number for s big enough. The exponential con-vergence of −L along the trajectory implies that the integral of k grad −L (ˇ γ ( t )) k L ( Y ) is finite.The length of path [ˇ γ ( t )] in the L metric is bounded above by such integral, as by theflow equations k grad −L (ˇ γ ( t )) k L ( Y ) is exactly the L norm of its derivative, and the resultfollows. (cid:3) Proof of Theorem 4.4.
This follows as the one in the book with modifications (tobe made in the reducible case) analogous to those we discussed above for Lemma 4.6. Theadditional complication comes from the function Λ q , which can be dealt with as in the endof Section 2. It is useful to remark that this function has limits at both ±∞ on a giventrajectory. This is because its value at a configuration [ b ] on a reducible critical submanifolddepends only on its blowdown π ∗ ([ b ]). (cid:3) To get rid of the energy bound in the assumption of Proposition 4.4, we need someregularity assumptions on the moduli spaces, which assure some strong finiteness results onthe set of non empty moduli spaces as in the following lemma (see Proposition 16 . . . . Proposition . Suppose that a regular Morse-Bott perturbation q has been fixed. Thenfor given critical submanifolds [ C − ] and [ C + ] there are only finitely many relative homotopyclasses z for which the moduli space ˘ M + z ([ C − ] , [ C + ]) is non-empty. Furthermore: • if c ( s ) is torsion then for a given [ C − ] and any d there are only finitely many pairs ([ C + ] , z ) for which ˘ M + z ([ C − ] , [ C + ]) is non-empty and of dimension at most d . • if c ( s ) is not torsion, suppose that the perturbation has been chosen so that there areno reducible solutions (see Section . in the book). Then there are only finitely manytriples ([ C − ] , [ C + ] , z ) such that the moduli space ˘ M + z ([ C − ] , [ C + ]) is non-empty (withoutrestrictions on the dimension). Before giving a proof of Proposition 4.9 we recall a useful definition from the book.Suppose that a reducible critical point a lies over the configuration ( B, ∈ C k ( Y ), andcorresponds to the element λ ∈ Spec( D q ,B ). In this case, we define ι ( a ) = ( | (Spec( D q ,B ) ∩ [0 , λ ) | , if λ > , / − | (Spec( D q ,B ) ∩ [0 , λ ) | , if λ < , where of course eigenvalues are counted with multiplicity. If a is irreducible, we set ι ( a ) = 0.This definition is set up so that if [ a ] and [ a ′ ] are two critical points whose blow down is thesame critical point [ α ] ∈ B k ( Y ), then by Lemma 3.16gr z ([ a ] , [ a ′ ]) = 2( ι ( a ) − ι ( a ′ ))for the trivial homotopy class z . Furthermore the value of ι is constant on a critical sub-manifold [ C ], hence we can univocally define the value value ι [ C ]. Proof of Proposition 4.9.
Suppose there is [ ˘ γ ] ∈ ˘ M z ([ C − ] , [ C + ]), and suppose thatthe resting manifolds are [ C − ] = [ C ] , [ C ] , . . . , [ C n − ] , [ C n ] = [ C + ] . The fact that the moduli space M z ([ C ] , [ C ]) is Smale-regular implies thatdim[ C ] + gr z ([ C ] , [ C ]) ≥ , where the inequality is strict in the not boundary obstructed case. The regularity conditionimplies that the evaluation maps ev + : M z ([ C ] , [ C ]) → [ C ]ev − : M z ([ C ] , [ C ]) → [ C ]are transverse, and as they have non disjoint image we have that (cid:0) dim[ C ] + gr z ([ C ] , [ C ]) (cid:1) + (cid:0) dim[ C ] + gr z ([ C ] , [ C ]) (cid:1) ≥ dim[ C ] . By induction on the number of components and using the additivity of the relative grading(Lemma 3.7) we then prove gr z ([ C − ] , [ C + ]) ≥ − dim[ C − ] . The proof of Lemma 16 . . C such that for every[ C − ] , [ C + ] and z , and any broken trajectory [˘ γ ] ∈ ˘ M + z ([ C − ] , [ C + ]), we have the energy bound E q (˘ γ ) ≤ C + 8 π ( ι [ C − ] − ι [ C + ]) . (cid:3) . COMPACTNESS AND FINITENESS 65 Remark . Unlike the classical case, when the spin c structure is not torsion the modulispaces of the form M z ([ C ] , [ C ]) might be not empty.The rest of the present section (and chapter) is dedicated to understand in detail thestructure of the moduli spaces of unparametrized broken trajectories. We first introduce akey definition from the book. Definition . A topological space N d is a d -dimensional space stratified by manifolds if there are closed subsets N d ⊃ N d − ⊃ · · · ⊃ N ⊃ N − = ∅ such that N d = N d − and each space N e \ N e − (for 0 ≤ e ≤ d ) is either empty or homeo-morphic to a manifold of dimension e . We call N e \ N e − the e -dimensional stratum . We willalso call stratum any union of path components of N e \ N e − . Example . Spaces stratified by manifolds (even compact ones) allow some patholo-gies. Consider the space N obtained as the union over all n ∈ N of all circles C n withcenter in ( − /n,
0) and radius 1 /n and the segment joining (0 ,
0) and (1 , N = { (0 , , (1 , } , and N \ N is a 1-manifold with countably many path components.Consider a sequence of critical manifolds C = ([ C i ]) i =0 ,...,n and corresponding relativehomotopy classes z = ( z i ) i =1 ,...,n . We can then define the subspace˘ M z ( C )consisting of unparametrized broken trajectories such that the i th restpoint lies on the criticalmanifold [ C i ], and the i th component is in the relative homotopy class z i . When the pertur-bation is regular, this subspace has a natural manifold structure as the quotient of open setthe smooth fibered product M z ( C ) introduced in Section 4 consisting of n -uples such thateach component is non trivial by the action of R n given time translations on each component.The following result is then the analogue of Proposition 16 . . . . . in the book, andthe proofs applies verbatim. Proposition . Suppose we have fixed a regular perturbation q , and let M z ([ C − ] , [ C + ]) a d -dimensional moduli space containing irreducibles. Then the space of broken unparametrizedtrajectories ˘ M + z ([ C − ] , [ C + ]) is a compact ( d − -dimensional space stratified by manifolds, andthe top stratum consists of the irreducible part of ˘ M z ([ C − ] , [ C + ]) . Furthermore, the ( d − -dimensional stratum of ˘ M + z ([ C − ] , [ C + ]) is the union of pieces of three types: • strata of the form ˘ M ( z ,z ) ([ C − ] , [ C ] , [ C + ]) where none of the pairs of consecutive critical manifolds is boundary obstructed; • strata of the form ˘ M ( z ,z ,z ) ([ C − ] , [ C ′ ] , [ C ′′ ] , [ C + ]) where only the middle moduli space is boundary obstructed; • the intersection of ˘ M z ([ C − ] , [ C + ]) with the reducibles, in the case M z ([ C − ] , [ C + ]) con-tains both reducibles and irreducibles.The last case happens only when [ C − ] is unstable and [ C + ] is stable. Finally, we discuss the case of reducible trajectories. We will write M red z ([ C − ] , [ C + ]) for thesubset of M z ([ C − ] , [ C + ]) consisting of the reducible trajectories. Because of our transversalityhypothesis, this is either empty, all of M z ([ C − ] , [ C + ]), or the boundary of M z ([ C − ] , [ C + ]) inthe case [ C − ] is boundary unstable and [ C + ] is boundary stable. In any case we can introducea modified relative grading given by¯gr z ([ C − ] , [ C + ]) = gr z ([ C − ] , [ C + ]) − o [ C − ] + o [ C + ]where we define o [ C ] = ( , if [ C ] is boundary stable,1 , if [ C ] is boundary unstable.We can also introduce the spaces ˘ M red and ˘ M red+ , as the intersections of ˘ M and ˘ M + withthe reducibles. The situation for these moduli spaces is simpler, as we are essentially doingMorse theory on a closed manifold and there are no boundary obstructedness issues. Forexample M red z ([ C − ] , [ C + ]) is always a smooth manifold without boundary, and its dimensionis given by ¯gr z ([ C − ] , [ C + ]) + dim[ C − ] − . One also has the following result, the counterpart of Proposition 16 . . Proposition . Suppose M red z ([ C − ] , [ C + ]) is non empty and of dimension d . Then thespace of unparametrized reducible trajectories ˘ M red+ z ([ a ] , [ b ]) is a compact ( d − -dimensionalspace stratified by manifolds. The top stratum consists of ˘ M red z ([ C − ] , [ C + ]) alone, and the ( d − -dimensional stratum consists of the space of unparametrized broken trajectories withexactly two components.
5. Gluing
In this section we discuss a gluing result describing the structure of the space of un-parametrized broken trajectories along a stratum. One would like these spaces to look liketopological manifolds with boundary and corners. On the other hand this is in general falsefor the moduli space we are dealing with, and we will show that they have in general a slightlymore complicated type of structure. Our characterization will be enough for the applications,and in particular to define Floer homology in the next chapter.This section consists of two parts. In the first part we state the gluing result, introducingthe notion of thickened moduli space as in Chapter 19 in the book, while in the second one wediscuss the existence of stable ad unstable manifolds as in Chapter 18. As the second part isthe only part of the proof that requires some adaptations, we will describe it in quite detail.We start by introducing a useful definition.
Definition . Consider a pointed topological space (
Q, q ), let π : S → Q be a contin-uous map and consider S ⊂ π − ( q ). We say that π is a topological submersion along S if . GLUING 67 for every s ∈ S we can find a neighborhood U ⊂ S and a neighborhood Q ′ ⊂ Q of q witha homeomorphism ( U ∩ S ) × Q ′ → U commuting with π . Example . Suppose Q is (0 , ∞ ] n − and q = ( ∞ , . . . , ∞ ) . If we have a topological submersion π : S → Q along π − ( ∞ ) then total space S is locally homeomorphic to the product π − ( ∞ ) × (0 , ∞ ] n − ,i.e. it locally looks like the product with a topological manifold with corners.The following is the main gluing theorem, the counterpart of Theorem 19 . . Theorem . Consider a sequence of critical submanifolds C = ([ C i ]) i =0 ,...,n and corre-sponding relative homotopy classes z . Suppose that the moduli space ˘ M z ([ C ] , [ C n ]) containsirreducibles, and define O ⊂ { , . . . , n } as the set of indices i such thet the pair ([ C i − ] , [ C i ]) is boundary obstructed. Then there arean open neighborhood ˘ W ⊃ ˘ M z ( C ) inside ˘ M + z ([ C ] , [ C n ]) and a continuous map S : ˘ W → (0 , ∞ ] n − with the following properties. (1) There is a topological embedding j of ˘ W into a space E ˘ W equipped with a map alsodenoted by S to (0 , ∞ ] n − such that S ◦ j = S . (2) The map S : E ˘ W → (0 , ∞ ] n − is a topological submersion along the fiber over ∞ . (3) The image of j is the zero set of a continuous map δ : E ˘ W → R O vanishing on the fiber over ∞ . Hence the fiber over ∞ in both ˘ W and E ˘ W is identifiedwith the stratum ˘ M z ( C ) . (4) If ˘ W o ⊂ ˘ W and E ˘ W o ⊂ E ˘ W are the subset where none of the components of S isinfinite, then the restriction of j to ˘ W o is an embedding of smooth manifolds, and therestriction of δ to E ˘ W o is transverse to zero. (5) Let i ∈ O and δ i be the corresponding component of δ . Then for all z ∈ E ˘ W wehave: • if i ≥ and S i − ( z ) = ∞ then δ i ( z ) ≥ • if i ≤ n − and S i ( z ) = ∞ then δ i ( z ) ≤ . Definition . In the same setting as the theorem above, we call E ˘ W a thickening ofthe moduli space ˘ M + z ([ C ] , [ C n ]) along the stratum ˘ M z ( C ). In the simplest case in which none of the intermediate pairs is boundary obstructed, thethickening coincides with the open neighborhood ˇ W , so the theorem tells us that along thestratum the moduli space looks like the product of the stratum and a topological manifold withcorners (see Example 5.2). On the other hand, in the presence of boundary obstructednessthe local structure becomes more complicated, and a good image to have in mind is that ofExample 4.12. The simplest (but central) example is the case in which the stratum consistsof broken trajectories with three components, the middle one being boundary obstructed. Inthis case we have the following sharper statement (see Corollary 19 . . Lemma . In the setting of Theorem 5.3, suppose n = 3 and O = { } . Then δ is strictlypositive if S = ∞ and S is finite, and δ is strictly negative if S is finite and S is infinite. This is an important point in the definition of codimension- δ -structures , see Definition19 . . extended moduli space (seeSection 19 . M ([ C − ] , [ C + ])can be seen as the fibered product of the two Hilbert manifolds˜ M ( R ≤ × Y, [ C − ])˜ M ( R ≥ × Y, [ C + ]) , consisting of solutions on the negative (resp. positive) half cylinder which are asymptoticto a point in [ C − ] (resp. [ C + ]), where the maps are the restrictions R ± to the boundary˜ B σ ( { } × Y ). We have the projection to the boundary π ∂ : ˜ B σ ( Y ) → ∂ B σ ( Y )[ B, s, φ ] [ B, , φ ]and we define the extended moduli space E ˜ M ([ C − ] , [ C + ])as the fibered product of the composite maps π ∂ ◦ R ± . One should think of an element [ γ ] in E ˜ M ([ a ] , [ b ]) as a trajectory defined on the whole line but having a discontinuity δ = s ([ γ + ] | { }× Y ) − s ([ γ − ] | { }× Y )in the s coordinate across { } × Y . There is no action by translation on such an element [ γ ],it still has a well defined relative homotopy class z , and we can partition the moduli spaceaccordingly. The usual moduli space arises as the fiber over zero of the map˜ M ([ C − ] , [ C + ]) ֒ → E ˜ M ([ C − ] , [ C + ]) δ −→ R . If ([ C − ] , [ C + ]) is boundary obstructed then δ is identically 0, hence the extended modulispace coincides with the usual one. In the boundary unobstructed case if ˜ M ([ C − ] , [ C + ])is regular, then E ˜ M ([ C − ] , [ C + ]) is regular in a neighborhood of ˜ M ([ C − ] , [ C + ]) and containsit as a smooth codimension 1 submanifold. In both cases, E ˜ M z ([ C − ] , [ C + ]) has dimensiongr z ([ C − ] , [ C + ]) + dim[ C − ] + 1. . GLUING 69 The thickening E ˘ W of a neighborhood ˘ W is then the space in which we allow the com-ponents to belong to the respective extended moduli space. From the discussion above, itfollows that ˘ W sits inside it as the zero locus of all the maps δ , and its codimension is exactlythe number of consecutive pairs which are boundary obstructed.An important feature is that the extended moduli space still is still equipped with con-tinuous evaluations maps ev ± : E ˜ M z ([ C − ] , [ C + ]) → [ C ± ] , which are smooth in a neighborhood of ˜ M z ([ C − ] , [ C + ]). The following is the main additionalresult we will need. Proposition . Referring to the statement of Theorem 5.3, the evaluation maps ev ± on ˘ W extend to continuous evaluation maps ev ± : E ˘ W → [ C ± ] which are smooth in a neighborhood of ˘ W o ⊂ E ˘ W o . Furthermore, suppose we are giventhree critical submanifolds [ C − ] , [ C ] and [ C + ] , and sequences of critical submanifolds andrelative homotopy classes between them ( C , z ) and ( C , z ) . For m = 1 , , let ˘ W m the openneighborhoods of the strata M z m ( C m ) and E ˘ W m their thickening. Then the evaluation maps ev + : E ˘ W → [ C ]ev − : E ˘ W → [ C ] are transverse in a neighborhood of ˘ W o × ˘ W o ⊂ E ˘ W o × E ˘ W o . Proof.
The first part follows from the construction of the extended moduli spaces, seeSection 19 . W o × ˘ W o by theregularity assumptions on the moduli spaces (see Definition 3.14), hence they are transversein a neighborhood. (cid:3) To conclude the first part of this section, we just state the case of the reducible modulispaces (see Section 19 . Theorem . Suppose the moduli space M red z ([ C − ] , [ C + ]) is d -dimensional and non-empty, so that the space of unparametrized broken reducible trajectories ˘ M red+ z ([ C − ] , [ C + ]) is a ( d − -dimensional space stratified by manifolds. If M ′ ⊂ ˘ M red+ z ([ C − ] , [ C + ]) is a compo-nent of the codimension- e stratum, then along M ′ the space ˘ M red+ z ([ C − ] , [ C + ]) is the productof the stratum and a C manifold with an e -dimensional corner. We now turn our attention to the new analytical issues in the proof of Theorem 5.3,namely the parametrization of the space of solutions on a cylinder which are close to a givenconstant solution.
We first recall the notion of spectral decomposition for a k - asafoe operator L on a vectorbundle E → Y (see Chapter 17 in the book for the details). This naturally arises whenstudying Atiyah-Patodi-Singer boundary value problems. On the pulled-back vector bundleon the negative half cylinder Z = R ≤ × Y , which we call E , we can consider the translationinvariant operators D ± = ddt ∓ L : L ( Z ; E ) → L ( Z ; E ) . The restriction map to the boundary { } × Y extends for every k ≥ r : L k ( X ; E ) → L k − / ( Y ; E )which is also surjective with left inverse (see Theorem 17 . . spectral subspaces of L H ± ⊂ L / ( Y ; E )obtained as the boundary values of the kernel of D ± . Then one has the following lemma (seeLemma 17 . . Lemma . Suppose L is a k - asafoe hyperbolic operator. Then one has the direct sumof closed subspaces L / ( Y ; E ) = H + ⊕ H − . Furthermore, for each integer ≤ j ≤ k + 1 , we have L j − / ( Y ; E ) = ( H + ∩ L j − / ) ⊕ ( H − ∩ L j − / ) . As the operators that arise in our setting are slightly more general, we introduce thefollowing definition, see Lemma 1.7.
Definition . A k - asafoe operator L is called almost hyperbolic if the intersectionbetween its spectrum and the imaginary axis is { } , and the generalized 0-eigenspace coincideswith the kernel.If L is almost hyperbolic it has finite dimensional kernel H = Ker L and for each h ∈ H ⊂ L k ( Y ; E ) the translation invariant configuration s h satisfies the equation( d/dt + L ) s h = 0 . Notice though that these sections are not in L k ( Z ; E ). One can then define a decomposition,called generalized spectral decomposition , L / ( Y ; E ) = H + ⊕ H ⊕ H − where H + (respectively H − ) is the the positive (negative) spectral subspace of L − δ ( L + δ )for δ > L ± δ . Aninteresting space to study for our purposes is(2.15) L k,δ ( Z ; E, H ) = (cid:8) s | s − s h ∈ L k,δ ( Z ; E ) for some h ∈ H (cid:9) = H + L k,δ ( Z ; E ) ⊂ L k, loc ( Z ; E ) , where on the negative half cylinder we use the weight function f ( t ) = e − δt . This space comeswith the linear map Π : L k,δ ( Z ; E, H ) → H
0. GLUING 71 sending a configuration to its (unique) limit point. The norm of a configuration is defined by k s k = k s − Π ( s ) k L k,δ + k Π ( s ) k . We then have the following result.
Lemma . Consider an operator of the form D = d/dt + L where L is k - asafoe andalmost hyperbolic. Let Π : L ( Y ; E ) → H − the the projection with kernel H ⊕ H + .Then the operator D ⊕ (Π ◦ r ) ⊕ Π : L j,δ ( Z ; E, H ) → L j − ,δ ( Z ; E ) ⊕ ( H − ∩ L j − / ( Y ; E )) ⊕ H is an isomorphism for ≤ j ≤ k for δ > sufficiently small. Finally, the image of ker D under r in L j − / ( Y ; E ) is precisely H − ∩ L j − / ( Y ; E ) ⊕ H . Proof.
Using the same trick as in Proposition 3.4, the operator D ⊕ (Π ◦ r ) ⊕ : L j,δ ( Z ; E ) → L j − ,δ ( Z ; E ) ⊕ ( H − ∩ L j − / ( Y ; E ))is equivalent to the operator( D + δ ) ⊕ (Π ◦ r ) : L j ( Z ; E ) → L j − ( Z ; E ) ⊕ ( H − ∩ L j − / ( Y ; E ))for which the statement follows from Proposition 17 . . H − is the negativespectral subspace for the hyperbolic operator L + δ . The result then follows from the definitionof the space L j,δ ( Z ; E, H ). (cid:3) Example . We are especially interested in the following situation. The Hilbert vectorbundle K σj → C σk ( Y )carries a smooth family of operators with real spectrum, the Hessians { Hess σ q } . At a Morse-Bott critical point a , such an operator is almost hyperbolic, as it is a direct summand ofthe weighted extended Hessian (see Lemma 3.3), hence we have the (generalized) spectraldecomposition K σk − / , a = K + ⊕ T a C ⊕ K − . By quotienting by T a C , this also induces a spectral(2.16) N σk − / , a = N + a ⊕ N − a . on the normal bundles.Consider now a critical point a ∈ C ⊂ C σk ( Y ), and consider an L k -compatible productchart ( U , ϕ ) of a neighborhood U of a . As pointed out in Remark 2.3, we can consider such achart ϕ as the restriction of a L k − / -compatible prduct chart, which we also call ϕ . Define Z ∞ = ( R ≤ × Y ) ∐ ( R ≥ × Y ) . Given an open neighborhood [ U ] ⊂ [ C ] of [ a ] we can define the configuration space B τk,δ ( Z ∞ , [ U ]) = { γ ∈ B τk, loc | γ ∈ B τk,δ ( Z ∞ , [ b ]) for some [ b ] ∈ [ U ] } consisting of configurations exponentially asymptotic to the same configuration [ b ] ∈ [ U ] onboth ends (up to gauge equivalence). We can define the subspace M ( Z ∞ , [ U ]) ⊂ B τk,δ ( Z ∞ , [ U ])consisting of solutions to the Seiberg-Witten equations. Given [ b ] ∈ [ U ] we will write M ( Z ∞ , [ b ]) ⊂ M ( Z ∞ , [ U ])for the subset of solutions converging to [ b ]. As usual, we can also introduce the versions ofthe moduli spaces with the tildes. Similarly, we define the finite cylinders Z T = [ − T, T ] × Y. Theorem 17 . . M ( Z T ) on finite cylinders. Proposition . For finite T , the space ˜ M ( Z T ) ⊂ ˜ B τk ( Z T ) is a closed Hilbert subman-ifold. The subset M ( Z T ) is a Hilbert submanifold with boundary, identified with the quotientof ˜ M ( Z T ) by the involution i . The manifolds Z ∞ and all Z T for T > Y ∐ ¯ Y , so we have the continuousrestriction maps R : ˜ M ( Z ∞ , [ U ]) → ˜ B σk − / ( Y ∐ ¯ Y ) R : ˜ M ( Z T , [ U ]) → ˜ B σk − / ( Y ∐ ¯ Y ) . We will need to introduce suitable charts for the configuration spaces on the cylinders. Weexplain an idea that unfortunately does not immediately work. Suppose we are given a L k -compatible product chart ( ϕ, U ) which is the restriction of a L k − / -compatible product chart.This defines a correspondence between elements in a neighborhood of zero ˜ U T ⊂ T τk,γ a ( Z T )and elements in a neighborhood of γ a in ˜ C τk ( Z T ). Indeed, forgetting for a moment aboutregularity issues, we can think of an element of v ∈ T τγ a ( Z T ) as a path ˇ v ( t ) with values in T σ a ( Y ) together with a path of imaginary valued 1-forms ˇ c ( t ) on Y . By suitably restricting˜ U T , we can suppose that the configuration ˇ v ( t ) always lies in the domain U of the chart(because of the continuity of the restriction maps). In a similar fashion, we can think of anelement as a path ˇ γ ( t ) with values in C σ ( Y ) together with a path of imaginary valued 1-forms c on Y . The identification then just sends the path ˇ v ( t ) to the path ϕ (ˇ v ( t )) in C σ ( Y ), andleaves ˇ c ( t ) unchanged. The issue with this approach is that even though the chart we haveconstructed in Lemma 2.2 (which has the additional desirable property of being somehowgauge invariant) is the restriction of a L k − / -compatible product chart (see Remark 2.3), theestimates on the L k norm of the derivative do not hold on the domain of this larger chart.We fix this issue in the next result. Lemma . There exists an L k -compatible product chart ( ϕ, U ) around a which is therestriction of an L k − / -compatible product chart such that the construction above gives riseto a well defined chart ϕ : ˜ U T ⊂ T τk,γ a → ˜ C τk ( Z T ) onto a neighborhood of the constant solution γ a . The similar statement holds on the infinitecylinder (with the weighted Sobolev spaces). . GLUING 73 Proof.
As in the proof of Lemma 2.2 we can locally identify the slice through a with itstangent space T a C ⊕ N σ, a k , and the intersection of the critical submanifold C with the slicewith the image of the graph of a smooth function f : T a C → N σk defined in a neighborhood of the origin. This induces the map (defined in an L k neighborhoodof the origin) ˜ f : T σk, a ( Y ) → T σk, a ( Y )( v t , v n , v j ) (cid:0) v t , v n + f ( v t ) , v j (cid:1) where we have identified the tangent space with the direct sum T a C ⊕ N σk, a ⊕ J σk, a . The nicefeature of this map is that as in Remark 2.3 it is defined for each 1 ≤ j ≤ k in an L j neighborhood of the origin, and furthermore norms of the derivatives and their inverses inthe L k norm is bounded in this neighborhood. Here we use the fact that T a C does not dependon j ≤ k . This implies that the analogue of the fiberwise construction discussed above at thelevel of the tangent spaces induces a diffeomorphism˜ f : ˜ U T ⊂ T τk,γ a → T τk,γ a from a neighborhood of the origin onto a neighborhood of the origin. The desired chart isdefined by composing this map with the chart form Section 18 . i : T τk,γ a → ˜ C τk ( Z T )( a, r, ψ ) ( A + a ⊗ , s + r, φ )where γ a = ( A , s , φ ) and ˇ φ ( t ) = ˇ φ + ψ ( t ) p k ψ ( t ) k . The result follows because the chart i is also defined fiberwise. (cid:3) This section is devoted to the proof of the following theorem. Here we use the notation N = K σk − / , a ( Y ) /T a C . From this the proof of Theorem 5.3 follows with no essential modifications as in Chapter 19in the book.
Theorem . Fix an L k − / -compatible product chart ( U , ϕ ) in a neighborhood of a asin Lemma 5.13. Then there exists T such that for all T ≥ T we can find smooth parametriza-tions u ( T, − ) : B ( T a C ) × B ( N ) → ˜ M ( Z T ) u ( ∞ , − ) : B ( T a C ) × B ( N ) → ˜ M ( Z ∞ ) which are diffeomorphisms from a product of balls B ( T a C ) × B ( N ) onto a neighborhood of [ γ a ] . The map u ( ∞ , − ) respects the decomposition of the moduli space in the sense that forevery ξ ∈ B ( T a C ) one has u ( ∞ , ( ξ, − )) ( B ( N )) ⊂ M ( Z ∞ , [ ϕ ( ξ, , u ( ∞ , ( ξ, γ ϕ ( ξ, , ] . Furthermore, these parametrizations can be chosen so that the map µ T : B ( T a C ) × B ( N ) → B σk − / ( Y ∐ Y ) h Ru ( T, h ) is a smooth embedding of B ( T a C ) × B ( N ) for every T ∈ [ T , ∞ ] with the following additionalproperties. The function µ T : [ T , ∞ ) × B ( T a C ) × B ( N ) → B σk − / ( Y ∐ Y ) is smooth for finite T and furthermore µ T C ∞ loc −→ µ ∞ as T → ∞ . Finally there is an η > independent of T such that the images of the maps u ( T, − ) can betaken to contain all solutions [ γ ] ∈ M ( Z T ) with k γ − γ b k L k ( Z T ) ≤ η for some b ∈ U .In the case that a is reducible, the parametrizations u ( T, − ) for T ∈ ( T , ∞ ] are equivariantfor the Z / Z action of i . The parametrization u ( ∞ , − ) provided by the theorem also respects the fibered productstructure on M ( Z ∞ , [ U ]). This can be identified with (cid:8) ([ γ + ] , [ γ − ]) | lim t →∞ [˘ γ + ( t )] = lim t →−∞ [˘ γ − ( t )] (cid:9) ⊂ M ( R ≥ × Y, [ U ]) × M ( R ≤ × Y, [ U ])and and the map u ( ∞ , − ) provides local diffeomorphisms B ( T a C ) × B ( N + ) → M ( R ≥ × Y, [ U ]) B ( T a C ) × B ( N − ) → M ( R ≤ × Y, [ U ]) . We can think of the boundary values of these moduli spaces as the (local) stable and unstablemanifolds in a neighborhood of [ a ].The proof of Theorem 5.14 follows the analogous result in the Morse setting very closely.We first discuss a more abstract version for general operators of the form d/dt + L on thecylinders Z T , and then apply it to our special case. Here L is an almost hyperbolic k - asafoe operator. Suppose T >
2, and consider a smooth even function g T,δ on [ − T, T ] such that g T,δ ( t ) = ( e δ ( t + T ) if t ≤ − e − δ ( t − T ) if t ≥ . The function σ ( t ) T,δ = g ′ T,δ ( t ) /g T,δ ( t ) is δ for t ≤ − − δ for t ≥
1, and we can choosethe family so that for each δ > T on the interval [ − , L k,δ norm defined to be k u k L k,δ = k g T,δ · u k L k . We denote this normed space by L k,δ ( Z T , E ). The new norm, which we call the weightedSobolev norm, is obviously equivalent to the original one on each finite cylinder, but it willbe useful to study the behaviour of the operators D = d/dt + L : L k,δ ( Z T ; E ) → L k − ,δ ( Z T ; E ) . GLUING 75 for T going to infinity. Inside this space there is the subspace H consisting of the constantsections s h for h ∈ H = ker L . It is important to remark that the canonical identification(2.17) H → H sending an element h to the constant section s h has norm growing exponentially in the timeparameter T . There is the map Π : L k,δ ( Z T , E ) → H given by L k,δ projection, or, equivalently L δ projection as the elements in H are constant.We still denote by Π the composition of such a map with the identification (2.17). We denotethe kernel of such map by L k,δ ( Z T ; E ) ⊥ . On the infinite cylinder, we introduced the space L k,δ ( Z ∞ ; E, H ) in (2.15). In this case, we will consider the operator D = d/dt + L : L k,δ ( Z ∞ ; E, H ) → L k − ,δ ( Z ∞ ; E )Suppose we are also given a bounded linear operatorΠ : L k − / ( Y ∐ Y ; E ) → H for some Hilbert space H . This induces by restriction to the boundary (which we omit fromthe notation) a map Π : L k ( Z T ; E ) → H Π : L k ( Z ∞ ; E ; H ) → H. For simplicity, we introduce the notations E Tδ = L k,δ ( Z T ; E ) F Tδ = L k − ,δ ( Z T ; E )and E ∞ δ = L k.δ ( Z ∞ ; E, H ) , F ∞ δ = L k − ,δ ( Z ∞ ; E ) . The key assumption in what follows is that the linear operator(2.18) ( D, Π , Π) : E ∞ δ → F ∞ δ ⊕ H ⊕ H is invertible. This also implies the invertibility on weighted spaces with weight δ ′ sufficientlyclose to δ . The problem we are interested in is non-linear, and in our abstract setting wesuppose this non-linearity arises as a map α : C ∞ ( Z T ; E ) → L ( Z T ; E )obtained from a map α : C ∞ ( Y ; E ) → L ( Y ; E ) by restriction to slices { t } × Y . We willassume that α defines a smooth map α : L k ([ − , × Y ; E ) → L k − ([ − , × Y ; E )with α ( h ) = 0 for every h ∈ H and D α = 0. This implies that α defines smooth maps α : E Tδ → F Tδ α : E ∞ δ → F ∞ δ . We are then interested in the study of the maps F T = D + α : E Tδ → F Tδ F ∞ = D + α : E ∞ δ → F ∞ δ and especially in the spaces of solutions M ( T ) = ( F T ) − (0) ⊂ E T M ( ∞ ) = ( F ∞ ) − (0) ⊂ E ∞ δ . Our abstract version of the Theorem 5.14 is the following.
Proposition . Suppose the hypothesis above are satisfied, and in particular that themap (2.18) is invertible. Then for T ≥ T the sets M ( T ) and M ( ∞ ) are Hilbert submanifoldsof E T and E ∞ in a neighborhood of . There exist η > and smooth maps u ( T, − ) : B η ( H ) × B η ( H ) → M ( T ) T ∈ [ T , ∞ ] which are diffeomorphisms onto their image and preserve the product structure, i.e. for every T ∈ [ T , ∞ ] we have (Π , Π) u ( T, ( h , h )) = ( h , h ) u ( T, ( h , s h . Furthermore, for T ∈ [ T , ∞ ] the map obtained by composing with the restriction to theboundary µ T : B η ( H ) × B η ( H ) → L k − / ( Y ∐ Y )( h , h ) ru ( T, h , h ) is a smooth embedding. As a function on [ T , ∞ ) × B η ( H ) × B η ( H ) the map ( T, h , h ) µ T ( h , h ) is smooth for finite T , and µ T C ∞ loc −→ µ ∞ for T → ∞ . Finally, there is an η ′ > independent of T such that the images of the maps u ( T, − ) containall solutions u ∈ M ( T ) with k u k L k,δ ≤ η ′ . The strategy of the proof follows the classical case. We first prove the existence of thesolution u ( ∞ , h , h ) for k ( h , h ) k small. Lemma . There exist η , C > such that for every ( h , h ) with k h k , k h k ≤ η = η / C there exists a unique u = u ( ∞ , h , h ) in B η ( E ∞ δ ) satisfying F ∞ ( u ) = 0Π u = h. Furthermore the map u ( ∞ , − ) : B η ( H ) × B η ( H ) → B η ( E ∞ ) is smooth, sends ( h , to the constant section s h and satisfies k u ( ∞ , h , h ) k ≤ C k ( h , h ) k . . GLUING 77 Proof.
This follows from the application of the inverse function theorem (Proposition18 . . α : E ∞ δ → F ∞ δ is C and has vanishing derivative at the origin so it is uniformly Lipschitz with small Lipschitzconstant on small balls around zero, i.e. for any ε > η > u, u ′ ∈ E ∞ δ we have k u k , k u ′ k ≤ η = ⇒ k α ( u ) − α ( u ′ ) k ≤ ε k u − u ′ k . We can then just apply the inverse function theorem to the map( F ∞ , Π , Π ) : E ∞ δ → F ∞ ⊕ H ⊕ H . The fact that u ( ∞ , ( h , s h follows from the fact that α ( s h ) = 0. (cid:3) We then focus on the operators F T acting on finite cylinders. First we study their lin-earizations. Lemma . There exists a T such that for all T ≥ T the operator P T = D ⊕ Π : E T, ⊥ δ → F Tδ ⊕ H is invertible. Furthermore, for T → ∞ the operator norm k ( P T ) − k is bounded by a constantindependent of T . This implies in particular that the whole linearization of the operator F T is invertible.On the other hand it is clear that the norm of the inverses grows exponentially because ofthe constant sections H . The key point of the lemma is the fact that on the complement ofthis subspace the norm of the inverse is bounded. Proof.
This follows from modifying the usual patching argument (see the proof ofLemma 18 . . N ∞ the inverse of P ∞ . On Z T , pick a smooth partition ofunity β , β with β ( t ) = ( t ≤ −
10 for t ≥ β ( t ) = β ( − t )and smooth functions φ , φ so that φ ( t ) = ( t ≤ T / −
10 for t ≥ T / φ ( t ) = φ ( − t ) . We can also suppose that the non constant part of these functions does not depend on T . Weconstruct an almost right inverse for P T with the following patching argument. Call N ∞ theinverse of the linear operator P ∞ : E ∞ δ, → F ∞ δ ⊕ H, where E ∞ δ, is the subspace of E ∞ δ consisting of configurations asymptotic to zero. We can thendefine ρ : F Tδ → F ∞ δ v ρ ( v ) = ( τ ∗− T β v on [0 , ∞ ) × Yτ ∗ T β v on ( −∞ , × Y. and π : E ∞ δ → E Tδ u φ τ ∗ T u + + φ τ ∗− T u − where u + and u − are the parts of u on the two components R ≥ × Y and R ≤ × Y . We nowdefine the map ˜ N T : F Tδ ⊕ H → E Tδ ( v, h ) (Id − Π ⊥ ) π ◦ N ∞ ( ρ ( v ) , h )which has the property that P T ◦ ˜ N T = 1 + K T where the operator norm of K T goingto 0. In fact, ˜ N T ( u ) solves the equation outside the intervals [ − T / − , − T /
T / − , T / T / − , T / u ddt φ · τ ∗ T P ∞ + τ ∗− T β u. This map has norm bounded above by the quantity Ce − δT for some time independent constant C . This is because the multiplication by β (seen as a map from the finite cylinder to theinfinite one) has norm bounded by a time independent constant, while the multiplication by ddt φ (seen as a map from the infinite cylinder to the finite one) has norm decreasing as e − δT .In fact weight function for the finite cylinder is approximatively e δT at zero and e δT/ at T /
2, while the (translation of) the weight function on the infinite cylinder is e δT at zero and e δT/ at T /
2. The operator ˜ N T has bounded norm for the same reason. The operator P T isinjective as its kernel (on the whole space) consists of H , so the existence of the right inversefor T ≥ T implies that it is invertible in the same range. Its inverse N T has bounded normbecause k N T − ˜ N T k is going to 0 and the operator norm of ˜ N T remains bounded. (cid:3) We can then deduce the existence of solutions on a finite cylinder.
Corollary . Let η , η and C as in Lemma 5.16. Then there exists T such that forevery T ∈ [ T , ∞ ] and ( h , h ) ∈ H ⊕ H with k h k ≤ η there exists a unique u = u ( T, h , h ) in B η ( E Tδ ) satisfying F T ( u ) = 0(Π , Π ) u = ( h , h ) . The map u ( T, − ) : B η ( H ⊕ ˜ H ) → B η ( E T ) is smooth and satisfies u ( T, ( h , s h . Finally, we have the estimate k u ( T, h , h ) k ≤ C ( e δT k h k + k h k ) . . GLUING 79 Proof.
This is again an application of the inverse function theorem, but one needs someextra care to obtain maps defined on a time independent ball in H . Consider the non-linearmap (cid:0) F T ( s h + − ) , Π (cid:1) : E T, ⊥ δ → F Tδ ⊕ H, which sends 0 to 0 as α ( h ) = 0 and has linearization at the origin given by the operator( d/dt + L + D h α, Π) : E T, ⊥ δ → F Tδ ⊕ H. We can find a small ball B η ( H ) for which this operator has norm very close to that of ddt + L , independently of time, as the difference is given by a small operator acting fiberwise.In particular, the previous lemma tells us that the linearization is invertible and its norm isbounded by a fixed constant uniformly in time and h ∈ B η ( H ). Furthermore we can choosesuch ball small enough so that all the non-linear parts are also uniformly Lipschitz with fixedsmall constant on a ball of radius η . Then for each fixed h the inverse function theoremprovides us with a (unique) solution u h ( T, h ) ∈ L k,δ ( Z T ) with the additional property that k u h ( T, h ) − s h k ≤ C ′ k h k for some constant C ′ independent of h and T . We can interpret this as a map B η ( H ) × B η ( H ) → L k,δ ( Z T ) . The smoothness of this map follows from applying the inverse function theorem to the wholemap ( F T , Π , Π). Notice that in this case the ball on which the inverse function theoremapplies has size decreasing exponentially fast. (cid:3)
Notice that the map u depends on δ via the choice of the projection Π . To underline thiswe respectively denote these by u δ and Π δ , as it will be important in the next result. We needto compare the solution u δ ( T, h , h ) on the finite cylinder Z T with the solutions u ( ∞ , h, h )(which are independent of δ ). As above we denote the two components of the solution on theinfinite cylinder as u + ( h , h ) and u − ( h , h ). We then define for ( h , h ) the section U ( T, h , h ) = τ ∗ T u + ( T, h , h ) + τ ∗ T u − ( T, h , h ) − h , This section is close to u δ ( T, h, h ), as the next lemma shows. Lemma . For each η < η , consider the function ξ δ ( T, − ) : B η ( H ) × B η ( H ) → E Tδ ( h , h ) u δ ( T, h , h ) − U ( T, h , h ) . Then there is δ > so that the previous result still holds and this function converges to zeroin the C ∞ loc topology for T → ∞ . Proof.
Let us write(2.19) ( F T , Π δ , Π) U ( T, h , h ) = ( ζ ( h , h ) , h + ν δ ( h , h ) , h + g ( h , h )) . Fix a δ > ζ and g (which are independentof δ ) and their derivatives all have decay of the form K ( h , h ) e − ( δ ′ + δ ) T for 0 < δ ′ ≤ δ where K is a continuous function. The proof of Lemma 18 . . g (where we can pick δ ′ to be δ ), and shows that the normof the map to the unweighted spaces ζ : B η ( H ) × B η ( H ) → L k ( Z T )and its derivatives are bounded by functions of the form K ( h , h ) e − δ T . In particular, itsnorm seen as a map with values in L k,δ − δ ′ ( Z T ) is bounded by K ′ ( h , h ) e − ( δ + δ ′ ) T . For any δ > ν δ has norm decreasing as K ( h , h ) e − δT . This follows from the fact thatthe constant sections involved in the orthogonal projection have norm growing like e δT .The result then follows by picking δ = δ − δ ′ with δ ′ small enough so that the previ-ous results still hold. Indeed from the estimates provided by inverse function theorem thelinearization of the local inverse( F T , Π δ , Π) − : B ( F Tδ ⊕ H ⊕ H ) → E Tδ has norm growing like C ( h , h ) e δT at each point (0 , h , h ), and is defined in a ball of radius C ( h , h ) e − δT . Our configuration in equation (2.19) belongs to this ball for T big enough, andfor the same reason the result follows. (cid:3) With this in hand, we are finally ready to complete the proof of Proposition 5.15, followingthe same arguments of the book.
Proof of proposition 5.15.
There are two things left to check: the convergence of therestriction maps µ T and the smoothness of such a map on the product [ T , ∞ ) × B η ( H ) × B η ( ˜ H ). For the first one, the previous proposition tells us that it is sufficient to study theconvergence of ˜ µ T ( h, h ) = rU ( T, h, h ) . On the other hand, if h = ( h , ˜ h ), the component in L k − / ( Y ; E ) is the sum˜ µ T ( h ) = u + ( h ) | { }× Y + u − ( h ) | {− T }× Y − h = µ ∞ ( h ) + ( u − ( h ) | {− T }× Y − h )and the second term converges to zero in the C ∞ loc topology on the ball because u − ∈ L k,δ ( Z ≤ ; E, H ). The same argument applies to the other boundary component. Finally,the proof of smoothness is obtained by pulling back the whole family to a fixed cylinder Z T ,see the book. (cid:3) We now show how to deduce Theorem 5.14 from Proposition 5.15. First of all, we willwork with in the slices S τk, a ( Z T ) ⊂ C τk ( Z T ) S τk,δ, a ( Z ∞ , U ) ⊂ C τk,δ ( Z ∞ , U )both defined by the equations h a, n i = 0 at ∂Z − d ∗ a + is Re h iφ , φ i + i | φ | Re µ Y h iφ , φ i = 0 . GLUING 81 where as usual we write A = A + a ⊗
1. As usual, U is a neighborhood of a in the intersectionof the critical set C and the slice S σ a . On the half infinite cylinder, we are using a slice which isdifferent from the one we adopted in Section 3, see the discussion before Proposition 3.1. Thisequation defines a slice only for a choice of δ > Q γ obtained bystudying the gauge fixed Seiberg-Witten equations can be written as( V, c ) Ddt ( V, c ) + L γ ( t ) ( V, c ) . where L γ ( t ) is not the usual extended Hessian, and not its weighted counterpart. In particular,this family of operators is constant at the constant trajectory γ a . We will study a smallneighborhood of [ γ a ] in M ( Z T ) by studying solutions of the gauge-fixing equation togetherwith the perturbed Seiberg-Witten equations. To reduce ourselves to the linear setting ofProposition 5.15, we introduce a chart ϕ induced by a suitable L k -compatible product chart(which is the restriction of an L k − / -compatible product chart) as in Lemma 5.13. We cansuppose that the neighborhood ˜ U T of the origin on which the chart is defined contains all theconfigurations that have distance at most η > γ b where b is acritical point in U for some constant η is independent of T large enough.We have the restriction map r : C τk ( Z T ) → C σk − / ( Y ∐ Y ) × L k − / ( Y ∐ Y ; i R )where the second component records the normal component of the connection at the boundary.We will use the usual decomposition at a critical point T σk − / , a = J σk − / , a ⊕ K σk − / , a , where the second summand is T a U ⊕ N σk − / , a and we have spectral decomposition N = N σk − / , a = N + ⊕ N − as in equation (2.16). We can define the subspaces of T σk − / , a ⊕ L k − / ( Y ; i R ) H − Y = { } ⊕ { } ⊕ N − ⊕ L k − / ( Y ; i R ) H − Y = { } ⊕ { } ⊕ N + ⊕ L k − / ( Y ; i R )and define the projections Π − Y : T σk − / , a ⊕ L k − / ( Y ; i R ) → H − Y Π − Y : T σk − / , a ⊕ L k − / ( Y ; i R ) → H − Y with kernels respectively J σk − / , a ⊕ ( T a U ⊕ N + ) ⊕ { } J σk − / , a ⊕ ( T a U ⊕ N − ) ⊕ { } , and we can set H = H − ¯ Y ⊕ H − Y Π = Π − ¯ Y ⊕ Π − Y . Finally we let H = T a U and we have the map for T = ∞ Π : T τk,δ,γ a → H simply sends a configuration to its limit point. Given a path γ in C τk ( Z T ) such that therestrictions to the boundary lie in the domain of the chart U , we are interested in the systemof equations given by F τ q γ = 0Coul τ a γ = 0(Π , Π) ◦ ( ϕ − ◦ r ) γ = ( h , h )where ( h , h ) ∈ H ⊕ H . Hence given any element γ lying in a small neighborhood of γ a so that both restrictions lie in the domain U of the chart ϕ , we can alternatively write theequations as ( Q γ a + α )˜ γ = 0(Π , Π) ◦ r ◦ ˜ γ = ( h , h )where Q γ a is the linear part, α is the remainder of the terms (and defined slicewise) and˜ γ is the configuration in T τk,γ a ( Z T ) corresponding to γ under the local chart. The notationis justified because Q γ a is the linearization of the Seiberg-Witten equations with Coulombgauge condition arising in Section 4, because of the condition D ϕ = Id in the definition ofan L k − / -compatible product chart.We then turn to study this problem with the abstract gluing result proved above. Hereneither the domain T τk,γ a nor the range V τk − are in the form required by Proposition 5.15, butthey can be converted to spaces of sections of a finite dimensional vector bundle by the samedevice as Section 1. We next verify the key hypothesis of Proposition 5.15. Lemma . The linearized equations on the infinite cylinder Z ∞ are invertible at for δ > small enough. Proof.
This is essentially a parametric version of the argument that settles the Morsecase. One just has to look at the operator on each component, and we will focus on theoperator on Z ≤ given by( Q γ a , Π − Y ◦ r, Π ) : T τk,δ,γ a → V τk − ,δ,γ a ⊕ H − Y ⊕ H . If we write Q γ a = d/dt + L and call H ± L the spectral subspaces of the almost hyperbolicoperator L (which is the weighted extended Hessian at the point for t = ∞ , see Lemma 3.3),Lemma 5.10 tells us that the operator( Q γ a , Π − L ◦ r, Π ) : T τk,δ,γ a → V τk − ,δ,γ a ⊕ ( H − L ∩ L k − / ) ⊕ H is an isomorphism, where Π − L is the negative spectral projection. If we decompose now thedomain a the direct sum C ⊕ K , where K is the kernel of Q γ a , we can write the operator (cid:20) Q γ a | C x (Π − L ◦ r ) ⊕ Π | K . (cid:21) On the other hand Lemma 5.10 tells us that the image of K under r in L k − / ( Y ; E ) isexactly H − L ⊕ H . Furthermore the proof of Lemma 17 . . . THE MODULI SPACE ON A COBORDISM 83 case to show that Π − Y is an isomorphism on H − L , so the matrix (cid:20) Q γ a | C x (Π − Y ◦ r ) ⊕ Π | K . (cid:21) also defines an invertible operator. This is exactly the operator in equation (5). (cid:3) Hence Proposition 5.15 provides us a solution γ = u ( T, h, h ) to these equations for any T ∈ [ T , ∞ ] and k h k , k h k ≤ η . The boundary conditions include also the normal componentof the connections at the boundary, h a, n i = c on ¯ Y h a, n i = c on Y. By restricting to boundary conditions with c = c = 0 we get trajectories in Neumann-gauge,and hence we obtain a parametrization of the solutions in the slice u ( T, − ) : B η ( T a U ) × B η ( ˜ N ) → S τk,γ a u ( ∞ , − ) : B η ( T a U ) × B η ( ˜ N ) → S τk,δ,γ a such that there exists η > T ≥ T the image of u ( T, − ) contains allsolutions in S τk,γ a with k γ − γ b k L k,δ ≤ η for some b ∈ U . The final step to prove Theorem 5.14is to extend the result to a uniform neighborhoods in the moduli space of solutions defined bythe inequality k γ − γ b k L k,δ ≤ η where γ is not necessarily in the slice. This is proved in thesame fashion as Proposition 18 . . δ > k γ − γ a k L k ( Z T ) and k γ − γ a k L k,δ ( Z T ) are equivalent on a small neighborhood n k γ − γ a k L k ( Z T ) ≤ η o of γ a in M ( Z T ) by a constantindependent of time, for some constant η also independent of time. This follows from theexponential decay of a solution to the Seiberg-Witten equations always close to a constantone, as discussed in Section 2.
6. The moduli space on a cobordism
In this section we briefly discuss the adaptation of the theory we have developed so far tothe moduli spaces of solutions to the perturbed Seiberg-Witten equation on a general man-ifold with (possibly disconnected) boundary. This generalizes the content of Chapter 24 inthe book. The proofs of the results we are going to state can be easily obtained from thosein the Morse case, using the same techniques we have used throughout the present chapter.Let X be a compact connected oriented Riemannian 4-manifold with non empty (andpossibly disconnected) boundary Y = ∐ Y α , and let us suppose that the metric is cylindrical in the neighborhood of the boundary, so itcontains an isometric copy of I × Y where I = ( − C, ∂C = { } × Y . A spin c structure s X on X determines a spin c structure s on Y . We define the configuration space and thespace of tame perturbations B σk ( Y, s ) = Y B σk ( Y α , s α ) P ( Y, s ) = Y P ( Y α , s α ) . Suppose from now on that a given tame perturbation q = { q α } which is Morse-Bott hasbeen fixed on the boundary.Our approach to the construction of maps induced by cobordisms (and in particular ofthe module structures) will be different from the one in the book, and it will heavily rely onthe structure of moduli spaces on manifolds with more than one end. In that case, we willpartition the boundary Y in two classes: outgoing boundary components, which will have theboundary orientation, and incoming boundary components, with the opposite orientation.In the present section however, we will suppose that all the boundary components have theboundary orientation in order to keep the discussion as uniform as possible. Notice that theorientation reversal changes the role of boundary stable and unstable critical points on theincoming ends, which may cause some confusion with the gradings.The main difference with the setup we have worked so far is that in the case of a generalcobordism we cannot rely on the τ model, which is only defined for cylinders. Recall fromSection 2 in Chapter 1 that we have the configuration space C σ ( X, s X ) = (cid:8) ( A, s, φ ) | s ≥ , k φ k L ( X ) = 1 (cid:9) ⊂ A × R × Γ( X ; S + )and its Hilbert completion C σk ( X, s X ). The quotient by the gauge group action is denoted by B σk ( X, s X ) and is a Hilbert manifold with boundary. The Seiberg-Witten equations define asmooth section F σ of the vector bundle V σk − → C σk ( X, s X ) , and we introduce perturbations as follows. Pick a cut-off function β equal to 1 near t = 0and 0 near t = − C , and a bump function β with compact support in ( − C, I × Y , any tame perturbation p in P ( Y, s ) defines the sectionˆ p : C k ( X, s X ) → V k ˆ p = β ˆ q + β ˆ p , where q is the fixed Morse-Bott perturbation. This section can be extended as in Section 3of Chapter 1 to the blown-up setting, and we can hence define F σ p = F σ + ˆ p σ : C σk ( X, s X ) → V σk − The perturbed Seiberg-Witten equations F σ ˆ p = 0 are invariant under G k +1 ( X ), and we havethe moduli spaces of solutions M ( X, s X ) ⊂ B σk ( X, s X ) M ( X, s X ) = n ( A, s, φ ) | F σ ˆ p = 0 o (cid:14) G k +1 ( X ) . Similarly, we have the larger moduli space˜ M ( X, s X ) ⊂ ˜ B σk ( X, s X ) . . THE MODULI SPACE ON A COBORDISM 85 obtained by dropping the condition s ≥
0, and M ( X, s X ) is identified with its quotient by theinvolution i switching the sign of s . The unique continuation property of the Seiberg-Wittenequations (see Section 10 . σ and τ models ona finite cylinder tells us that there are well defined restriction maps to the cylindrical end M ( X, s X ) → M ( I × Y, s X ) ⊂ B τk ( I × Y, s X )and to the boundary R : M ( X, s X ) → B σk − / ( Y, s ) . Also, the section F σ ˆ p is transverse to zero, and ˜ M ( X, s X ) and M ( X, s X ) are respectively asmooth Hilbert manifold and a smooth Hilbert manifold with boundary (see Proposition24 . . X ∗ obtained byattaching cylindrical ends to X , namely X ∗ = X ∪ Y ZZ = [0 , ∞ ) × Y. On such a space we have the L k, loc configuration space C k, loc ( X ∗ , s X ) = A k, loc × L k, loc ( X ∗ ; S + ) . As we are not dealing with a Banach space anymore, we have to define the blow-up as follows(see Section 6 . S as the topological manifold obtained as thequotient of L k, loc ( X ∗ ; S + ) \ R + . We then define C σk, loc ( X ∗ , s X ) = (cid:8) ( A, R + φ, Φ) | Φ ∈ R ≥ φ (cid:9) ⊂ A k, loc × S × L k, loc ( X ∗ ; S + ) , which comes with a canonical blow down map π . If we call O ( −
1) the complex tautologicalline bundle on S , one can define the bundle V σk − = O ( − ∗ ⊗ π ∗ ( V k − ) → C σk, loc ( X ∗ , s X ) , and its section F σ : O ( − → π ∗ ( V k − ) F σ ( A, R + φ, Φ)( ψ ) = (cid:18) ρ ( F + A t ) − (ΦΦ ∗ ) , D + A ψ (cid:19) . We can consider again the perturbed equation, given by the continuous gauge-invariant section F σ p = F σ + ˆ p σ , where the perturbing term ˆ p σ is defined as before on the collar I × Y and isextended to be ˆ q σ on the cylindrical end Z . The unique continuation property tells us againthat there is a restriction map (cid:8) [ γ ] ∈ B σk, loc ( X ∗ , s X ) | F σ p ( γ ) = 0 (cid:9) → B τk, loc ( Z, s ) , and clearly this restriction satisfies the equation on the cylinder. Definition . For a critical submanifold[ C ] = Y [ C α ] ⊂ B σk ( Y, s ) , define the moduli space M ( X ∗ , s X ; [ C ]) ⊂ B σk, loc ( X ∗ , s X )as the set of all [ γ ] with F σ p ( γ ) = 0 and such that its restriction is asymptotic to a configurationof [ C ] on the cylindrical end Z .We can also consider the union over all spin c structures B σk, loc ( X ∗ ), and the moduli space M ( X ∗ ; [ C ]) = a s X M ( X ∗ , s X ; [ C ]) ⊂ B σk, loc ( X ∗ ) , where we implicitly restrict ourselves to the union to the spin c structures inducing the one towhich [ C ] belongs. Similarly, we introduce the set of reducible elements M red ( X ∗ , s X ; [ C ]) ⊂ M ( X ∗ , s X ; [ C ])as the configurations with representatives ( A, s, φ ) with s = 0. The configuration space canbe decomposed along the elements z ∈ π ( B σ ( X ; [ C ]))and so can the moduli space M ( X ∗ ; [ C ]) = [ z M z ( X ∗ ; [ C ]) . Also, given an element z ∈ π ( B σ ( Y ); [ C ] , [ C ′ ]), we obtain by concatenation a new element z ◦ z ∈ π ( B σ ( X ; [ C ′ ]) . Finally, there are natural continuous evaluation mapsev α : M ( X ∗ ; [ C ]) → [ C α ] ⊂ B σk ( Y α , s α )to each component of the critical submanifold.The clearest way to discuss regularity in this framework is to pass to a fiber productdescription. Both manifolds X and Z have boundary Y , and we have well defined restrictionmaps R + : M ( X, s X ) → B σk − / ( Y, s ) R − : M ( Z ; [ C ]) → B σk − / ( Y, s ) . Letting Fib( R + , R − ) be the fibered product of these two maps, we have the restriction map ρ : M ( X ∗ , s X ; [ C ]) → Fib( R + , R − ) . In fact Lemma 24 . . . THE MODULI SPACE ON A COBORDISM 87 Definition . Let consider a solution [ γ ] in M ( X ∗ , s X ; [ C ]). If [ γ ] is irreducible, we saythat the moduli space is Smale-regular at [ γ ] if the maps of Hilbert manifolds( R − , R + ) : M ( X, s X ) × M ( Z ; [ b ]) → B σk − / ( Y, s ) × B σk − / ( Y, s )is transverse to the diagonal at ρ [ γ ]. In the reducible case, we consider instead the map( R − , R + ) : M red ( X, s X ) × M red ( Z ; [ b ]) → ∂ B σk − / ( Y, s ) × ∂ B σk − / ( Y, s ) . We say that M ( X ∗ , s X ; [ C ]) is Smale-regular if it is regular at every point. Finally, we saythat a perturbation ˆ p is Smale-regular if all the moduli spaces are Smale-regular.One has the following result (see Proposition 24 . . Lemma . Let [ C ] = Q [ C α ] be a critical submanifold. If the moduli space M ( X ∗ , s X ; [ C ]) is non empty and Smale-regular, then it is • a smooth manifold consisting only of irreducibles, if any [ C α ] is irreducible; • a smooth manifold consisting only of reducibles, if any [ C α ] is boundary stable; • a smooth manifold with (possibly empty) boundary if all [ C α ] are boundary stable.In the last case, the boundary consists of the reducible elements of the moduli space. Definition . Suppose that exactly c + 1 of the [ C α ] are boundary unstable, with c ≥ C ] is boundary obstructed with corank c .We can define the grading as follows. Let [ γ ] be any element of B σk,z ( X ∗ ; [ C ]) and γ agauge representative. Suppose [ γ ] is asymptotic to the critical point [ b ] and let [ γ b ] be thecorresponding constant trajectory in B τk,δ ( Z, s ; [ C ]). We have the operator Q σγ = D γ F σ p ⊕ d σ, † γ on X and the translation invariant operator Q γ b on Z . There are restriction maps r + : ker( Q σγ ) → L k − / ( Y ; iT ∗ Y ⊕ S ⊕ i R ) r − : ker( Q γ b ) → L k − / ( Y ; iT ∗ Y ⊕ S ⊕ i R ) . Then the operator r + − r − : ker( Q σγ ) ⊕ ker( Q γ b ) → L k − / ( Y ; iT ∗ Y ⊕ S ⊕ i R )is Fredholm (see Proposition 24 . . z ( X ; [ C ]) to be its index.The point of the definition is that is makes sense even when the moduli space is empty. Wethe define the relative grading to begr z ( X, [ C ]) = Ind z ( X ; [ C ]) − dim[ C ]where the las term denotes the sum of the dimensions of the critical manifolds. This gradinghas the simple additivity propertygr z ◦ z ( X ; [ C ]) = gr z ( X ; [ C ]) + gr z ([ C ] , [ C ]) , where the last term is the sum over all components. Notice that in the case X is a finitecylinder (hence X ∗ is an infinite cylinder), the definition coincides with the usual one. This isbecause of the orientation convention the new index is obtained by considering the operator as acting on the (unproperly called) weighted Sobolev space on the doubly infinite cylinderdefined by the function f ( t ) = e δt . In particular its index differs by the one we used in Section 4 by dim[ C − ] by simple a spectralflow argument. We also have the following (see Proposition 24 . . Proposition . If the moduli space M z ( X ; [ C ]) is non-empty and Smale-regular, itsdimension is gr z ( X ; [ C ]) + dim[ C ] in the boundary unobstructed case. If the moduli space isboundary obstructed with corank c , then its dimension is gr z ( X ; [ C ]) + dim[ C ] + c . In order to have a nice characterization of the compactifications of the moduli spaceswe will need a stronger transversality assumption on the evaluation maps, as in Section 4.Suppose we have for each α a sequence of critical submanifolds C α = ([ C α ] , [ C α ] , . . . , [ C αn α ]) , and corresponding homotopy classes of relative homotopy classes z α . We denote the productsimply by C and z . We can then consider the space M z ( X ∗ , C )consisting of tuples ([ γ ] , [ γ αi ]) with[ γ ] ∈ M z ( X ∗ ; [ C ])[ γ αi ] ∈ M z αi ([ C αi − ] , [ C αi ]) for 1 ≤ i ≤ n α , such that the evaluations agree, i.e. we haveev α [ γ ] = ev − [ γ α ] , ev + [ γ αi ] = ev − [ γ αi +1 ] for i = 1 , . . . , n α − . The space M z ( X ∗ , C ) is naturally equipped with an evaluation mapev : M z ( X ∗ , C ) → Y α [ C αn α ] . We then introduce the following definition, which has to be interpreted in an inductive fashionas Definition 3.14 in Section 4.
Definition . Suppose we are given a Smale-regular perturbation ˆ p . We then say that p is regular if the following holds. For every sequence of critical manifolds C and correspondingrelative homotopy classes z (see the notation above), and any other critical submanifold[ C + ] = Q [ C α + ] and relative homotopy class z α + ∈ π ( B σk ( Y ) , [ C αn α ] , [ C + ]), the evaluationsev : M z ( X ∗ , C ) → Y [ C αn α ] Y ev α − : Y M z α + ([ C αn α ] , [ C α + ]) → Y [ C αn α ]are transverse smooth maps.We now discuss the transversality result in the general case of a family of perturbationsand metrics, which will be needed in the rest of the present work. Let P be a smooth finite . THE MODULI SPACE ON A COBORDISM 89 dimensional manifold (possibly with boundary) parametrizing a smooth family of Riemannianmetrics g P on X , all of which contain an isometric copy of I × Y . Similarly, let p P ∈ P ( Y, s )be a smooth family of perturbations, and write p p = β ( t ) q + β ( t ) p p as usual. Let M ( X ∗ , s X ; [ C ]) p the moduli space corresponding to the metric g p and perturba-tion p p , and consider the total space M ( X ∗ , s X ; [ C ]) P = [ p { p } × M ( X ∗ , s X ; [ C ]) p ⊂ P × B σk, loc ( X ∗ , s X ) , where the identification with a fixed spin bundle S ± p is implicitly made so that the configu-ration space can be thought to be metric independent. The notions of Smale-regularity andregularity readily extend to this context by considering the restriction maps and the evalua-tion maps from the moduli spaces parametrized by the strata P \ ∂P and ∂P (and not for afixed p ). The following is the key transversality result, which is proved as Proposition 24 . . Proposition . Suppose we have a fixed regular Morse-Bott pertubation q on Y . Let g P be a smooth family of metrics as above and p P a family of tame perturbations. Supposethat the family parametrized by the boundary ∂P is regular. Then there is a new family ofperturbations ˜ p P over P such that ˜ p p = p p for all p ∈ ∂P and such that the corresponding parametrized moduli space is regular at every point. Remark . This result readily generalizes to families of metrics and perturbationsparametrized by polyhedra. This will be useful in some other aspects of the theory.There is a natural way to introduce the compactification of the moduli space on a generalmanifold with cylindrical ends, analogous to the one for trajectories defined in Chapter 4.
Definition . Consider a critical submanifold [ C ]. A broken X - trajectory asymptoticto [ C ] consists of pairs ([ γ ] , [˘ γ ]) where: • [ γ ] belongs to a moduli space M z ( X ∗ , [ C ]); • [ ˘ γ ] consists of an unparametrized broken trajectory [ ˘ γ α ] in ˘ M + z α ([ C α ] , [ C α ]) for every α such that its negative evaluation coincides with the evaluation of [ γ ] at the end Y α .The homotopy class the broken trajectory is given by z = z ◦ z ∈ π ( B σ ( X, [ C ]) , and we denote by M + z ( X ∗ , [ C ]) the space of broken X -trajectories in this homotopy class.This space is topologized in a way analogous to the spaces ˘ M + ([ C − ] , [ C + ]), see Section 24 . M z ( X ∗ , [ C ]) sits inside M + z ( X ∗ , [ C ]) as the specialcase in which each [ γ α ] has zero components. Furthermore, there is a natural continuousevaluation map ev : M + z ( X ∗ , [ C ]) → [ C ] . It is important to notice that even though when X is a finite cylinder X ∗ is an infinitecylinder, the compactification we have just constructed is quite different from the space ofbroken unparametrized trajectories Example . Here we deal with a simple (reducible) example. Consider a non degener-ate perturbation q , and two boundary stable critical points a and a over the same reduciblecritical point α and corresponding to the smallest and second smallest eigenvalues. Let z therelative homotopy class corresponding to the path connecting a and a in C σk ( Y ). Then themoduli space M red z ([ a ] , [ a ])is identified with a copy of C P with two points removed, see Section 14 . M red z ([ a ] , [ a ]) is diffeomorphic to S , and hence coincides with ˘ M red+ z ([ a ] , [ a ]).We can compare this space with its compactification given by the space of I × Y -trajectories.This perturbation does not fall in the class studied above (it is translation invariant), buteverything still makes perfectly sense. In particular, the compactification M red+ z ([ a ] , [ a ]) isidentified with S × [0 , { , } × S = M z ([ a ] , [ a ]) × ˘ M red+ z ([ a ] , [ a ]) a ˘ M red+ z ([ a ] , [ a ]) × M z ([ a ] , [ a ])and z and z are the corresponding trivial homotopy classes.Given a family of metrics g P and perturbations p P as above, we can form the parametrizedspace of broken X -trajectories M + ( X ∗ , [ C ]) P = [ p { p } × M + ( X ∗ , [ C ]) p which can be given a natural topology using the fact that the metric and perturbation on thecylindrical part are independent of p . The main compactness and finiteness theorem is thenthe following (see Theorem 24 . . Theorem . Suppose that the families of metric and perturbations { ( g p , p p ) } p ∈ P isregular. Then for each [ C ] the family of moduli spaces M + z ( X ∗ , [ C ]) P is proper over P . Forfixed [ C ] , this family of moduli spaces is non empty for only finitely many components z ∈ π ( B σk ( X, [ C ])) . We discuss the structure of the compactifications as spaces stratified by manifolds (seeProposition 24 . .
10 in the book). The space M + z ( X ∗ , [ C ]) P can be written as the disjointunion of the following subspaces. For any other critical submanifold [ C ] and strata M α ofthe stratification of ˘ M + ([ C α ] , [ C α ]), we consider the subspace M ′ of pairs ([ γ ] , [ ˘ γ ]) such that • [ γ ] ∈ M ( X ∗ , [ C ]) P ; • [ ˘ γ α ] ∈ M α ; • for each α , the evaluations ev α [ γ ] and ev − [ ˘ γ α ] agree.The space M ′ has a natural structure of smooth manifold (with boundary, if P has boundary)induced by its fibered product description, as the evaluation maps are transverse by the regu-larity assumption. These subspaces define a decomposition of the space in smooth manifolds,and this defines a structure of stratified space, as the next result states. For a typical element([ γ ] , [ ˘ γ ]) ∈ M z ( X ∗ , [ C ]) P × Y α ˘ M + z α ([ C α ] , [ C α ]) . THE MODULI SPACE ON A COBORDISM 91 we write n α for the number of components of [ ˘ γ α ], and denote its i th component by [˘ γ αi ] Proposition . Suppose a regular family of perturbations and metrics parametrizedby P is fixed. If M z ( X ∗ , [ C ]) P contains irreducible solutions and has dimension d , then M + z ( X ∗ , [ C ]) P is a d -dimensional space stratified by manifolds, with top stratum the irre-ducible part of M z ( X ∗ , [ C ]) P . The ( d − -dimensional stratum in M + z ( X ∗ , [ C ]) P consists ofelements of the following types. • The elements with n α = 1 for exactly one index α = α ∗ and all the others n α zero. Inthis case neither [ γ α ∗ ] or [ γ ] are boundary obstructed. • The elements with n α = 2 for exactly one index α = α ∗ and all the others n α zero. Inthis case, [ γ α ] is boundary obstructed but [ γ α ] and [ γ ] are not. • The elements with [ γ ] boundary obstructed of corank c . In this case n α = 1 for exactly c + 1 indices α , and n α = 0 in the other cases. If n α = 1 then the trajectory [˘ γ α ] is notboundary obstructed. • The unbroken reducible solutions, if the moduli space contains both reducibles and irre-ducibles. • The unbroken irreducible solutions lying over ∂P , if P has boundary.In the first three cases above, if any of the moduli spaces involved contains both reducibles andirreducibles then only the irreducibles contribute to the ( d − -dimensional stratum. As in the case of broken trajectories, the gluing properties along a stratum involve theconstruction of local thickenings of the moduli spaces, which have the additional propertiesthat the evaluation maps extend to them. The local structure along the strata is slightlymore general than that of Theorem 5.3 and Proposition 5.6 because of the higher corankboundary obstructed trajectories. The following is Definition 24 . . KM07 ], and is usefulto describe the structure along the codimension one strata of the moduli space.
Definition . Let N be a d -dimensional space stratified by manifolds and M d − aunion of ( d −
1) dimensional components. We say that N has a codimension- c δ -structure along M d − if there is an open set W ⊃ M d − a topological embedding j : W → EW and amap S = ( S , . . . , S c +1 ) : EW → (0 , ∞ ] c +1 with the following properties:(1) the fiber along ∞ is identified with j ( M d − ) and S is a topological submersion alongit (see Definition 5.1);(2) the subset j ( W ) ⊂ EW is the zero set of a continuous map δ : EW → Π c whereΠ c ⊂ R c +1 is the hyperplace { δ ∈ R c +1 | P δ i = 0 } ;(3) if e ∈ EW has S i = ∞ for some index i , then δ i ≤ S ( e ) = ∞ ;(4) on the subset of EW where all the S i are finite, δ is smooth and transverse to zero.The main example to have in mind is given by EW = { x | x i ≥ i } ⊂ R c +1 , S i = 1 /x i and δ i = cx i − X j = i x j . In this case the zero locus of δ is given by the half-line W ⊂ EW where all the x i are equal.With this definition the gluing theorem is then the following. Proposition . Suppose a regular family of perturbations and metrics parametrizedby P is fixed. Then the space M P ( X ∗ ; [ C ]) has a codimension- c δ -structure stratum along eachcodimension one stratum. Here we have discussed the simplest possible case, but it is not hard to generalize thisresult to obtain a statement analogous to that of Theorem 5.3. Also as usual the case of thereducible moduli spaces is significantly easier.HAPTER 3
Floer homology for Morse-Bott singularities
In this chapter we construct monopole Floer homology when the singularities of the per-turbed Chern-Simons-Dirac functional are Morse-Bott. There are in literature many differentapproaches to the definition of the homology on a smooth finite dimensional manifold equippedwith a Morse-Bott function. In the present work we will follow that of Fukaya ([
Fuk96 ]),which was also introduced for gauge-theoretic purposes (see also Hutchings’ lecture notes[
Hut ]). First, he introduces a variant of the chain complex of a smooth manifold consistingof smooth maps from simplicial complexes which are transverse to a given countable collec-tion of smooth maps. Then he defines the chain complex for the original manifold as thedirect sum of the modified chain complexes of the critical submanifolds where the simplicialcomplexes are required to be transverse to the evaluation maps of the moduli spaces. Thedifferential is given by the push-forward of these chains via the flow defined through a fiberproduct construction.The choice of Fukaya’s construction is mainly motivated by the fact that the transversalitythat we can generically achieve is not sufficient to pursue other constructions which requirethe evaluation maps to be submersions. Furthermore our compactified moduli spaces arenot smooth compact manifolds with corners. Also, the boundary obstructedness phenomenamakes other approaches unfeasible. On the other hand, Fukaya’s approach can be readilyadapted to our setting, and that is the content of the first two section of the present chapter.In Section 1 we introduce the notion of abstract δ -chain , and construct a variant of the singu-lar homology of a smooth manifold M obtained by considering smooth maps from stratifiedspaces with nice properties into M which are transverse to a given collection of smooth maps.In Section 2 we use these in order to define the Morse-Bott chain complexes associated to athree manifold, and prove the basic properties of the invariants following Chapter 22 of thebook. In Section 3 we construct the maps induced by a cobordism and use these to prove theinvariance of the homology, which shows among the other things that the new approach givesrise to the same invariants defined in the book. It is worth noting that our definition of themodule structure will follow the construction in [ Blo10 ] which fits our problem better.From now on we will work over F , the field with two elements.
1. Homology of smooth manifolds via stratified spaces
In this section we discuss an alternative construction of the homology of a smooth manifoldfollowing the treatment of [
Lip14 ]. The next definition is rather awkward but should not
934 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES surprise the reader, as it is made to fit the results of Theorem 5.3 and Proposition 6.14 in theprevious chapter.
Definition . A topological space N d is a d -dimensional abstract δ -chain if it is astratified space N d ⊃ N d − ⊃ · · · ⊃ N ⊃ N − = ∅ of dimension d with the following additional structures. We are given a finite partition ofeach stratum N e \ N e − = m e a i =1 M ei for each e = 0 , . . . d , and we call the closure of each M ei an e dimensional face . We denote thetop stratum of a face ∆ by ˚∆. The set of faces satisfies the following combinatorial property:whenever a codimension e face ∆ ′′ is contained in a codimension e − e − ′′ .Furthermore, each pair of faces ∆ ′ ⊂ ∆ has an associated finite set N (∆ , ∆ ′ ), togetherwith a subset O (∆ , ∆ ′ ) ⊂ N (∆ , ∆ ′ ), where the O is for obstructed, and their local structuresatisfy the following properties. There is an open neighborhood of ˘ W (∆ , ∆ ′ ) of ˚∆ ′ inside ∆together with a topological embedding j inside a space E ˘ W (∆ , ∆ ′ ) endowed with a topologicalsubmersion S : E ˘ W (∆ , ∆ ′ ) → (0 , ∞ ] N (∆ , ∆ ′ ) , called the local thickening , such that:(1) the map S is a topological submersion along the fiber over ∞ , and this is identifiedwith ˚∆ ′ ;(2) the image j ( ˘ W ) ⊂ E ˘ W is the zero set of a map δ : E ˘ W → R O (∆ , ∆ ′ ) vanishing along the fiber of S over ∞ ;(3) calling ˘ W o ⊂ ˘ W and E ˘ W o ⊂ E ˘ W the subsets where none of the components of S isinfinite, the restriction of j to ˘ W o is a smooth embedding, and the restriction of δ to E ˘ W o is transverse to zero.This collection of local thickenings is compatible in the following sense. Whenever we havethree faces ∆ ′′ ⊂ ∆ ′ ⊂ ∆, we have a canonical inclusion N (∆ ′ , ∆ ′′ ) ֒ → N (∆ , ∆ ′′ )with identifications O (∆ ′ , ∆ ′′ ) = O (∆ , ∆ ′′ ) ∩ N (∆ ′ , ∆ ′′ )and (0 , ∞ ] N ((∆ ′ , ∆ ′′ ) = { ( x , . . . , x N (∆ , ∆ ′′ ) ) | x α = ∞ if α N (∆ ′ , ∆ ′′ ) } ⊂ (0 , ∞ ] N ((∆ , ∆ ′′ ) so that we also have the identifications E ˘ W (∆ ′ , ∆ ′′ ) ≡ E ˘ W (∆ , ∆ ′′ ) ∩ S − (cid:16) (0 , ∞ ] N (∆ ′ , ∆ ′′ ) × {∞} (cid:17) S (∆ ′ , ∆ ′′ ) ≡ S (∆ , ∆ ′′ ) | E ˘ W (∆ ′ , ∆ ′′ ) : E ˘ W (∆ ′ , ∆ ′′ ) → (0 , ∞ ] N (∆ ′ , ∆ ′′ ) δ (∆ ′ , ∆ ′′ ) ≡ δ (∆ , ∆ ′′ ) | E ˘ W (∆ ′ , ∆ ′′ ) : E ˘ W (∆ ′ , ∆ ′′ ) → R O (∆ ′ , ∆ ′′ ) . . HOMOLOGY OF SMOOTH MANIFOLDS VIA STRATIFIED SPACES 95 In the last line we are implicitly stating that the restriction of δ (∆ , ∆ ′′ ) has image contained inthe subspace corresponding to R O (∆ ′ , ∆ ′′ ) under the identifications above. Finally, whenever∆ ′ ⊂ ∆ has codimension one, for each face ∆ the data above defines a codimension- c δ -structure along ∆ ′ in the sense of Definition 6.13 of Chapter 2, in the sense that there is anidentification of R O (∆ , ∆ ′ ) with the subspace Π c ⊂ R c +1 .Thinking about our moduli spaces the set N (∆ , ∆ ′ ) describes the parameters along whichthe trajectories in ∆ can break in order to become trajectories in ∆ ′ , and the subset O (∆ , ∆ ′ )keeps track of which of these are boundary obstructed. Example . It is clear that any manifold with corners is an abstract δ -chain by takingthe thickening of the neighborhood ˘ W to be the neighborhood itself. In this case, eachsubset O (∆ , ∆ ′ ) is empty. Also, the disjoint union of abstract δ -chains (with the obviousdecomposition in faces) and each face of an abstract δ -chain are again abstract δ -chains. Fora more interesting example any space of broken unparametrized trajectories ˘ M + z ([ C − ] , [ C + ]) isan abstract δ -chain, where the partition of the strata is given by fixing the resting submanifoldsand the relative homotopy classes of the components. The only part which does not followfrom Theorem 5.3 and Corollary 5.5 is the combinatorial condition on the faces, which canbe easily checked case by case. On the other hand, it is important to notice that there is notin general any similar combinatorial property for higher codimension faces. Remark . We will consider two thickenings E ˘ W and E ˘ W of two neighborhoods ˘ W and ˘ W of a face to be equivalent if they coincide (up to isomorphism respecting all the givenstructures) in a smaller neighborhood contained in both. In other words, we will alwaysconsider germs of thickenings.The following result is the version of Stokes’ theorem that we will need, and its prooffollows with no modifications as in Section 21 . Proposition . Let ∆ be a -dimensional abstract δ -chain. Then the union of its zerodimensional faces consists of an even number of points. We also have the following definition.
Definition . A homeomorphism ϕ : ∆ → ∆ between two abstract δ -chains is an isomorphism if the following hold: • it maps strata to strata, and each of these maps is a diffeomorphisms; • it induces a bijection between faces; • given faces ∆ ′′ ⊂ ∆ ′ and ∆ ′′ ⊂ ∆ ′ such that ϕ (∆ ′ ) = ∆ ′ and ϕ (∆ ′′ ) = ∆ ′′ , thehomeomorphism ϕ | ˘ W (∆ ′ , ∆ ′′ ) : ˘ W (∆ ′ , ∆ ′′ ) → ˘ W (∆ , ∆ ′′ )extends to a homeomorphism Eϕ | E ˘ W (∆ ′ , ∆ ′′ ) : E ˘ W (∆ ′ , ∆ ′′ ) → E ˘ W (∆ ′ , ∆ ′′ )commuting with the maps S and δ and is a diffeomorphism in a neighborhood of j ( ˘ W o ) ⊂ E ˘ W o . As in the previous remark, we are only interested in the germ of the extensions of the home-omorphism ϕ to the thickening.To define homology, we need to define maps from a given abstract δ -chain inside a smoothmanifold. This is done in order to fit the properties of evaluation maps, following Proposition5.6 in Chapter 2. Definition . Let X be a (possibly non compact) smooth manifold without boundary.A δ -chain in X is a pair σ = (∆ , f ) where • ∆ is an abstract δ -chain, and f : ∆ → X is a continuous map; • the restriction of f to each stratum of ∆ is a smooth map; • for each pair of faces ∆ ′ ⊂ ∆ ′′ , the map extends to a continuous map Ef (∆ ′ , ∆ ′′ ) onthe local thickening E ˘ W (∆ ′ , ∆ ′′ ) which is smooth in a neighborhood of ˘ W (∆ , ∆ ′′ ); • the collection of extensions to the local thickenings is compatible, in the sense that forevery triple ∆ ⊃ ∆ ′ ⊃ ∆ ′′ the map Ef (∆ ′ , ∆ ′′ ) is the restriction of Ef (∆ , ∆ ′′ ) underthe identification of Definition 1.1.As before, we consider the germ of the extensions of the map to the thickening. We saythat two δ -chains σ = (∆ , f ) and σ ′ = (∆ ′ , f ′ ) are equivalent if there exists an isomorphism ϕ : ∆ → ∆ ′ such that f ′ ◦ ϕ = f . We denote the isomorphism class of σ by [ σ ] = [∆ , f ].We also introduce the notion of transversality in our context. Definition . We say that two δ -chains σ = (∆ , f ) and σ = (∆ , f ) are transverse if the following hold: • the restrictions to each pair of strata f | M and f | M are transverse smooth maps; • for each pair of faces ∆ ′ ⊃ ∆ ′′ and ∆ ′ ⊃ ∆ ′′ , the extensions Ef (∆ ′ , ∆ ′′ ) and Ef (∆ ′ , ∆ ′′ ) are transverse in a neighborhood of ˘ W o × ˘ W o .We denote their fibered product as σ × σ .We have the following easy result. Lemma . The fibered product σ × σ has a natural structure of a δ -chain in X . Proof.
We show that the fibered product is an abstract δ -chain. The strata of the fiberedproduct are the fibered products of the strata on each factor, and they inherit a naturalsmooth structure because of the transversality hypothesis. Similarly, the local thickenings ofthe fibered product are the fibered products of the local thickenings, and the correspondingsets of components are obtained by concatenation. It is then easy to prove that the desiredproperty hold. For example, the map( δ , δ ) : E ˘ W × X E ˘ W → R O × R O is transverse to zero in a neighborhood of ˘ W o × X ˘ W o because the maps Ef and Ef areextensions of maps which are already transverse. (cid:3) We are now ready to define a variant of the singular chain complex of a smooth (possiblynon compact) manifold X . Suppose we are given a countable collection of pairs F = { σ α = (∆ α , f α ) } . HOMOLOGY OF SMOOTH MANIFOLDS VIA STRATIFIED SPACES 97 of δ -chains in X . A geometric F -transverse chain of X of dimension d is a d -dimensional δ -chain σ which is transverse to all the chains in the family F . We define ˜ C F d ( X ) to be the F -vector space generated by all geometric transverse chains of dimension d up to isomorphism after we quotient out by the relations(∆ , f ) + (∆ ′ , f ′ ) ∼ (∆ ∐ ∆ ′ , f ∐ f ′ ) . Remark . Notice that this definition has some set theoretic issues, as the collection ofall abstract δ -chains is not a set. Nevertheless we can restrict ourselves to consider a smallercollection which form a set. For example for this section we can consider all the manifoldswith corners contained in a fixed Hilbert space (with the induced smooth structure). In therest of the work, we can consider a slightly bigger family closed under fibered products withthe compactified moduli spaces of trajectories.We denote its elements by [ σ ] = [∆ , f ]. We can then define the linear map˜ ∂ : ˜ C F d ( X ) → ˜ C F d − ( X )[∆ , f ] X ∆ ′ [∆ ′ , f | ∆ ′ ] , where the sum is taken over all codimension one faces ∆ ′ ⊂ ∆. This map well defined becausethe restriction of f to each face is still a smooth transverse map by definition. It is clear fromthe definition of geometric stratified space (and in particular the combinatorial condition onthe set of faces) that ∂ is zero, hence the pair ( ˜ C F∗ ( X ) , ∂ ) is a chain complex. The keydefinition from [ Lip14 ] is the following.
Definition . A d -dimensional chain [ σ ] ∈ ˜ C F d ( X ) is called F - small if it has a repre-sentative ∐ (∆ i , f i ) such that the subset S f i (∆ i ) is contained in the image f (∆) of a δ -chainof dimension j < d which is transverse to F . We say that a chain [ σ ] is negligible if both [ σ ]and ∂ [ σ ] are small.For example, the chain represented by the only map to the point[0 , → ∗ is negligible. Define the subspace N F d ( X ) ⊂ ˜ C F d ( X )as the subspace generated by all negligible chains. As ∂ [ σ ] = 0 is clearly small, we havethat the boundary of a negligible chain is again negligible. We define then ( C F∗ ( X ) , ∂ ) to bethe quotient of ( ˜ C F∗ ( X ) , ∂ ) by the subcomplex generated by negligible chains, and denote itshomology by H F∗ ( X ). Remark . It is useful to notice that C F k ( X ) is trivial for k ≥ dim( X ) + 2.The following is the main result of the present section. Proposition . The homology H F∗ ( X ) is canonically isomorphic to the singular ho-mology H ∗ ( X ; F ) . Before proving this result we state the main transversality result.
Lemma . Suppose we are given a countable family of δ -chains F . Given any δ -chain (∆ , f ) , a natural number k and ε > there exists another δ -chain (∆ , f ′ ) which transverseto all the δ -chains in the family F and is ε -close to the original one in the C k topology oneach stratum. The map f ′ is homotopic to f , and there is a chain (∆ × [0 , , F ) such that therestrictions of F to ∆ × { } and ∆ × { } are respectively f and f ′ . Finally, if f is F -small(negligible), we can choose f ′ to be also F -small (negligible). Proof.
The only tricky point in the result is to preserve smallness (negligibility) in thesmall perturbation. Choose a smooth map F : X × P → X where ( P, p ) is a pointed smooth connected manifold such that • each x in X the differential of the map F ( x, − ) defined on P is a submersion; • the map F ( − , p ) is the identity.Given a δ -chain (∆ , f ) we can consider the family of chains with values in X parametrized by p in P given by (∆ , F ( f ( − ) , p )). If the original chain was small (negligible) then each of theseis small (negligible), and the result follows by a standard application Sard’s theorem. (cid:3) Remark . If we suppose that our original chain (∆ , f ) was already transverse, theproof shows that we can also arrange the chain (∆ × [0 , , F ) to be transverse by choosing ageneric path in the manifold P . Proof of Proposition 1.12.
We first consider the case in which F = ∅ . FollowingChapter 4 [ Sch93 ], we just need to show that the homology groups satisfy the classicalEilenberg-Steenrod axioms for homology for a restricted family of pairs of manifolds. We calla pair of smooth manifolds (
X, A ) admissible if X is without boundary and A is either a closedsubmanifold or a codimension zero submanifold whose boundary is a closed submanifold. It isimmediate to extend the definition of the homology to the relative case, obtaining the group H ∅∗ ( X, A ) as the homology of the quotient complex C ∅∗ ( X ) /C ∅∗ ( A ). It is also clear that asmooth map of pairs ϕ : ( X, A ) → ( Y, B )induces a map in homology ϕ ∗ : H ∅∗ ( X, A ) → H ∅∗ ( Y, B )by composition. If fact, if [ σ ] is small, so its image is contained in f (∆) for a smallerdimensional δ -chain [∆ , f ], then the image of ϕ ∗ [ σ ] is contained in ϕf (∆), hence is also small.The desired result then follows once we prove that the functor H ∅∗ satisfies the followingaxioms: • the existence of a natural, long exact homology sequence for the pair; • the homotopy invariance; • the invariance under excision; • the dimension axiom.The proof of the first two axioms is straightforward. To prove the dimension axiom, weneed to show that every chain of dimension bigger or equal than one with values in a spaceconsisting of a single point is negligible. This is obvious if the dimension is at least two, andin the case of a one dimensional chain it follows from Proposition 1.4. In order to prove the . HOMOLOGY OF SMOOTH MANIFOLDS VIA STRATIFIED SPACES 99 excision invariance we need to show that for any collection U = { U j } of subspaces of X suchthat their interiors form an open cover of X the inclusion C ∅ , U∗ ( X ) ֒ → C ∅∗ ( X )induces an isomorphism in homology, where the left hand side is the subspace generated by δ -chain whose image is contained in some of the U j . Given any δ -cycle σ = (∆ , f ), by applyingLemma 1.13 we can find by compactness a finite collection of closed balls { B i } which coverthe image of f (∆) such that the submanifolds { ∂B i } and σ are transverse, and each { B i } iscontained in the interior part of some set of U . Setting [ B i \ [ ∂B i = a D j we have that X j [ f − ( ¯ D j ) , f | f − ( ¯ D j ) ]is a cycle in C ∅ , U∗ ( X ) whose image in C ∅∗ ( X ) is homologous to σ via the (subdivision of) σ × I ,hence the induced map in homology is surjective. Suppose now σ is a δ -cycle in C ∅ , U∗ ( X )which is the boundary of a chain τ in C ∅∗ ( X ). Using Lemma 1.13 as we did before we canperturb it to a new chain ˜ τ in C ∅ , U∗ ( X ). Furthermore ∂ ˜ τ is a cycle homologous to σ in C ∅ , U∗ ( X )for perturbation small enough, so also injectivity follows.Finally the case of a non empty countable family F follows from the fact that the naturalinclusion C F∗ ( X ) ֒ → C ∅∗ ( X )induces an isomorphism in homology. This is showed as in the proof of the excision propertyabove by applying Lemma 1.13, with the additional observation in Remark 1.14 in order toshow injectivity. (cid:3) Remark . In general, given any two homology theories satisfying the Eilenberg-Steenrod axioms, any isomorphism between the homologies of the point will give rise to anatural equivalence between the homology theories. In our case we have a canonical isomor-phism, because the only automorphism of F is the identity.Cohomology is defined in an analogous way, by considering non necessarily compact ab-stract δ -chains ∆ and proper maps f with values in our smooth manifold X . In particular, wedefine ˜ C q F ( X ) as the space generated by dim X − q dimensional such objects. The boundaryof a cochain is the boundary of the respective abstract δ -chain with the restriction of themaps. We can similarly define the notion of F -small cochain, and define the cochain complex( C ∗F ( X ) , δ ) and its homology H ∗F ( X ). The analogue of Proposition 1.12 says that this is theusual singular cohomology of the space, and with this approach Poincar´e duality for compactmanifolds is tautological.The most important feature of this construction is that one can define an intersectionpairing between homology and cohomology classes(3.1) H F q ( X ) ⊗ H q F ′ ( X ) → F . Given a homology and a cohomology class, one can represent them by a chain and a cochaintransverse to each other (using Proposition 1.13), and define their pairing as their intersectionnumber. This is well defined because of Proposition 1.4.
00 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES
Remark . In general one can define the intersection between classes of differentdimensions, by requiring extra transversality conditions for small and negligible faces (whichare vacuous in the context above), see [
Lip14 ].The following is the analogue of the universal coefficient theorem.
Proposition . The intersection pairing 3.1 is perfect.
The proof of this fact can be achieved in a way completely analogous to the proof ofPoincar´e duality for the singular theory by defining a cap product (with the simplices of afixed triangulation of the manifold). As we will need only the case in which X is a point orthe real projective plane (which can be easily worked out by hand), we will not spell out thedetails of the proof.
2. Floer homology
In this section we define monopole Floer homology using the transverse δ -chains discussedin the previous section. The whole construction of the invariants carries over as in Chapter22 of the book with only minor changes, leading to a priori proofs of the basic properties. Inparticular, we will not rely on the isomorphism with the Kronheimer and Mrowka’s invari-ants (except for the duality results), which will be discussed together with the functorialityproperties of the invariants later in the chapter. This will be useful in the next chapter, whenwe will construct the Pin(2)-theory. We start with a definition. Definition . A Morse-Bott perturbation q is admissible if it is regular in the sense ofDefinition 3.14 in Chapter 2 and in the case c ( s ) is not torsion there are no reducible criticalpoints.Notice that we have implicitly fixed a Riemannian metric g on Y . Given an admissibleperturbation q , we will define three versions of the Floer homology groups d HM ∗ ( Y, s ) , d HM ∗ ( Y, s ) , HM ∗ ( Y, s ) . As the notation suggests, these groups will not depend on the choice of the Riemannianmetric g and the perturbation q , but we will postpone this result to the next section. Theconstruction closely follows Fukaya’s approach ([ Fuk96 ]). Let C ⊂ B σk ( Y, s ) denote the set ofcritical submanifolds of the blow-up gradient of the Chern-Simons-Dirac functional. This canbe written as a disjoint union C o ∪ C s ∪ C u , consisting respectively of irreducible, boundary-stable and boundary-unstable critical sub-manifolds. We have the countable family F of δ -chains (cid:0) M + z ([ C ] , [ C ′ ]) , ev − (cid:1) for each pair of critical submanifolds [ C ] , [ C ′ ] and relative homotopy class z . We define thevector spaces over F given by the direct sum of the chain complexes of the critical submanifolds . FLOER HOMOLOGY 101 (defined in the previous section) C o = M [ C ] ∈ C o C F∗ ([ C ]) C s = M [ C ] ∈ C s C F∗ ([ C ]) C u = M [ C ] ∈ C u C F∗ ([ C ])where the transversality condition is with respect to the family F defined above, and we setˇ C = C o ⊕ C s ˆ C = C o ⊕ C u ¯ C = C s ⊕ C u . As in the Morse case, the vector spaces ˇ C, ˆ C and ¯ C have a grading with values in a set J with an action by Z , which is defined as follows (see Section 22 . I = [ t , t ], configurations a , a ∈ C σk ( Y, s ) and perturbations q , q ∈ P ,we can consider the space C of pairs ( γ, p ) where: • γ ∈ C τk ( I × Y ) is a configurations such that the restriction to { t i }× Y is gauge-equivalentto a i for i = 1 , • p is a continuous path in the Banach space P with p ( t i ) = q for i = 1 , P γ, p = ( Q γ, p , − Π +1 , Π − ) T τ ,γ ( I × Y ) → (cid:0) V τ ,γ ( I × Y ) ⊕ L ( I × Y ; i R ) (cid:1) ⊕ H +1 ⊕ H − where Q γ, p is the linearization of the Seiberg-Witten equations together with gauge fixing ofSection 4 in Chapter 2 (considered on a finite cylinder) and Π +1 , Π − are spectral projections(See Section 20 . g and spin c structure s on Y we define thegrading set as J ( Y, s ) = ( B σk ( Y, s ) × P × Z )) / ∼ where we identify ([ a ] , q , m ) and ([ b ] , q , n ) if there exists ( γ, p ) connecting ([ a ] , q ) to ([ b ] , q )such that the operator P γ, p has index n − m . The map([ a ] , q , m ) ([ a ] , q , m + 1)descends to the quotient defining a Z action on J ( s ). For a fixed Morse-Bott perturbation q ,it is clear that if [ a ] and [ b ] are in the same critical submanifold [ C ] then we can identify([ a ] , q , n ) ∼ ([ b ] , q , n ) . For a δ -chain [ σ ] in C F d ([ C ]) we can then define its grading asGr[ σ ] = ([ a ] , q , d ) / ∼∈ J ( s ) , for any choice of a point [ a ] in [ C ]. By the additivity of the index and grading for any path z joining [ C ] to another critical submanifold [ C ′ ] that for every [ σ ′ ] in C F d ′ ([ C ])Gr[ σ ] = Gr[ σ ′ ] + gr z ([ C ] , [ C ′ ]) + ( d − d ′ ) ∈ J ( s ) .
02 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES
For a δ -chain σ in a reducible critical submanifold we can also introduce a modified gradingby defining Gr[ σ ] = ( Gr[ σ ] , [ σ ] ∈ C s Gr[ σ ] − , [ σ ] ∈ C u . We can decompose each of C o , C s and C u using the grading Gr as a direct sum over thecomponents C oj , C sj and C uj generated by critical points of grading j ∈ J ( s ), and then defineˇ C j = C oj ⊕ C sj ˆ C j = C oj ⊕ C uj ¯ C j = C sj ⊕ C uj +1 (notice that the last subspace is homogeneous for the modified grading). Remark . Even though there is no distinguished element in J ( s ), there is a canonicalmap J ( s ) → Z / Z which we can use to define a canonical mod two grading, see Section 22 . J ( s ) with a more geometric object, namely the set of homotopyclasses of plane distributions on Y , see Chapter 28 in the book. In the case c ( s ) is torsion,one can also define absolute Q -gradings (see Section 28.3 in the book). We will discuss thiscase in the next chapter, where this additional structure will turn out to be decisive.We now define the differential. Given an F -transverse δ -chain [ σ ] = [∆ , f ] in a criticalsubmanifold [ C ], for every moduli space ˘ M + z ([ C ] , [ C ′ ]) the fibered product σ × (cid:16) ˘ M + z ([ C ] , [ C ′ ]) , ev − (cid:17) is an abstract δ -chain which naturally defines δ -chain in [ C ′ ] via the evaluation mapev + : σ × ˘ M + z ([ C ] , [ C ′ ]) → [ C ′ ] . Such a map is transverse to all the evaluation maps ev − with codomain [ C ′ ] because σ is bydefinition transverse to all the fibered products of the moduli spaces, so the δ -chain is again F -transverse. For simplicity, we will denote this δ -chain simply by σ × ˘ M + z ([ C ] , [ C ′ ]). We canthen define the operators ∂ oo : C o ∗ → C o ∗ ∂ os : C o ∗ → C s ∗ ∂ us : C u ∗ → C s ∗ ∂ uo : C u ∗ → C o ∗ . FLOER HOMOLOGY 103 defined on a generator [ σ ] of C F∗ ([ C ]) as ∂ oo [ σ ] = ∂ [ σ ] + X [ C ′ ] ∈ C o [ σ × ˘ M + ([ C ] , [ C ′ ])] ∂ os [ σ ] = X [ C ′ ] ∈ C s [ σ × ˘ M + ([ C ] , [ C ′ ])] ∂ us [ σ ] = X [ C ′ ] ∈ C s [ σ × ˘ M + ([ C ] , [ C ′ ])] ∂ uo [ σ ] = X [ C ′ ] ∈ C o [ σ × ˘ M + ([ C ] , [ C ′ ])]where in the first two cases [ C ] consists of irreducibles while in the last two it consists ofboundary unstable critical points. Notice that in the first case we also have the summandcorresponding to the differential in the δ -chain complex ∂ of the critical manifold introducedin Section 1. Lemma . The maps above are well defined.
Proof.
As remarked above, the fact that the fibered product is transverse to all theevaluation maps follows from the regularity of the moduli spaces. Also, if [ σ ] is negligiblethen also the fibered product [ σ × ˘ M + z ([ C ] , [ C ′ ])] is negligible, because of the transversalitycondition in Definition 1.10. Finally, we want to show that all but finitely many simplices[ σ × ˘ M + z ([ C ] , [ C ′ ])] are negligible, so that the sums are finite. Because there are only finitelymany critical submanifolds in the blow down, we can focus in the case of reducible criticalsubmanifolds. Suppose [ σ × ˘ M + z ([ C ] , [ C ′ ])] with [ C ′ ] reducible (and blowing down to [ C ]) is nonempty. Then if [ C ′′ ] is a critical submanifold also blowing down to [ C ] and corresponding toan eigenvalue negative enough, then Lemma 3.16 in Chapter 2 tells us that [ σ × ˘ M + z ([ C ] , [ C ′′ ])]has dimension bigger than dim[ C ′′ ] + 2, hence it is negligible. (cid:3) Similarly, in the case when both critical submanifolds are reducible, we can define theoperators ¯ ∂ ss : C s ∗ → C s ∗ ¯ ∂ su : C s ∗ → C u ∗ ¯ ∂ us : C u ∗ → C s ∗ ¯ ∂ uu : C u ∗ → C u ∗
04 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES defined on an element [ σ ] of C F∗ ([ C ]) as¯ ∂ ss [ σ ] = ∂ [ σ ] + X [ C ′ ] ∈ C s [ σ × ˘ M red , + ([ C ] , [ C ′ ])]¯ ∂ su [ σ ] = X [ C ′ ] ∈ C u [ σ × ˘ M red , + ([ C ] , [ C ′ ])]¯ ∂ us [ σ ] = X [ C ′ ] ∈ C s [ σ × ˘ M red , + ([ C ] , [ C ′ ])]¯ ∂ uu [ σ ] = ∂ [ σ ] + X [ C ′ ] ∈ C u [ σ × ˘ M red , + ([ C ] , [ C ′ ])]where in the first two cases [ C ] consists of boundary stable critical points while in the lasttwo it consists of boundary unstable critical points. Notice that the two maps ∂ us , ¯ ∂ us : C u ∗ → C s ∗ involve different moduli spaces. Finally, we define the operatorsˇ ∂ : ˇ C ∗ → ˇ C ∗ ˆ ∂ : ˆ C ∗ → ˆ C ∗ ¯ ∂ : ¯ C ∗ → ¯ C ∗ respectively as ˇ ∂ = (cid:20) ∂ oo ∂ uo ¯ ∂ su ∂ os ¯ ∂ ss + ∂ us ¯ ∂ su (cid:21) ˆ ∂ = (cid:20) ∂ oo ∂ uo ¯ ∂ su ∂ os ¯ ∂ uu + ¯ ∂ su ∂ us (cid:21) ¯ ∂ = (cid:20) ∂ ss ∂ us ∂ su ∂ uu . (cid:21) These operators have degree −
1, and are chain maps as stated by the next result.
Proposition . The squares ¯ ∂ , ˇ ∂ and ˆ ∂ are zero as operators on ¯ C, ˇ C and ˆ C . Proof.
The proof follows from the characterization of the codimension one strata inProposition 4.13 in the same way as Proposition 22 . . ∂ given by A = ∂ oo ∂ oo + ∂ uo ¯ ∂ su ∂ os : C o → C o . . FLOER HOMOLOGY 105 Given an irreducible critical manifold [ C − ] and a transverse δ -chain σ in C F∗ ([ C − ]), we havethat A [ σ ] = ( ∂ F ) [ σ ]+ X [ C + ] ⊂ C o ∂ F [ σ × ˘ M + ([ C − ] , [ C + ])]+ X [ C + ] ⊂ C o [ ∂ F [ σ ] × ˘ M + ([ C − ] , [ C + ])]+ X [ C + ] ⊂ C o X [ C ] ∈ C o [ σ × ˘ M + ([ C − ] , [ C ] , [ C + ])]+ X [ C + ] ⊂ C o X [ C ] ∈ C s X [ C ] ∈ C u [ σ × ˘ M + ([ C − ] , [ C ] , [ C ] , [ C + ])] , where we used that the strata of the compactified moduli spaces are defined as the fiberedproducts. It is clear that the term in the first row is zero. On the other hand, the codimen-sion one faces of the δ -chain [ σ × ˘ M + z ([ C − ] , [ C + ])] are by definition the fibered product of acodimension one face in one factor and the whole space in the other. Hence by Proposition4.13 in Chapter 2 its boundary in C F∗ ([ C + ]) is given by the sum[ ∂ F [ σ ] × ˘ M + z ([ C − ] , [ C + ])]+ X [ C ] ∈ C o [ σ × M + z ([ C − ] , [ C ] , [ C + ])]+ X [ C ] ∈ C s X [ C ] ∈ C u [ σ × M + z ([ C − ] , [ C ] , [ C ] , [ C + ])]where the last term involves boundary obstructed moduli spaces. This completes the proof. (cid:3) Definition . We define the monopole Floer homology groups of Y as the homologiesof the three graded chain complexes ( ˇ C ∗ , ˇ ∂ ) , ( ˆ C ∗ , ˆ ∂ ) and ( ¯ C ∗ , ¯ ∂ ): d HM ∗ ( Y, s ) = H ( ˇ C ∗ , ˇ ∂ ) d HM ∗ ( Y, s ) = H ( ˆ C ∗ , ˆ ∂ ) HM ∗ ( Y, s ) = H ( ¯ C ∗ , ¯ ∂ ) . The choice of metric and perturbation ( g, p ) is implicit in our notation, and we will makeit explicit when needed.The rest of the section is dedicated to the study of the basic properties of these objects,which will follow as those in Chapter 22 in the book in exactly the same way. We exposethe main constructions (without going too deep into details) as they will be useful for ourpurpose in the next chapter. Before doing this, we show that in the case of a non-degenerateperturbation our construction gives the same result. Lemma . Suppose the admissible perturbation q is chosen so that all critical pointsare non-degenerate. Then the three flavors of monopole Floer homology coincide as gradedgroups with those defined in the book.
06 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES
Proof.
In the case the perturbation is non degenerate the chain complex is exactlythe one defined in the book. Indeed by Proposition 1.4 if the manifold is a point only zerodimensional chains can be not negligible. Hence the chain complex of each critical submanifoldconsists of a single F in degree zero, and the moduli spaces of dimension one or higher do notcontribute as any fibered product is necessarily negligible. (cid:3) Exact sequences.
The monopole Floer homology groups should be thought as half-infinite-dimensional homology groups of the moduli space of configurations( B σk ( Y, s ) , ∂ B σk ( Y, s )) , and many of the basic properties of the usual homology groups can be performed. We firststate the exact sequence for the pair in homology. Proposition . For any ( Y, s ) , there is an exact sequence · · · i ∗ −→ d HM k ( Y, s ) j ∗ −→ d HM k ( Y, s ) p ∗ −→ HM k − ( Y, s ) i ∗ −→ d HM k − ( Y, s ) j ∗ −→ . . . The maps in the sequence are induced by the chain maps i : ¯ C ∗ → ˇ C ∗ , j : ˇ C ∗ → ˆ C ∗ , p : ˆ C ∗ → ¯ C ∗ given in components by i = (cid:20) ∂ uo ∂ us (cid:21) , j = (cid:20) ∂ su (cid:21) , p = (cid:20) ∂ os ∂ us (cid:21) , see Section 22 . Filtrations.
The chain complex we have introduced for a Morse-Bott perturbation ismuch bigger than the usual one for a non degenerate perturbation, as it is for example notnecessarily finitely generated in each dimension. When the perturbation satisfies a strongertransversality hypothesis, we can somehow handle the problem better by means of a spectralsequence (see for example [
Fuk96 ] or [
AB95 ]).
Definition . Suppose the spin c structure is torsion. We say that an admissible Morse-Bott perturbation q is weakly self-indexing if for each pair of critical submanifolds [ C − ] , [ C + ]and relative homotopy class z connecting them such that M z ([ C − ] , [ C + ]) is not empty we havethat gr z ([ C − ] , [ C + ]) ≥ ( Proposition . Suppose a spin c structure with c ( s ) torsion and a weakly self-indexingMorse-Bott perturbation q are fixed. Then there is a spectral sequence E ∗ ji where i is a positiveinteger and j is an element of J ( s ) such that E ji = M Gr[ C ]= j, [ C ] ⊂ C o ∪ C s H i ([ C ]; F ); and converges to d HM i + j ( Y, s ) . There are similar spectral sequences for the other flavors. . FLOER HOMOLOGY 107 Proof.
For a fixed critical submanifold [ C ] there is a natural increasing filtration on thechain complex ˇ C ∗ given by F k ˇ C ∗ = M gr z ([ C ] , [ C ]) ≤ k C F∗ ([ C ]) , where the sum is taken over irreducible and boundary stable critical submanifolds. Thedifferential respects this filtration because of the weakly self indexing hypothesis, and thespectral sequence in the proposition is exactly the one induced by this filtration. The spectralsequence converges for the same reason why the differential is well defined, see Lemma 2.3. (cid:3) Example . We study the case of S with the round metric, see Section 22 . q so that no irreducible critical point is introduced. The critical submanifolds consist of aninfinite sequence of projective spaces all lying over the same critical reducible critical point.By the results of Lemma 3.16 in Chapter 2, it is clear that such a perturbation is weaklyself-indexing, and for dimensional reasons the sequence collapses at the first page. Hence,we obtain again the calculation of Section 22 . c structure) induced by the values ofthe perturbed Chern-Simons-Dirac functional −L . Let these be α < · · · < α m , and consider the energy filtration given by G k ˇ C ∗ = M − L ([ C ]) ≤ α k C F∗ ([ C ]) , which is well defined because the value of −L is always decreasing along a trajectory. Noticethat this filtration does not preserve gradings.This filtration turns in handy when we need to restrict our attention to δ -chains whichare transverse to a larger family of evaluation maps, as described in the next definition. Definition . We say that a countable family of δ -chains F ′ containing the F , theone given by the evaluation maps of the moduli spaces of broken unparametrized trajectories,is a compatible extension if • each σ = (∆ , f ) in F ′ \ F is transverse to all evaluation maps ev + of the moduli spacesof trajectories; • for each such σ and any moduli space M = ˘ M + ([ C ] , [ C ′ ]) the fibered product( σ × ( X, ev + ) , ev − )is also an element of F ′ .We are especially interested in the case when the family F ′ \ F is consists of the modulispaces on a cobordism equipped with the evaluation on the incoming end. This forms acompatible extension exactly when the perturbation is regular in the sense of Definition 6.6in Chapter 2.
08 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES
Lemma . Consider compatible extension F ′ ⊃ F . Then the subspace defined as ˇ C ′∗ ( Y, s ) = ˇ C ∗ ( Y, s ) ∩ (cid:16)M C F ′ ∗ ([ C ]) (cid:17) . is a subcomplex and the inclusion map is a quasi-isomorphism. The same statement holds forthe other versions. Proof.
It is clear from the definition that the this is a subcomplex. To prove that theinclusion in a quasi isomorphism, consider first the case when the spin c structure is torsion.We can then consider the energy filtration for both chain complexes. The inclusion map i isthen a map of filtered chain complexes, and our goal is to show that the induced map on the E page(3.2) i ∗ : H ( G j ˇ C ′∗ / G j − ˇ C ′∗ ) → H ( G j ˇ C ∗ / G j − ˇ C ∗ )is an isomorphism. In fact, this implies that the map induced on the E ∞ page is an isomor-phism, hence so is the map induced in homology. We have the splitting as chain complexes G j ˇ C ∗ / G j − ˇ C ∗ = M − L ([ C ])= α j M π ([ C ])=[ C ] C F∗ ([ C ]) because we are only considering the trajectories with on which −L is constant, hence we justneed to study each of the summands. In the irreducible case, i ∗ is an isomorphism becauseof Proposition 1.12. In the reducible case the chain complexes M π ([ C ])=[ C ] C F ′ ∗ ([ C ]) ֒ → M π ([ C ])=[ C ] C F∗ ([ C ])have a filtration given by the ordering of the corresponding eigenvalues. The map inducedon the first page of the associated spectral sequence is again an isomorphism thanks toProposition 1.12, so the result follows from the same reason as above.The case of a non-torsion spin c structure is slightly more complicated because there isnot an energy filtration as the Chern-Simons-Dirac functional is circle valued. To tackle thisissue, we follow the nice approach of [ FS92 ] by constructing an auxiliary chain complex withan absolute Z -grading. Consider any value ϑ ∈ R / (4 π Z ) such that there are no criticalsubmanifolds [ C ] with −L ([ C ]) = ϑ . We define the chain complex (cid:16) ˇ C ϑ ∗ ( Y, s ) , ˇ ∂ ϑ (cid:17) whose underlying vector space is ˇ C ( Y, s ) but the differential only counts moduli spaces con-sisting of unparametrized broken trajectories ˘ γ such that −L never achieves the value ϑ alongthem. There are no differences in the proof of the fact that this is actually a chain complex.Furthermore on such a complex there is a well defined energy filtration, hence we can proveas above that its homology does not depend on the choice of the family F ′ . The result willfollow if we can find a filtration on ( ˇ C ∗ ( Y, s ) , ˇ ∂ ) such that the associated spectral sequence isconvergent and the E page of the is the homology of this modified chain complex.Let d ∈ N + the generator of the image of the evaluation of c ( s ) on H ( Y, Z ). For a fixedcritical submanifold [ C ] we can define the Z -valued gradinggr ϑ ([ C ]) = gr z ϑ ([ C ] , [ C ]) ∈ Z , . FLOER HOMOLOGY 109 where z θ is a relative homotopy class whose image under −L avoids ϑ . This is well definedbecause the quantity(3.3) E q ( z ) + 4 π gr z ([ C ] , [ C ′ ])is independent of the homology class, see Lemma 16 . . Z /d Z grading of ˇ C ∗ ( Y, s ) can be lifted to a (not canonical) absolute Z -grading,which we denote by the underlined index. For l ∈ Z /d Z the (increasing) filtration for s congruent to l modulo d we set G s ˇ C l = M j ≥ ˇ C s − jd . It follows from the definition of the grading gr ϑ and the invariance of the quantiy (3.3) thatthis is a filtration on the Z /d Z -graded chain complex, i.e.ˇ ∂ (cid:0) G s ˇ C l (cid:1) ⊂ G s − ˇ C l − . The E page coincides with ( ˇ C ϑ ∗ ( Y, s ) , ˇ ∂ ϑ ), and the filtration is bounded because of the finite-ness properties of the moduli spaces (Proposition 4.9 in Chapter 2), so the associated spectralsequence converges. (cid:3) Cohomology and duality.
The main drawback of our approach to the Morse-Bott case isthat it is not clear how to define a duality map at the chain level because of the transversalityconditions needed to define the intersection product. For this reason, we will define thecohomology tautologically as the homology of − Y , i.e. Y with the orientation reversed, anddefine the pairing via intersection. The fact that this pairing is perfect will follow fromthe invariance of the homology, as in the non-degenerate case this is simply the universalcoefficient theorem.Recall that a spin c structure ( S, ρ ) on Y defines the spin c structure ( S, − ρ ) on − Y . Wecan identify the configuration spaces C k ( Y, s ) and C k ( − Y, s ), and if we choose the perturbation − q on Y we have the relation −L ( Y ) = −−L ( − Y ) . We can then identify the critical submanifolds, with the notion of boundary stable and bound-ary unstable switched. We also have the canonical identification(3.4) M z ( Y ; [ C − ] , [ C + ]) = M − z ( − Y ; [ C + ] , [ C − ]) . There is a map o : J ( − Y, s ) → J ( Y, s )([ a ] , n ) → (cid:0) [ a ] , − n − N (Hess σ q , a ) (cid:1) , where N (Hess σ q , a ) is the dimension of the generalized zero eigenspace, which changes the signof the Z action, i.e.(3.5) o ( j + n ) = o ( j ) − n.
10 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES
We then define the cochain complexes asˇ C j ( Y, s ) = ˆ C o ( j ) ( − Y, s )ˆ C j ( Y, s ) = ˇ C o ( j ) ( − Y, s )¯ C j ( Y, s ) = ¯ C o ( j ) ( − Y, s ) , where the differentials ˇ δ, ˆ δ and ¯ δ are the ones induced by this identification. In particular,the relation 3.5 implies that these differentials have degree 1. From this, we can define the Floer cohomology groups d HM ∗ ( Y, s ) = H ( ˇ C ∗ , ˇ ∂ ) d HM ∗ ( Y, s ) = H ( ˆ C ∗ , ˆ ∂ ) HM ∗ ( Y, s ) = H ( ¯ C ∗ , ¯ ∂ ) . It is important to remark that a priori the chain complexes of Y and − Y involve different δ -chains. In fact using the identification in equation (3.4) in the former case they are requiredto be transverse to the maps ev − while in the latter case to the maps ev + .Because of the definition of the map o it is natural to interpret the groups ˇ C ∗ , ˆ C ∗ and¯ C ∗ as the direct sum the cochain complexes of the critical submanifolds with the naturalgrading. Indeed, at a critical point a the quantity N (Hess σ q , a ) is the dimension of the criticalsubmanifold to which the point belongs to. Hence via intersection theory we can define anatural pairing d HM j ( Y, s ) ⊗ d HM j ( Y, s ) → F or, equivalently, a linear map d HM j ( Y, s ) → Hom( d HM j ( Y, s ) , F ) . To see that this is well defined, we need to show that given a cohomology class [ σ ] and ahomology class [ τ ] we can find a cocycle σ in ˇ C k and a cycle τ in ˇ C k representing themwhich are transverse, so that they have a well defined intersection number. By definition,any cocycle σ is transverse to all the evaluation maps ev + . In particular, we can define acompatible extension F ′ for the chain complex ˇ C ∗ computing the homology (see Definition2.11) where F ′ \ F = n ( ˘ M + ([ C ] , [ C ′ ]) × σ, ev − ) o . The existence of a representative τ for the homology class which is transverse to σ follows thenby applying Lemma 2.12 to the family F ′ . A similar argument implies that the intersectionnumber only depends on the classes, and not by the representatives. Finally, this pairingis perfect because of the invariance properties and the universal coefficient theorem in thenon-degenerate case. Completions.
Let G ∗ be an abelian group graded by the set J equipped with a Z action.Let O α ( α ∈ A ) be the set of free Z -orbits in J , and fix an element j α ∈ O α for each α .Consider the subgroups G ∗ [ n ] = M α M m ≥ n G j α − m , . INVARIANCE AND FUNCTORIALITY 111 which form a decreasing filtration of G ∗ . We define the negative completion of G ∗ as thetopological group G • ⊃ G ∗ obtained by completing with respect to this filtration (which isclearly independent of the choice of the j α ). We then define the negative completions d HM • ( Y, s ) , d HM • ( Y, s ) , HM • ( Y, s )of the Floer groups defined above. These also satisfy the properties discussed above, and theyare natural objects to study when dealing with functoriality properties.
3. Invariance and functoriality
The aim of this final section is to construct maps on monopole Floer homology inducedby cobordisms. These are the essential ingredient to study invariance and functoriality. Ourdefinition of the chain complexes is different from the one in the book, and we cannot followthe approach there in order to construct maps induced by a cobordism X and a cohomologyclass u d on its configuration space. On the other hand there is an alternative approach (whichcan be found for example in [ Blo10 ]) leading to the same result which requires the additionof some incoming cylindrical ends of the form( −∞ , × ( S )to the given cobordism W from Y − to Y + . In this case, we will obtain maps d HM • ( W ) : d HM • ( Y − ) ⊗ d HM • ( S ) ⊗ . . . d HM • ( S ) → d HM • ( Y + ) , where each factor still depends on the choice of metric and perturbation. Using these, we willbe able to prove invariance and functoriality.It is important to notice that in general given a manifold with more than one incomingend Y , . . . , Y m and one outgoing end Y + there is not a well-defined induced map d HM • ( Y ) ⊗ · · · ⊗ d HM • ( Y m ) → d HM • ( Y + ) , as there are substantial issues with the combinatorics of the codimension one strata. On theother hand, it is shown in [ Blo13 ] that the combinatorics of the degeneration and gluing ofmoduli spaces work out in order to define the map d HM • ( Y ) ⊗ d HM ( Y ) ⊗ · · · ⊗ d HM • ( Y m ) → d HM • ( Y + )and similarly for the map d HM • ( Y ) ⊗ d HM ( Y ) ⊗ · · · ⊗ d HM • ( Y m ) → d HM • ( Y + ) . In particular, Corollary 4 .
15 in that paper provides an extremely general family (with possiblymore incoming and outgoing ends at the same time) in which there is a natural induced map.In our case the situation is much more simple due to the absence of irreducible solutions on S for the right choice of metric and perturbation (see Example 2.10).Suppose we are given a cobordism W from Y − to Y + and consider a metric on W whichis cylindrical near the boundary components. Suppose we are also given a finite number ofmarked points p = { p , . . . , p m } in the interior. Removing a small ball around the markedpoint we can think of each marked point p i to give rise and incoming end S i . We also suppose
12 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES that each of these has a metric with positive scalar curvature (e.g. the round metric), andthat the metric is a product near the end. We will denote the manifold obtained by addingcylindrical ends by W ∗ p . We will refer to the ends corresponding to the marked points aspunctures. Choose on each S i a small Morse-Bott perturbation q i with only one reduciblesolution in the blow down and no irreducible solutions. Finally choose as in Section 6 ofChapter 2 an additional perturbation ˆ p on the cobordism so that all moduli spaces are regular(see Definition 6.6 in that section).To define the maps we need to construct the fibered products with the moduli spaces onthe cobordisms, hence we need to assume more transversality conditions on the set of δ -chainwe are considering. For each ( m + 1)-uple of critical submanifolds in the incoming end[ C − ] , [ C ] , . . . , [ C m ]respectively in Y − and the puncture S i we have a countable family of smooth maps F W = { ( f i , ∆ i ) } f i : ∆ i → [ C − ] × m Y i =1 [ C i ] , where ∆ i is a moduli space on the cobordism M + z ([ C − ] , [ C ] , . . . , [ C m ] , W ∗ p , [ C + ])and the map f i is the product of the evaluation maps ev − to each end. We define the vectorspace C o • ( Y − , p ) as the subspace of C o • ( Y − ) ⊗ C u • ( S ) ⊗ · · · ⊗ C u • ( S m )generated by tensor products of F -transverse δ -chains [ σ − ] ⊗ [ σ ] ⊗ · · · ⊗ [ σ m ] where [ σ − ] ∈ C F∗ ([ C − ]) and [ σ i ] ∈ C F∗ ([ C i ]) such that σ − × σ × · · · × σ m : ∆ − × ∆ × · · · × ∆ m → [ C − ] × m Y i =1 [ C i ]is transverse to all the maps in the family F W . To simplify the notation, we will set C = ([ C ] , . . . , [ C m ])[ σ ] = [ σ ] ⊗ · · · ⊗ [ σ m ] . Similarly, we can define the vector spaces C u • ( Y − , p ) ⊂ C u • ( Y − ) ⊗ C u • ( S ) ⊗ · · · ⊗ C u • ( S m ) C s • ( Y − , p ) ⊂ C s • ( Y − ) ⊗ C u • ( S ) ⊗ · · · ⊗ C u • ( S m ) , defined by the same transversality hypotheses. We can then define the vector spacesˇ C • ( Y − , p ) = C o • ( Y − , p ) ⊕ C s • ( Y − , p )ˇ C • ( Y − , p ) = C o • ( Y − , p ) ⊕ C u • ( Y − , p )ˇ C • ( Y − , p ) = C s • ( Y − , p ) ⊕ C u • ( Y − , p ) . The vector spaces C u • ( S i ) are the underlying vector spaces of the chain complex ˆ C • ( S i ).These complexes associated to the three sphere have a natural Z valued grading so that thehomology is canonically identified with the graded ring F [[ U ]], where U has degree − . INVARIANCE AND FUNCTORIALITY 113 is obtained by a total shift of −
1. This implies that ˇ C • ( Y − , p ) and its companions have anatural J ( Y − ) grading. The next result follows using the energy filtration as in Lemma 2.12(which deals with the case with no punctures). Lemma . The subspace ˇ C ∗ ( Y − , p ) of ˇ C ∗ ( Y − ) ⊗ ˆ C ∗ ( S ) ⊗ · · · ⊗ ˆ C ∗ ( S m ) is a subcomplex,and the inclusion map is a quasi-isomorphism. In particular in many occasions we will need to restrict to a smaller class of δ -chainssatysfying additional transversality hypothesis, and this result (and some slight modificationsof it) will tell us that this will not effect the final result.We can define the maps on the completions m oo : C o • ( Y − , p ) → C o • ( Y + ) m os : C o • ( Y − , p ) → C s • ( Y + ) m uo : C u • ( Y − , p ) → C o • ( Y + ) m us : C u • ( Y − , p ) → C s • ( Y + )by the formulas m oo ([ σ − ] ⊗ [ σ ]) = X [ C + ] ∈ C o ( Y + ) X z ∈ π ([ σ − ] , [ σ ]) × M + z ([ C − ] , C , W ∗ p , [ C + ]) m os ([ σ − ] ⊗ [ σ ]) = X [ C + ] ∈ C s ( Y + ) X z ∈ π ([ σ − ] , [ σ ]) × M + z ([ C − ] , C , W ∗ p , [ C + ]) m uo ([ σ − ] ⊗ [ σ ]) = X [ C + ] ∈ C o ( Y + ) X z ∈ π ([ σ − ] , [ σ ]) × M + z ([ C − ] , C , W ∗ p , [ C + ]) m us ([ σ − ] ⊗ [ σ ]) = X [ C + ] ∈ C s ( Y + ) X z ∈ π ([ σ − ] , [ σ ]) × M + z ([ C − ] , C , W ∗ p , [ C + ]) . It is important to remark that these sums might be potentially infinite, but they arenevertheless well defined on the completions of the chain complexes because of the finitenessresults in Theorem 6.11 in Chapter 2. Similarly, we can define maps¯ m ss : C s • ( Y − , p ) → C s • ( Y + )¯ m su : C s • ( Y − , p ) → C u • ( Y + )¯ m us : C u • ( Y − , p ) → C s • ( Y + )¯ m uu : C u • ( Y − , p ) → C u • ( Y + )
14 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES using the reducible moduli spaces via the sums¯ m ss ([ σ − ] ⊗ [ σ ]) = X [ C + ] ∈ C s ( Y + ) X z ∈ π ([ σ − ] , [ σ ]) × M red+ z ([ C − ] , C , W ∗ p , [ C + ])¯ m su ([ σ − ] ⊗ [ σ ]) = X [ C + ] ∈ C s ( Y + ) X z ∈ π ([ σ − ] , [ σ ]) × M red+ z ([ C − ] , C , W ∗ p , [ C + ])¯ m us ([ σ − ] ⊗ [ σ ]) = X [ C + ] ∈ C u ( Y + ) X z ∈ π ([ σ − ] , [ σ ]) × M red+ z ([ C − ] , C , W ∗ p , [ C + ])¯ m uu ([ σ − ] ⊗ [ σ ]) = X [ C + ] ∈ C u ( Y + ) X z ∈ π ([ σ − ] , [ σ ]) × M red+ z ([ C − ] , C , W ∗ p , [ C + ]) . We then put the pieces together to define the operators between the chain complexesˇ m : ˇ C • ( Y − , p ) → ˇ C • ( Y + )ˆ m : ˆ C • ( Y − , p ) → ˆ C • ( Y + )¯ m : ¯ C • ( Y − , p ) → ¯ C • ( Y + )by the formulas(3.6) ˇ m = (cid:20) m oo m uo ¯ ∂ su ( Y − , p ) + ∂ uo ( Y + ) ¯ m su m os ¯ m ss + m us ¯ ∂ su ( Y − , p ) + ∂ us ( Y + ) ¯ m su (cid:21) ˆ m = (cid:20) m oo m uo ¯ m su ∂ os ( Y − , p ) + ¯ ∂ su ( Y + ) m os ¯ m uu + ¯ m su ∂ us ( Y − , p ) + ¯ ∂ su ( Y + ) m us (cid:21) ¯ m = (cid:20) ¯ m ss ¯ m us ¯ m su ¯ m uu (cid:21) . In the next proposition, we use the natural identification between at the homology levelprovided by Lemma 3.1.
Proposition . The operator ˇ m is a chain map, i.e. it satisfies the relation ˇ ∂ ( Y + ) ◦ ˇ m = ˇ m ◦ ˇ ∂ ( Y − , p ) and the induced map in homology d HM • ( W, p ) : d HM • ( Y − , s − ) ⊗ F [[ U ]] ⊗ · · · ⊗ F [[ U m ]] → d HM ( Y + , s + ) is independent of the choice of the punctures and of the metric on W (isometric to the fixedcylindrical one in a collar of the boundary and standard around the punctures) and perturba-tion. The same statement holds also for the operators ˆ m and ¯ m . Proof.
We first show that the map is a chain map. We will only show that the componentof ˇ m given by B = ∂ oo m oo + m oo ∂ oo + ∂ uo ¯ ∂ su m os + m uo ¯ ∂ su ∂ os + ∂ uo ¯ m su ∂ os : C o • ( Y − , p ) → C o • ( Y + ) . INVARIANCE AND FUNCTORIALITY 115 is zero. Consider [ σ − ] ⊗ [ σ ] ∈ C o • ( Y − , p ). For a given irreducible critical submanifold [ C + ] ⊂B σk ( Y + ), we have that the component of B ([ σ − ] ⊗ [ σ ]) in the summand C F∗ ([ C + ]) is given by ∂ (cid:2) ( σ − , σ ) × M + ([ C − ] , C , W ∗ p , [ C + ]) (cid:3) + X [ C ′ + ] ∈ C o ( Y + ) (cid:2) ( σ − , σ ) × M + ([ C − ] , C , W ∗ p , [ C ′ + ]) (cid:3) × ˘ M + ([ C ′ + ] , [ C + ])+ (cid:2) ( ∂ [ σ − ] , σ ) × M + ([ C − ] , C , W ∗ p , [ C + ]) (cid:3) + "X C ′ ( σ − , ˆ ∂ [ σ ]) × M + ([ C − ] , C ′ , W ∗ p , [ C + ]) + X [ C ′− ] ∈ C o ( Y − ) h(cid:16) ( σ × ˘ M + ([ C − ] , [ C ′− ]) , σ (cid:17) × M + ([ C ′− ] , C , W ∗ p , [ C + ]) i + X [ C ′ + ] ∈ C s + X [ C ′′ + ] ∈ C u + h ( σ − , σ ) × M + ([ C ′− ] , C , W ∗ p , [ C ′ + ]) × ˘ M red+ ([ C ′ + ] , [ C ′′ + ]) × ˘ M + ([ C ′′ + ] , [ C + ]) i + X [ C ′− ] ∈ C s − X [ C ′′− ] ∈ C u − h(cid:16) σ × ˘ M + ([ C − ] , [ C ′− ]) × ˘ M + ([ C ′− ] , [ C ′′− ]) , σ (cid:17) × M + ([ C ′′− ] , C , W ∗ p , [ C + ]) i + X [ C ′− ] ∈ C s − X [ C ′ + ] ∈ C u + h ( σ × ˘ M + ([ C − ] , [ C ′− ]) , σ ) × M + ([ C ′− ] , C , W ∗ p , [ C ′ + ]) × ˘ M + ([ C ′ + ] , [ C + ]) i . Here, in the second term of the third row we sum over the sequences C ′ that differ from C byat most one element (notice that for dimensional reasons very few terms actually contributeto the sum). The term ∂ oo m oo corresponds to the first two rows and m oo ∂ oo corresponds to thethird and fourth row. The remaining three terms correspond each to one of the last threelines. The claim follows because the first term is equal to the sum of all the others. Thisfollows as in the proof of Proposition 2.4 from the fact that the codimension 1 faces of thefibered product of δ -chains is given by the fibered products of a codimension 1 face is onefactor with the other factor, together with the classification of codimension one strata ofthe moduli space of solutions on a cobordism (see Theorem 6.12). Notice that the fact thatthe differentials in the factors corresponding to the punctures only involve reducible modulispaces drastically simplifies the formula. The proof in the other cases is essentially the sameway, see also Lemma 25 . . g , ˆ p ) and( g , ˆ p ), consider a path P connecting them which such that the corresponding parametrizedmoduli spaces are regular in the sense of Definition 6.6. We will consider the chain complexˇ C • ( Y − , p ) P which is the subchain complex ofˇ C • ( Y − ) ⊗ C u • ( S ) ⊗ · · · ⊗ C u • ( S m )consisting of tuples of δ -chains which are transverse to all evaluation maps arising from themoduli spaces with data P (hence also to the ones arising from ( g , ˆ p ) and ( g , ˆ p )). Theinclusion of this chain complex in each of the two subcomplexes of ˇ C ∗ ( Y − , p ) correspondingto a choice of metric and perturbation is a quasi-isomorphism (as in Lemma 3.1). Also, thetwo choices of metrics and perturbation give rise to two chain mapsˇ m , ˇ m : ˇ C • ( Y − , p ) P → ˇ C • ( Y + ) ,
16 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES and our goal is to show that these are chain homotopic. The chain homotopy is the mapˇ m ( P ) : ˇ C • ( Y − , p ) P → ˇ C • ( Y + )obtained by the same formulas 3.6 by substituting the moduli spaces on the cobordisms bythe parametrized counterparts M + ([ C − ] , C , W ∗ p , [ C + ]) P . The discussion above can be carried over with the only difference in this case each modulispace has an additional codimension one face given by M + ([ C − ] , C , W ∗ , [ C + ]) ∂P . The sameproof as above shows thatˇ ∂ ( Y + ) ◦ ˇ m ( P ) + ˇ m ( P ) ◦ ˇ ∂ ( Y − , p ) = ˇ m + ˇ m , which proves the result. (cid:3) Remark . It is important to remark that the groups we are dealing with depend onthe choice of metric and perturbations on the three manifolds, even though our notation doesnot make that explicit. The key point of the result is that the induced map does not dependon the choice of the metric and perturbation on the cobordism.Given a cobordism W from Y − to Y + , we define the maps d HM • ( U d | W ) : d HM • ( Y − ) → d HM • ( Y + )as follows. Consider on W marked points p = { p , . . . , p m } , and fix non-negative integers { d i } summing up to d . Then we set d HM • ( U d | W )( x ) = d HM • ( W, p )( x ⊗ U d ⊗ · · · ⊗ U d m m )where the map d HM • ( W, p ) is the above constructed above induced by the cobordism whichhas the marked points interpreted as incoming ends. Our next goal is to show that such amap is independent of the perturbation on the punctures, the number of punctures and thepartition of d . Lemma . The map d HM • ( U d | W ) defined above and its analogues are independent ofthe choices made. Proof.
We will use a metric stretching argument to show that the map is the one inducedby a single puncture and the class u d on the corresponding end. We will focus on the to case,as the other ones are essentially identical.Because of the metric independence on the maps, we can suppose we are in the followingspecial situation. Let p be any point in the interior of W , such that there is a ball B with positive scalar curvature around p and metric which is a product S × ( − ε,
0] near theboundary where S has the standard round metric. We can assume that the neighborhoodsaround each marked point p i we used to define the chain map ˇ m are all contained in such aball. For every T ≥
0, we can define the Riemannian manifold W ( T ) = B ∪ ([0 , T ] × ∂B ) ∪ ( W \ int B )obtained by inserting a cylinder of length T along the boundary of the ball. We can interpretthis family of manifolds also as a family of metrics parametrized by [0 , ∞ ) on the fixed manifold W , which is always standard near the punctures p i . On the additional tube [0 , T ] × ∂B , we . INVARIANCE AND FUNCTORIALITY 117 consider a non degenerate perturbation q translation invariant perturbation not introducingany irreducible solution. Notice that the perturbations are not supported in the boundaryanymore, but the details of the construction carry out without any problem in this slightlymore general setting. We can then consider the moduli spaces on the cobordism M ([ C − ] , C , W ∗ p , [ C + ]) [0 , ∞ ) parametrized by the family of metrics in [0 , ∞ ). We can find time independent perturbationson B and W \ int B so that these moduli spaces are regular, as it follows from their fiberedproduct description (see Proposition 26 . . T = ∞ M + ([ C − ] , C , W ∗ p , [ C + ]) ∞ . This is a stratified space consisting of quintuples([ˇ γ ] , [ γ B ] , [ˇ γ ] , [ γ X ] , [ γ + ])where [ˇ γ ] ∈ ˘ M + (cid:0) ([ C − ] , C ) , ([ C ′− ] , C ′ ) (cid:1) [ γ B ] ∈ M ( C ′ , B ∗ , p , [ C ])[ˇ γ ] ∈ ˘ M + ([ C ] , [ C ′ ])[ γ X ] ∈ M ([ C ′− ] , [ C ′ ] , X ∗ p , [ C ′ + ])[˘ γ + ] ∈ ˘ M + ([ C ′ + ] , [ C + ])are such that the evaluations on the corresponding ends coincide. Here [ C ] and [ C ′ ] are unstable critical submanifolds for the flow on S with the perturbation q . These objects alsohave a well defined relative homotopy class. We define the total compactified space as M + z ([ C − ] , C , W ∗ p , [ C + ]) = [ S ∈ [0 , ∞ ] { S } × M + z ([ C − ] C , W ∗ p , [ C + ]) S . The topology on the total space is defined in the same fashion as the one on the space ofbroken trajectories, and it is not hard to prove that this space is an abstract δ -chain whosecodimension one faces are given by: • the moduli space M + z ([ C − ] , C , W, [ C + ]) ; • the union over S ∈ [0 , ∞ ] of the codimension one faces of M + z ([ C − ] , C , W, [ C + ]) S corre-sponding to the same sequences of critical submanifolds and homotopy class. • the top strata of the fiber over ∞ .Regarding the last item, the analogue of Proposition 26 . . M ( C , B ∗ , p , [ C ]) × M ([ C − ] , [ C ] , X ∗ p , [ C + ]) .
18 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES
Using these compactifications, we can define the map H oo : C o • ( Y − , p ) → C o • ( Y + )[ σ ] ⊗ [ σ ] X [ C + ] ∈ C o ( σ, σ ) × M + z ([ C − ] , C , W ∗ p , [ C + ]) , and similarly the maps H os , H uo , H us and using the reducible counterparts of the moduli spacesthe operator ¯ H ss , ¯ H uu , ¯ H su , ¯ H us . As it should be clear, here we restrict to the subcomplexgenerated by tuples of δ -chains transverse to all the parametrized moduli spaces, which isquasi isomorphic to the total complex following Lemma 3.1. We then defineˇ H : ˇ C • ( Y − , p ) → ˇ C • ( Y + )ˇ H = (cid:20) H oo H uo ¯ ∂ su + ∂ uo ¯ H su H os ¯ H ss + H us ¯ ∂ su + ∂ us ¯ H su (cid:21) . The following identity holds, ˇ H ◦ ˇ ∂ + ˇ ∂ ◦ ˇ H = ˇ m + ˇ m ∞ , where ˇ m is the chain map induced by the original cobordism and ˇ m ∞ is analogous mapdefined by using the moduli spaces M + ([ C − ] , C , W ∗ p , [ C + ]) ∞ . It is important to notice thatalthough the latter are is not an abstract δ -chain, it consists of a union of them along codi-mension one faces and the whole construction carries over without any complication. Toverify the identity, we have for example that ∂ oo H oo + ∂ uo ¯ ∂ su H os + H oo ∂ oo + H uo ¯ ∂ su ∂ os + ∂ uo ¯ H su ∂ os = ( ˇ m ) oo + ( ˇ m ∞ ) oo , which follows as usual by the characterization of the codimension one strata. The map ˇ m ∞ is just the composition of two chain maps, namely the mapˆ m ( B , p ) : ˆ C • ( S ) ⊗ · · · ⊗ ˆ C • ( S m ) → ˆ C • ( S )induced by the punctured cobordism B and the mapˇ m ( W, p ) : ˇ C • ( Y − , p ) → ˇ C • ( Y + )that defines the map induced by the cobordism W with the single puncture p . We againdid not mention the transversality issue that can be handled in the usual way by restrictingto a quasi-isomorphic chain complex. To conclude, we just need to show that the chain mapˆ m ( B , p ) induces at the homology level the multiplication map F [[ U ]] ⊗ · · · ⊗ F [[ U m ]] → F [[ U ]] U d ⊗ · · · ⊗ U d m m U d + ··· + d m . The argument is very close to that of Lemma 27 . . B ) ∗ p there is only one anti-self dual connection A up to gauge equivalence, and no irreduciblesolutions because of positive scalar curvature. Each generator U d i i can be realized as a pro-jective subspace M i of the critical submanifold [ C i ] representing a generator in homology, andsimilarly U d is represented by projective subspace in a critical submanifold [ C + ]. By linearityof the Dirac operator D + A , we have that the top stratum of the moduli space M + ([ C ] , . . . , [ C m ] , ( B ) ∗ p , [ C + ])is a complex projective space with some hyperplanes removed. Furthermore, the image of theevaluation maps in [ C ] × · · · × [ C m ] . INVARIANCE AND FUNCTORIALITY 119 projects to a projective space in each factor and has the same dimension as the moduli space.The positivity of the scalar curvature implies that the L index of the operator D + A is zero.By simple dimensional considerations that the fibered product between Q M i and the modulispace on B represents the class U d . (cid:3) The aim is now to check functoriality and invariance for the construction. The firstverification is trivial.
Proposition . If X is a trivial cylindrical cobordism from ( Y ; g, q ) to itself, then themaps d HM • (1 | X ) , d HM • (1 | X ) , HM • (1 | X ) are the identity on the Floer homology groups. Proof.
For this computation we can simply consider the product cobordism I × Y from Y to itself with a translation invariant perturbation. We claim that the chain map is in this casethe identity. In fact the evaluation maps of a moduli space consisting of non translationallyinvariant solutions factors through a smaller dimensional space (the space of unparametrizedtrajectories), hence the fibered products with it will always be negligible. This implies thatwe can restrict ourselves to consider the translationally invariant moduli spaces M z ([ C ] , [ C ])(with z trivial), which induce the identity map at the chain level. (cid:3) The verification of the composition law is not as straightforward, and also uses a metricstretching argument similar to the one in Proposition 3.2.
Proposition . Let Y , Y and Y be -manifolds, and let W and W be cobordismsfrom Y to Y and from Y to Y respectively which are cylindrical near their boundaries. If d = d + d , we have the identities d HM • ( u d | W ) = d HM • ( u d | W ) ◦ d HM • ( u d | W ) where W = W ◦ W is the composite cobordism, and similarly for the from and bar versions. Proof.
The proof is essentially the same as in Chapter 26 in the book, to which we referfor the details. By the independence of the map, we can pick a single puncture with weight d i on each of the cobordisms. For every S ≥ W ( S ) = W ∪ ([0 , S ] × Y ) ∪ W , the composite cobordism with a cylinder of length S inserted in the middle. The manifolds W ( S ) can also be interpreted as a family of metrics of the fixed manifold X . As in the proof ofProposition 3.2, we can compactify the union of the moduli spaces for each metric, obtainingthe space M + ([ C − ] , [ C ] , [ C ] , W ∗ p ,p , [ C + ]) [0 , ∞ ] , where [ C i ] is a critical manifold on the end corresponding to p i , by adding a fiber over S = ∞ which we call M + ([ C − ] , [ C ] , [ C ] , X ( ∞ ) ∗ p ,p , [ C + ]) . This consists of tuples of the form([ ˘ χ ] , [ χ ] , [˘ γ ] , [ ˘ χ ] , [ χ ] , [˘ γ ] , [ ˘ χ ])where
20 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES • each [ ˘ χ i ] is a broken trajectories on Y i ; • [ χ ] is a solution on ( W ) ∗ p , and similarly [ χ ] is a solution on ( W ) ∗ p ; • each [˘ γ i ] is a broken trajectory on the end corresponding to the puncture p i ,such that the evaluation maps on the corresponding ends agree. The topology on the totalspace is defined in a similar fashion to that on a space of broken trajectories (see Section5 in Chapter 2). As before, this space is obtained by gluing abstract δ -chains along theircodimension one faces, and the top strata of the fiber over ∞ consist of objects of the form: • M × M ; • ˘ M × M × M ; • M × ˘ M × M ; • M × M × ˘ M where the ˘ M i indicate the typical moduli space on Y i , and M and W are the typical modulispaces on the cobordisms with prescribed asymptotic at the puncture. As usual, in the lastthree cases, the middle moduli space is boundary obstructed, and we use the fact that thereare no boundary obstructed trajectories on the three-sphere. The proof then follows the samelines of Lemma 3.4. In particular, one can define the operators K oo , K os , K uo , K us and ¯ K ss , ¯ K su , ¯ K us , ¯ K uu via the fibered products with the moduli spaces M + ([ C − ] , [ C ] , [ C ] , W ∗ p ,p , [ C + ]) [0 , ∞ ] , anddefine the operator ˇ K : ˇ C • ( Y , { p , p } ) → ˇ C • ( Y )ˇ K = (cid:20) K oo K uo ¯ ∂ su + m uo ¯ m su + ∂ uo ¯ K su K os ¯ K ss + K us ¯ ∂ su + m us ¯ m su + ∂ us ¯ K su (cid:21) , where the terms m uo ¯ m su and m us ¯ m su are the composition of the maps induced by the twocobordisms. It is then not hard to check that this map satisfies the identityˇ K ◦ ˇ ∂ ( Y , { p , p } ) + ˇ ∂ ( Y ) ◦ ˇ K = ˇ m ( W, { p , p } ) + ˇ m ( W , p ) ◦ m ( W , p ) , hence defines a chain homotopy between the two chain maps in consideration (see Chapter26 in the book for the details). (cid:3) We recollect the invariance and functoriality results we have just proved in the followingcorollaries.
Corollary . For different choices of metrics and perturbation ( g, q ) the Floer ho-mology groups are canonically isomorphic, hence they are canonically isomorphic to the onesdefined in the book. These identifications also preserve the maps induced by cobordisms. Proof.
The canonical isomorphism is the map induced by the trivial cobordism I × Y (with the appropriate metrics cylindrical near the boundary). Such a map is well definedbecause of Proposition 3.2, and is hence an isomorphism by Lemma 3.5. Furthermore, byLemma 2.6 the construction gives the same result as the construction in the book, and infact the maps induced by a cobordism are the same because in the case of a non degenerateperturbation only the zero dimensional moduli spaces come into definition of the differentials. (cid:3) . INVARIANCE AND FUNCTORIALITY 121 Recall from Definition 3 . . cob whose objects are compact,connected, oriented 3-manifolds and whose morphisms are isomorphism classes of cobordisms. Corollary . The Floer groups define covariant functors from the cobordism category cob to the category of groups d HM • : cob → group d HM • : cob → group HM • : cob → group . Remark . There is a subtlety in the last result (see also the discussion after Corollary23 . . { G a } a ∈ A with isomorphisms φ a a satifying the compatibilitycondition φ a a φ a a = φ a a and morphisms are collections of maps satifying the obvious compatibility relations. On theother hand there is a natural functor from this category to the category of groups: to a family( { G a } a ∈ A , φ ) we assign the subgroup of Y a ∈ A G a consisting of collections { g a } a ∈ A with g b = φ ab g a (which is isomorphic to each of the G a ).Finally the maps induced by cobordisms can also be used in order to define on the Floerhomology groups of Y a graded module structure over the ring F [[ U ]]. Indeed, given ξ ∈ d HM • ( Y ) one defines the cap product U d ∩ ξ = d HM • ( U d | I × Y )( ξ )and the analogues for the other versions. The fact that this is a module structure followsdirectly from Proposition 3.6. Similarly one can define the cup product in cohomology. Remark . Notice that in order to make the discussion simpler we have not dealtwith the whole action of the cohomology group with F coefficients of the moduli space ofconfigurations of W , but only with the elements of the form U d . The action of the remainingterms, namely the ones of the formΛ ∗ ( H ( W ; Z ) / Tor ⊗ F )can be defined in an analogous way as follows. Given a degree one element x in the exteriorproduct, one chooses an embedded loop γ representing it. A tubular neighborhood of theloop is diffeomorphic to S × D , and we can suppose that the metric on it is the product ofthe standard ones. The boundary, which is a copy of S × S , has positive scalar curvature,and we consider on it the spin c structure s coming from the spin structure. We know fromChapter 36 in the book that d HM • ( S × S , s ) is canonically isomorphic to F [[ U ]] ⊕ F [[ U ]] {− } ,
22 3. FLOER HOMOLOGY FOR MORSE-BOTT SINGULARITIES and the group is trivial in other spin c structure (again, the actual absolute gradings areobtained by shifting by − c structure s , and nosolutions for the others. Call W γ the manifold obtained from W by removing the neighborhoodof γ . This has three boundary components, namely Y − and S × S as incoming ends and Y + asan outgoing end. The discussion of the present section then carries over without modificationsto show that one can define a map d HM • ( Y − ) ⊗ d HM • ( S × S ) → d HM • ( Y + )by considering the moduli spaces on the manifold with cylindrical ends W ∗ γ . The map d HM • ( x | I × Y ) is then defined by evaluating this map at the element(0 , ∈ F [[ U ]] ⊕ F [[ U ]] {− } = d HM • ( S × S ) . Similarly, we can define the action of an element x ∧ · · · ∧ x n ⊗ U d by deleting tubular neighborhoods of the loops γ i representing x i and a ball with weight d .The proofs that these are well defined are analogous to those discussed in the section.HAPTER 4 Pin(2) -monopole Floer homology
In this chapter we apply the theory we have developed in order to study a specific casearising from a natural action of conjugation on the set of spin c -structures. The final productwill be a set of invariants of three manifolds, called Pin(2) -monopole Floer homology , whichare the analogue of the Pin(2)-equivariant Seiberg-Witten-Floer homology recently definedby Manolescu in [ Man13a ].The action by conjugation has been known and exploited since the early days of Seiberg-Witten theory. In order to remark this we named our first section with the same name asthe corresponding section in the classical reference [
Mor96 ]. On the other hand, the ideato exploit this additional symmetry of the equations at a more refined level is the key pointbehind Manolescu’s breakthrough. In our setting, this will arise as an involution on the modulispaces of configurations, which will in turn imply that the critical points are Morse-Bott andsymmetrical in a very interesting way. Exploiting additional features at the algebraic level,and we will be able to define our new invariants building on the work of Chapter 3.
1. An involution in the theory
We start with some remarks at the level of Clifford algebras (see for example [
Mor96 ] formore details). The Lie group Spin(3) is naturally identified with the group SU(2) of unitarydeterminant one matrices. Furthermore its spin representation S corresponds the space ofquaternions H on which it acts via the natural action on the left of SU(2) on the latterwhen thought as C . We will always think of the complex structure on H as given by themultiplication on the right , hence we have the identification of complex vector spaces C → H ( z , z ) z + jz . In particular, the multiplication by j on the right induces a complex antilinear isomorphismof C , i.e. a complex isomorphism between C and its conjugate vector space.When we have a spin c structure s = ( S, ρ ) on a three manifold Y , we can also considerits conjugate spin c structure ¯ s = ( ¯ S, ρ ), where ¯ S is the conjugate bundle of S and the actionvia ρ of the real one-forms is the same one. In this case, the action of j induces a complexantilinear identification : s → ¯ s which can also be extended to an action : C ( Y, s ) → C ( Y, ¯ s )( B, Ψ) ( ¯ B, Ψ · j )where ¯ B is the conjugate connection of B . This also descends to an identification (still called ) between the moduli spaces of configurations B ( Y, s ). Furthermore, is the automorphismof C ( Y, s ) given by ( B, Ψ) ( B, − Ψ) , and therefore it acts as the identity on the moduli space of configurations B ( Y, s ).The most interesting case of this isomorphism is certainly the one in which the spin c structure s is actually isomorphic to its conjugate, in which case we call it self-conjugate . Inthis case the map is an involution of the configuration space. The following result gives usa classification of such spin c structures. Proposition . Given an oriented Riemannian three manifold Y , there always existsa self-conjugate spin c -structure s . Furthermore for a fixed one there is a one-to-one corre-spondence between: (1) isomorphism classes of self conjugate spin c structures s on Y ; (2) complex line bundles L isomorphic to their conjugate line bundle ¯ L ; (3) the -torsion of H ( Y ; Z ) . Proof.
This is achieved by the same construction as in Section 1 of Chapter 1, see alsoProposition 1 . . Y is trivial, Y admits a spinstructure, and the spin c structure it defines via the spin representation is clearly self-conjugate.Furthermore, given such a structure s = ( S , ρ ), for any line bundle L which is isomorphicto its conjugate we have that the spin c structure S = S ⊗ C Lρ = ρ ⊗ L is again self-conjugate, and in fact every self-conjugate spin c structure arises in this way foran appropriate choice of a line bundle isomorphic to its conjugate. Finally, such line bundlesare identified as those whose first Chern class is 2-torsion, because c ( ¯ L ) = − c ( L ). (cid:3) As we have seen every spin structure induces a self-conjugate spin c structure. By consid-ering the Bockstein exact sequence · · · → H ( X ; Z ) → H ( X ; Z / Z ) δ → H ( X ; Z ) · → H ( X ; Z ) → · · · we see that the 2-torsion classes in H ( X ; Z ) are exactly the image of the Bockstein homomor-phism δ . At the level of spin c structures, this tells us that the self-conjugate ones are exactlythose which come from a genuine spin structure. Furthermore, for a fixed spin structure s ,the spin structures s + x and s + x for x i ∈ H ( X ; F ) determine the same spin c structureif and only if δ ( x ) coincides with δ ( x ). Hence each self-conjugate spin c structure comesfrom exactly 2 b ( Y ) spin structures. For example, on S × S there are two spin structuresthat induce the same spin c structure while R P the two spin structures induce distinct spin c structures. . AN INVOLUTION IN THE THEORY 125 Given a self-conjugate spin c structure, we can then consider it as coming from a fixedspin structure s , so that it is comes with a canonical choice of the automorphism j of thespinor bundle S given by right multiplication. The action of j makes S into a quaternionicvector bundle and we can think of this additional structure as endowing the equations witha Pin(2)-symmetry, where Pin(2) = S ∪ j · S ⊂ H . There is a unique -invariant (hence Pin(2)-invariant) base connection B , namely the oneinduced by the Levi-Civita connection on the tangent bundle and the conjugation invariantconnection on the determinant line bundle. This has the additional property that the Diracoperator commutes with the action of j , i.e.(4.1) D B (Ψ · j ) = ( D B Ψ) · j. When working with a self-conjugate spin c structure, we will always suppose to have fixed thisas a base connection, and we can write the action of on the configuration space C ( Y, s ) as · ( B + b, Ψ) = ( B − b, Ψ · j ) . Furthermore, the Dirac operators D B : Γ( S ) → Γ( S )are compatible with the action of in the sense that(4.2) D B (Ψ · j ) = D ¯ B Ψ · j. This can be easily checked with a local computation. For a fixed orthonormal frame { e i } , if B = B + b one has D B (Ψ · j ) = X k ρ ( e k )( ∇ B e k + b ( e k ))(Ψ · j ) = X k ρ ( e k )( ∇ B e k (Ψ) · j + (Ψ · j ) · b ( e k ))= X k ρ ( e k )( ∇ B e k (Ψ) · j − (Ψ · b ( e k )) · j ) = D ¯ B Ψ · j where we used the fact that b is purely imaginary.As we have chosen the basepoint B , the Chern-Simons-Dirac functional L (which is welldefined up to gauge because c is torsion) is -invariant. Hence its gradient is -equivariant,and also defines an involution of the set of critical points. Because the action of j on C does not have eigenvectors, a fixed point of the action of on B ( Y, s ) is necessarily of theform [ B, b ( Y ) such fixed points corresponding to the fixed points of theinvolution on the moduli space of flat connections : T → T x
7→ − x. These are exactly the the flat connections whose representatives are gauge equivalent to theirconjugate or, equivalenty, the flat connections which have holonomy ± B ( Y, s ) does not depend on the actual choice of the spin structureinducing the given one. This is a manifestation of Schur’s Lemma, as such a j is pointwisean isomophism of irreducible SU (2)-modules, hence any other such j is necessarily gaugeequivalent to it. This independence statement implies that the fixed points are the gaugeequivalence classes of the canonical connections of the 2 b ( Y ) spin structures inducing the
26 4. Pin(2)-MONOPOLE FLOER HOMOLOGY given one. We will refer to these connections as the spin connections.Notice that the relation (4.1) implies that the operator D B is quaternionic linear, hencein particular eigenspaces are always even dimensional. This is true in a more general context,as in the next lemma. Lemma . Suppose the connection B is gauge equivalent to a spin connection, hence itis gauge equivalent to its conjugate via a gauge transformation u , i.e. ¯ B = u · B . Then theright multiplication by j ′ = ju − determines a quaternionic structure on the eigenspaces of D B , which are in particular alwayseven dimensional. Proof.
Suppose we have D B Ψ = Ψ · λ. Then because of the gauge invariance of the equations, the relation (4.2) and the fact that λ is real we have the identities(Ψ · j ) · λ = ( D B Ψ) · j = D ¯ B (Ψ · j ) = D u · B (Ψ · j ) = D B (Ψ · j · u − ) · u, hence D B (Ψ · j · u − ) = (Ψ · j · u − ) · λ. The lemma then follows from the fact that ju − anticommutes with the complex multiplica-tion by i , and has square −
1, so it defines a quaternionic structure on the eigenspace. (cid:3)
The map naturally induces an automorphism of the blow-up : C σ ( Y, s ) → C σ ( Y, s )( B + b, s, ψ ) ( B − b, s, ψ · j )which descends to a fixed point free involution : B σ ( Y, s ) → B σ ( Y, s ) , and it is immediate to check that the blown up gradient of the Chern-Simons-Dirac func-tional (grad L ) σ is equivariant under this action. Furthermore, the action induces a smoothinvolution in the completion of the configuration spaces in each Sobolev norm.The case of a four dimensional Riemannian manifold X is completely analogous. Inthis setting, the spin representation S = S + ⊕ S − can by identified with H ⊕ H , and asbefore there is an action from the right by quaternion multiplication by j , which determinesan isomorphism between S and its conjugate ¯ S . At a global level, everything regarding theinvolution for a given self-conjugate spin c -structure holds with the obvious translations. Theonly different point is that a self conjugate spin c structure does not exist unless the manifoldis spin (or equivalently, the second Stiefel-Whitney class w ( X ) is zero), as one can easilycheck using the construction of Proposition 1.1. This involution naturally induces involutionsof the various configurations spaces, and their properties are the same those described abovefor the configuration spaces of three manifolds.Similarly, a self conjugate spin c structure on a four manifold with boundary X induces aself conjugate spin c structure on its boundary, and the quaternionic structures are compatible. . EQUIVARIANT PERTURBATIONS AND MORSE-BOTT TRANSVERSALITY 127 The induced involutions on the configurations spaces are also compatible with the restrictionmaps.In the same way the cohomology ring of BS is important in the usual monopole Floerhomology, the cohomology ring of B Pin(2) will have a special role in our construction. Inparticular, it will arise as the Floer homology groups associated to S . It can be calculatedby means of the Serre spectral sequence associated to the fiber bundle R P ֒ → B Pin(2) → H P ∞ = B SU(2)coming from the identification SU(2) / Pin(2) = R P , which collapses at the E page for degreereasons. The ring on which our modules will be defined is obtained from this by completionand degree reversal. Definition . We denote by R the ring R = F [[ V ]][ Q ] / ( Q )where V and Q have degrees respectively − −
2. Equivariant perturbations and Morse-Bott transversality
As we saw in the previous section, when the spin c structure is self-conjugate, the mono-pole equations have an additional symmetry provided by , the action of the quaternionicmultiplication by j . We would like to exploit this symmetry in order to construct a Floerchain complex with additional algebraic properties, and in order to do so we need to ensurethat the perturbations are compatible with the extra action. The aim of the present sectionis to construct Pin(2)-equivariant tame perturbations, and use them to achieve Morse-Botttransversality in the sense of Chapter 3.From now on we suppose that the spin c structure s is self-conjugate and fix a base spinconnection B . Recall the construction of cylinder function from Section 4 of Chapter 1. Firstof all, notice that for a coclosed 1-form c ∈ Ω( Y ; i R ), the map r c : C ( Y, s ) → R ( B + b ⊗ , Ψ) Z Y b ∧ ∗ ¯ c has the symmetry r c ( · ( B, Ψ)) = − r c ( B, Ψ) . Also, for a given smooth section Υ of the bundle S → T × Y , one can define the G o ( Y )-invariant map ˜ q Υ : C ( Y, s ) → H ( B, Ψ) q Υ ( B, Ψ) − j · q Υ ( · ( B, Ψ))
28 4. Pin(2)-MONOPOLE FLOER HOMOLOGY where we recall q Υ ( B, Ψ) = Z Y h Ψ , Υ † ( B, Ψ) i ∈ C . This map is equivariant under the action of , as it easily follows from the property q Υ ( B, − Ψ) = − q Υ ( B, Ψ) . We stress again on the fact that j acts on H by left multiplication. Hence, given a finitecollection of coclosed 1-forms c , . . . , c n + t , the first n being coexact and the remaining t beinga basis of harmonic forms, and a collection Υ , . . . , Υ m of sections of S , we can define as inSection 4 of Chapter 1 the map ˜ p : C ( Y, s ) → R n × T × H m ( B, Ψ) (cid:0) r c ( B, Ψ) , . . . , r c n + t ( B, Ψ) , . . . , ˜ q Υ ( B, Ψ) , . . . , ˜ q Υ m ( B, Ψ) (cid:1) which is invariant under G o ( Y ) and equivariant under the remaining action of Pin(2). On theright hand side, j acts as minus the identity on the R and T factors and via right multiplicationon the H m factor. Definition . We call a real valued function f on C ( Y, s ) a Pin(2)- invariant cylinderfunction if it arises as g ◦ ˜ p where: • the map ˜ p : C ( Y, s ) → R n × T × H m is defined as above; • the function g is a Pin(2)-invariant function on R n × T × H m with compact support.Because of the symmetry of the problem, the perturbed functional −L = L + f for a Pin(2)-invariant cylinder function f will always have the fixed points of the involution on B ( Y, s ) ascritical points. The analogue of Proposition 4.3 in Chapter 1 also holds for Pin(2)-invariantcylinder functions, as reassumed in the following result. Proposition . If f is a Pin(2) -invariant cylinder function, then its gradient grad f : C ( Y ) → T is a tame perturbation (in the sense of Definition 4.1 in Chapter ) which is also -equivariant.Furthermore, for every compact subset K of a finite dimensional submanifold M of thebase configurations space B ok ( Y ) , both Pin(2) -invariant, one can find a collection of c µ , Υ ν and a neighborhood U ⊃ K such that p : B ok ( Y ) → R n × T × H m gives an embedding of U . Hence for any [ B, Ψ] ∈ B ∗ k ( Y ) and any non zero tangent vector v there exists a Pin(2) -invariant cylinder function whose differential D [ B, Ψ] f ( v ) is non-zero. Proof.
The formal gradient of a -invariant function is a -equivariant vector field. Fur-thermore, because our map ˜ p is determined in a very simple explicit way by the map p definedby means of the same family of coclosed forms and sections of S , it is immediate to see thatthe estimates of Chapter 11 in the book continue to hold with small adaptations, hence thegradient of f is a tame perturbation in the usual sense. The second part is clear as if wecompose the map ˜ p with the natural projection map on each H factor H → C z + jw z, . EQUIVARIANT PERTURBATIONS AND MORSE-BOTT TRANSVERSALITY 129 we obtain the corresponding map p , which in turn can be made into an embedding by Propo-sition 4.3 in Chapter 1. The last sentence then follows because the fixed points of the actionare reducible. (cid:3) Finally, as in the end of Chapter 1, and in particular Proposition 4.4, we can construct aBanach space e P with a linear map D : e P → C ( C ( Y ) , T ) λ q λ such that every q λ is a -equivariant tame perturbation and the image contains the formalgradient of each element of a countable family of Pin(2)-invariant cylinder function { f i } arising as follows. For every pair ( n, m ), choose a countable collection of ( n + m )-tuples( c , . . . , c n , Υ , . . . , Υ m ) which are dense in the C ∞ topology in the space of all such ( n + m )-tuples. Choose a countable collection of compact subsets K ⊂ R n × T × H m which is dense inthe Hausdorff topology and, for each K , a countable collection of Pin(2)-invariant functions g α ( K ) with support in K which are dense (in the C ∞ topology) in the space of Pin(2)-invariantfunctions with support in K and such the subset (cid:8) g α | g α | K ∩ ( R n × T ×{ } ) = 0 (cid:9) is dense (in the C ∞ topology) in the space of Pin(2)-invariant functions with support on K and vanishing on K ∩ ( R n × T × { } ). Then the family { f i } is obtained composing in allpossible ways the functions arising from our choices. We call such a pair ( e P , D ) (oftenlydenoted just by e P ) a large Banach space of -equivariant perturbations .We now focus on the transversality problem. Notice that the same proof of Lemma1.2 applies to show that if [ B,
0] is one of the 2 b ( Y ) fixed points of on B ( Y ), then forany -equivariant tame perturbation q the eigenspaces of the operator D B, q are naturallyquaternionic vector spaces, hence their spectrum (as complex-linear operators) cannot besimple. Furthermore, the points [ B,
0] are always critical points for the perturbed Chern-Simons-Dirac operator when the perturbing function is a Pin(2)-cylinder function, hence ourcritical points will never be non-degenerate in the sense of the book. In light of our discussion,it is natural to introduce the following definition (see also Proposition 12 . . Definition . Let a = ( B, r, ψ ) ∈ C σk ( Y ) be a critical point of the -invariant vectorfield (grad −L ) σ . We say that a is Pin(2) -non-degenerate if the following conditions hold:(1) if r = 0, then the corresponding point ( B, rψ ) ∈ C ∗ k ( Y ) is non-degenerate in the usualsense, i.e. grad −L is transverse to the subbundle J k − ⊂ T k − ;(2) if r = 0, then B ∈ A k is a non-degenerate zero of (grad −L ) red (i.e. (grad −L ) red istransverse to the subbundle J red k − ⊂ T red k − ) and furthermore, if λ is the eigenvalue of D B, q corresponding to ψ then λ is not zero, and • if B is not gauge equivalent to a spin connection, the λ eigenspace has complexdimension one; • if B is gauge equivalent to a spin connection, the λ eigenspace has complexdimension two.A Pin(2)-non degenerate reducible critical point ( B, , ψ ) with B not conjugate to a spinconnection is non-degenerate in the usual sense, and its gauge equivalence class is an isolated
30 4. Pin(2)-MONOPOLE FLOER HOMOLOGY singularity in B σk ( Y ). On the other hand when the connection B is spin, the eigenspace inwhich ψ lies is two dimensional, and its image in B σk ( Y ) is a two dimensional sphere arisingas the quotient by the action of S of the unit sphere of the eigenspace. This quotient is aHopf fibration. Furthermore, the next result tells us that in order to study our problem wecan rely on the machinery developed in Chapters 2 and 3. Proposition . Suppose a -equivariant perturbation q such that the critical points of (grad −L ) σ are Pin(2) -non-degenerate. Then the singularities are Morse-Bott.
Proof.
We will focus on the case of a reducible critical point a = ( B, , ψ ) where B isgauge equivalent to a spin connection, which is the only one which needs some adaptationsfrom the proof in Chapter 12 of the book. Let λ be the eigenvalue corresponding to a . Inthis case, as in the proof of Lemma 12 . . d Hess q ,α can bewritten as the map( b, r, ψ, c ) ( − dc + ∗ db + 2 D ( B , q ( b, , λr, Π ⊥ (cid:2) ( D q ,B − λ ) ψ + ρ ( b ) ψ + cψ + D q B , (( b, , (0 , ψ ))+ ( r/ D q B , ((0 , ψ ) , (0 , ψ )) (cid:3) , − d ∗ b + i | ψ | Re µ Y ( h iψ , ψ i )) . where Π ⊥ is the orthogonal projection to the real-orthogonal complement of ψ . We furtheranalyze this operator by decomposing c = iǫ + ˆ cψ = iǫ ψ + ˆ ψ where the ǫ j are constants, ˆ c and ˆ ψ are orthogonal respectively to the constants and to iψ (with the real L inner product). Exploiting S -equivariance and the fact that the configu-ration is a reducible critical point (hence ψ is an eigenvector of D q ,B ), one can write theoperator in the block lower triangular form rǫ ǫ ˆ cb ˆ ψ λ ∗ − ∗ − ∗ ∗ ∗ − d ∗ ∗ ∗ ∗ − d ∗ d q ∗ ∗ ∗ ∗ ∗ D q ,B − λ rǫ ǫ ˆ cb ˆ ψ where ∗ d q is the operator b
7→ ∗ db + 2 D ( B , q ( b, D q ,B − λ is acting on ( C ψ ) ⊥ . Also, notice that the tangent space T a C (in the sense ofSection 1 of Chapter 2) to the critical submanifold is contained in the latter space, as it issimply the complex orthogonal inside the eigenspace.The interesting part of the operator, i.e. the one corresponding to the Hessian Hess σ q , a , isthe operator on R ⊕ K ⊕ ( C ψ ) ⊥ . EQUIVARIANT PERTURBATIONS AND MORSE-BOTT TRANSVERSALITY 131 (where K is the space of coclosed imaginary 1-forms) given by λ ∗ ∗ d q | K ∗ ∗ D q ,B − λ . The Pin(2)-non-degeneracy of a implies that λ = 0 and that ∗ d q | K is an isomorphism, as itis the restriction of the operator D B (grad −L red ). On the other hand the operator, as D q ,B hasa two dimensional λ -eigenspace spanned (over the complex numbers) by ψ and T a C , and as T a C is contained in ( C ψ ) ⊥ , the operator D q ,B − λ descends to an isomorphism on D q ,B − λ : ( C ψ ) ⊥ / T a C → ( C ψ ) ⊥ / T a C. This is equivalent to the fact that Hess σ,ν q , a is an isomorphism, hence that the singularity isMorse-Bott. (cid:3) Remark . If the connection is the base one, the Dirac operator is genuinely quater-nionic, hence the tangent space T a C at a critical point a = ( B , , ψ ) is simply the complexspan of ψ · j .We now want to show that for a generic choice of the -equivariant perturbation q one canachieve Pin(2)-non-degeneracy at all critical points and regularity in the Morse-Bott sense(see Definition 3.14 in Chapter 2). We will prove the following transversality result (see alsoTheorem 12 . . Theorem . Let e P be a large Banach space of tame -equivariant perturbations. Thenthere is a residual subset of perturbations such that for every q in such a subset all the criticalpoints of (grad −L ) σ are Pin(2) -non-degenerate, hence Morse-Bott, and all moduli spaces areregular.
Proof.
The proof follows with few small complications in the same way as in Section12 . . B, , ψ ) where B is gauge equivalent to a spin connection. We show that in fact thesecritical points are non-degenerate for an open dense subset of perturbations in e P . From thiswe can then carry the proofs in Sections 12 . . -equivariant perturbation, the operators D B, q with B a spin connection have two dimensional eigenspaces and zero is not an eigenvalue.This set is also open, because we are considering only finitely many gauge equivalence classes.Following then Section 12 .
6, we show that if e P ⊥ ⊂ e P is the set of tame perturbations which vanish along the reducible locus of C ( Y ), we can finda generic subset of perturbations in e P such that D B, q has two dimensional eigenspaces. Bygauge invariance we just need to show this for one of the 2 b ( Y ) spin connections, and forsimplicity we only consider the case of the base one B (as the others can be obtained bychanging the quaternionic structure).
32 4. Pin(2)-MONOPOLE FLOER HOMOLOGY
In this case, we consider the space Op H consisting of self adjoint quaternionic linear Fredholm maps L k ( Y ; S ) → L k − ( Y ; S )of the form D B + h , where h is a quaternionic self-adjoint operator which extends to abounded operator on all L j for j ≤ k . This space is stratified by the dimension of the kernel,and the set of operators whose (quaternionic) spectrum is not simple is a countable image ofFredholm maps with negative index F n : Op H n × R → Op H ( L, λ ) L + λ, where Op H n ⊂ Op H is the space of operators with kernel of dimension exactly n . Our claim isthat the map ˜ M : P ⊥ → Op H q ⊥ D B, q ⊥ is transverse to both the stratification of the maps according to the dimension of the kerneland the Fredholm maps { F n } , from which the proof follows as in Theorem 12 . . q ⊥ = grad f is a be a perturbation in P ⊥ , where f = g ◦ ˜ p with g vanishing along R n × T × { } . If we denote by V the kernel of D B , q , the tangent space to the stratificationby the dimension of the kernel at the point D B , q ∈ Op H is the kernel of the compression map from Op H to the self adjoint quaternionic operators on V . Furthermore, V can be regarded as a subspace of the normal bundle to A k in C k ( Y ), andby Proposition 2.2 we can choose a p (defined by a collection of coclosed forms and sections inthe countable collection defining the large space of perturbations) whose differential embedsthis into a linear subspace of H m ⊂ T (0 , , ( R × T × H m )in a Pin(2)-equivariant fashion. We have that p ( B,
0) is necessarily(0 , , ∈ R n × T × H m by Pin(2)-equivariance. By choosing a Pin(2)-invariant function δg (which can we suppose inour defining collection) on R × T × H m vanishing along R × T × { } we can find a perturbation δ q ⊥ = grad( δg ◦ p ) ∈ P ⊥ such that the Hessian of ( δg ◦ p ) | V is any chosen Pin(2)-equivariant (i.e. quaternionic) self-adjoint endomorphism of V , which proves our claim. The second claim follows with the sameproof from the fact that the normal bundle to the image of F n at L + λ is naturally isomorphicto the space of traceless, self-adjoint quaternionic endomorphisms of Ker( L ).We then turn to the arrange that (grad −L ) red is a transverse section downstairs at these2 b ( Y ) points. Of course the set of perturbations in e P for which this condition holds is open.To show that it is dense, we need to study the surjectivity at a configuration ( B,
0) with B aspin connection of the map (grad −L ) red : K red k → K red k − b
7→ ∗ db + 2 D ( B, q ( b, . . EQUIVARIANT PERTURBATIONS AND MORSE-BOTT TRANSVERSALITY 133 where K red j is the space of coclosed 1-forms. As p ( B,
0) is necessarily (0 , ,
0) the second terminvolves the Hessian of the function g at the origin. The requirement that g is Pin(2)-invariantimplies that it is an even function when restricted to R n × T × { } . The result then follows bychoosing the map p so that its differential is an embedding on the kernel of the linearizationof (grad −L ) red and the fact that the Hessian of an even function on R n × T at the origin canbe any self-adjoint linear operator.The proof of transversality for moduli spaces follows with the similar adaptations fromthe one in Chapter 15 in the book and Theorem 3.17 in Chapter 2. It is clear that if the pathof configurations contains a point which is not reducible with the connection equivalent to aspin one then the proof of transversality in the classical case applies without any significantchange. The only observation to be made in order to run the transversality argument is thatif a trajectory γ connecting two critical points is such that there exists a time t and T > γ ( t )] = [ · ˇ γ ( t + T )]then the trajectory is constant (this is required in order to be able to find a perturbation thathas non zero inner product with an element in the cokernel of the map defining the universalmoduli space, see equation (15 .
3) in the book). On the other hand this relation implies that[ˇ γ ( t + 2 T )] is the same as [ˇ γ ( t )], hence the trajectory is constant.Finally, the only possible complication regards the trajectories that connect critical points[ a ] and [ b ] with blow down the reducible critical point [ B,
0] with B equivalent to a spinconnection, and such that the homotopy class is trivial. On the other hand, the proof oftransversality in this situation in Theorem 3.17 in Chapter 2 only uses only the fact that theoperator D q ,B is hyperbolic (and not that its spectrum is simple), so it still holds in our casewithout any modification. (cid:3) The set of perturbations we have introduced is enough to define the Floer chain complexes,but is not sufficient to prove the invariance and functoriality properties. The issue is that forsuch perturbations, on a spin cobordism the spin connection is always a solution of the Seiberg-Witten equations (in the blow down) hence transversality might not hold simply because ofindex reasons. A finite dimensional example of this phenomenon is that on the circle S there is no regular (in the sense of Morse homology) family of functions connecting cos ϑ and − cos ϑ which is conjugation invariant at each point. We introduce further perturbations inthe blow up as follows. Definition . Consider a self-conjugate spin c structure s on X . For k ≥
2, a k -tamePin(2) -equivariant asd -perturbation is a mapˆ ω : C σ ( X, s ) → L ( X, i su ( S + ))with the following properties:(1) the map ˆ ω is gauge invariant and -equivariant, where acts on the right hand side asthe multiplication by − ω ∈ C ∞ (cid:0) C σk ( X ) , L k +1 ( X, i su ( S + )) (cid:1) ;restricting to zero at the boundary of X ;
34 4. Pin(2)-MONOPOLE FLOER HOMOLOGY (3) for every integer j ∈ [ − k, k ], the first derivative D ˆ ω ∈ C ∞ (cid:0) C σk ( X ) , Hom( T C σk ( X ) , L k +1 ( X, i su ( S + ))) (cid:1) extends to a map D ˆ ω ∈ C ∞ (cid:0) C σk ( X ) , Hom( T C σj ( X ) , L j ( X, i su ( S + ))) (cid:1) ;(4) the image of ˆ ω is precompact in the L k +1 topology.We say that ˆ ω is tame if it is tame for every k ≥ C ∗ ( X, s ) via the blow down map. The conditions weimpose are very similar to those of a tame perturbations (Definition 4.1 in Chapter 1), andin fact they are introduced for the same reasons. In particular, condition (2) is require to setup the perturbed equations and condition (4) has the same role as condition (5) in Definition4.1 in Chapter 1. A slight difference is that our requirements involve the L k +1 norm, whichwill be needed in the proofs of compactness.Suppose now we are in the same setting as is Section 6 of Chapter 2, so our manifold X has cylindrical end Z = I × Y and we are given k -tame perturbations ˆ q , ˆ p and cut-offfunctions β, β . Given a k -tame Pin(2)-equivariant asd -perturbation ˆ ω we can define thegauge invariant and -equivariant section F σ p , ˆ ω = F σ + ˆ p σ + ˆ ω : C σk ( X, s X ) → V σk − , and the perturbed Seiberg-Witten equations F σ p , ˆ ω = 0. The definition the moduli spaces inthis context carries over without particular modifications.These moduli spaces satisfy the same properties discussed in Section 6 of Chapter 2. Wediscuss the most significant one, namely the proof of the compactness of the moduli spaceson X (see Theorem 24 . . X ǫ = X \ (( − ǫ, × Y ) , and we take X ′ ⊂ X ǫ . Proposition . Let γ n be a sequence in C σk ( X ′ ) of solutions to the perturbed equations F σ p , ˆ ω = 0 . Suppose that there is a uniform bound on the perturbed topological energy E top q ( γ n ) ≤ C , and that for each component Y α of Y there is a uniform upper bound − Λ q ( γ n | {− ǫ }× Y α ) ≤ C . Then there is a sequence of gauge transformations u n ∈ G k +1 ( X ) such that after passingto a subsequence the restrictions u n ( γ n ) | X ′ converge in the topology of C σk ( X ′ ) to a solution γ ∈ C σk ( X ′ ) . Proof.
We focus on the case on the specail case of a sequence of solutions ( A n , s n , φ n )on a cylinder Z = I × Y (where the perturbation p is time independent so it is not supportedin a collar). Indeed, the general case follows along the same lines of Theorem 24 . . ω induces a perturbation downstairs on the irreduciblelocus, and write π ( γ n ) = ( A n , Φ n ). . EQUIVARIANT PERTURBATIONS AND MORSE-BOTT TRANSVERSALITY 135 We introduce the perturbed analytical energy (compare Chapter 4 in the book) on generalmanifold X as E an q , ˆ ω ( A, Φ) = E top q + k F p , ˆ ω ( A, Φ) k . The imaginary valued 2-form ρ − Z (ˆ ω ( A, Φ)) is anti-self dual, so it can be written as ρ − Z (ˆ ω ( A, Φ)) = 12 ( dt ∧ ω t ( A, Φ) + ∗ ω t ( A, Φ))where ω t ( A, Φ) is a family of imaginary valued 1-forms on Y and ∗ is the Hodge star on thethree manifold. These elements are all in L k − because of Condition (1) in Definition 2.7. Inparticular we have ˆ ω ( A, Φ) = − ρ ( ω t ( A, Φ))as elements of sl ( S + ) ∼ = sl ( S ). Hence if ( A, Φ) is a configuration in temporal gauge that solves F σ p , ˆ ω = 0 then the path ( ˇ A ( t ) , ˇΦ( t )) is a flow line for the (partially defined) vector fieldgrad −L + (ˆ ω t , ω t depends on the configuration on the whole cylinder. We can then write, for aconfiguration in temporal gauge, E an q , ˆ ω ( A, Φ) = 2( −L ( t ) − −L ( t )) + Z t t k ddt ˇ γ ( t ) + grad −L + (ˆ ω t , k dt = Z t t k ddt ˇ γ k + k grad −L + (ˆ ω t , k dt + 2 Z t t h ddt ˇ γ, (ˆ ω t , i . This can be written in a gauge invariant fashion as E an q , ˆ ω ( A, Φ) = Z t t (cid:13)(cid:13)(cid:13)(cid:13) ddt ˇ A − dc (cid:13)(cid:13)(cid:13)(cid:13) dt + Z t t (cid:13)(cid:13)(cid:13)(cid:13) ddt ˇΦ + c Φ (cid:13)(cid:13)(cid:13)(cid:13) dt + Z t t k grad −L + (ˆ ω t , k dt + 4 Z t t tr (cid:18) ( ddt ˇ A − dc ) ρ − Z (ˆ ω ) (cid:19) dt. Because of condition (4) in Definition 2.7, the L -norm of ˆ ω ( A, Φ) is bounded independentlyof ( A, Φ), so by applying the Peter-Paul inequality to the last two terms we have that that E an q , ˆ ω ≥ E an q − C for some constant C independent of ( A, Φ) and as in equation (10 .
12) in the book12 E an q = Z t t (cid:13)(cid:13)(cid:13)(cid:13) ddt ˇ A − dc (cid:13)(cid:13)(cid:13)(cid:13) dt + Z t t (cid:13)(cid:13)(cid:13)(cid:13) ddt ˇΦ + c Φ (cid:13)(cid:13)(cid:13)(cid:13) dt + Z t t k grad −Lk dt. Then, Lemma 10 . . C ′ , C ′′ we have an inequalityof the form E an q , ˆ ω ( A, Φ) ≥ Z Z (cid:18) | F A t | + |∇ A Φ | + 14 ( | Φ | − C ′ ) (cid:19) − C ′′ ( t − t ) . From this we see that a bound on the topological energy implies as in the usual case an L bound on | F A tn | and |∇ A n Φ n | and an L bound on | Φ | . As in the proof of Theorem 10 . . p ( A n , Φ n ) havea subsequence converging in the L topology. Furthermore, condition (5) in Definition 2.7implies that we can pass to a subsequence for which ˆ ω ( A n , Φ n ) converges in the L k +1 topology.
36 4. Pin(2)-MONOPOLE FLOER HOMOLOGY
Hence because of the properness of the Seiberg-Witten map we can pass to a subsequencethat converges up to gauge in the interior domains in the L topology (see Theorem 5 . . L topology, using the fact thatˆ ω ( A n , Φ n ) converges in the L k +1 topology and the standard perturbation ˆ q ( A n , Φ n ) convergesin the L topology. We can continue like this to prove convergence in any interior domain inthe L k +1 topology.We need now to prove convergence in the blow up, so suppose we are given a sequenceof irreducible solutions ( A n , Φ n ) converging in the L k +1 topology so that the L norm of Φ n going to zero. We need to show that the renormalized spinors Φ n / k Φ n k L converge in the L k +1 topology. Because our new perturbations only affect the connection component, theproof of Proposition 10 . . n restricts to zero in a slice { t }× Y then it is identically zero. In particular we canpass to the τ -model description of the moduli spaces. The fact that ˆ ω ( A n , Φ n ) is convergentin the L k +1 topology implies that t grad −L (ˇ γ n ( t )) + (ˆ ω t ( γ n ) , L path in the Hilbert space L k ( Y ; iT ∗ Y ⊕ S ) converging in the topology of L paths.The proof of Proposition 10 . . ζ on C σk ( Y ) such that for any solution γ of the equations F p , ˆ ω ( γ ) we have that ddt Λ q (ˇ γ τ ( t )) ≤ ζ (ˇ γ ( t )) k grad −L (ˇ γ ( t )) + ( ω t ( γ ) , k L k ( Y ) . Notice that the right hand side is only defined for almost every t and is locally square in-tegrable. This is again because the additional perturbation only involves the connectioncomponents, while the proof of the inequality follows from differentiating the equation forthe spinor part. Together with the upper bound on Λ q , this implies that we have a uniformbound | Λ q (ˇ γ n ( t )) | ≤ M, so we can conclude as in the usual proof (as it again only involves manipulations with thespinor part). (cid:3) Remark . It is important to remark that for these perturbed equations the proof ofunique continuation (see Proposition 10 . . asd -perturbations which is large enough to achieve transversality. We introduce a simple class ofperturbations called squared projections , and show that they define tame asd -perturbations.Consider an embedded closed ball B together with a base point p . Consider a finite dimen-sional subspace of compactly supported spinors V ⊂ C ∞ c ( B ; S + )which is quaternionic, i.e. invariant under the action of j , and let π V be the L orthogonalprojection to the subspace V . On B there is a unique spin connection A . For any other . EQUIVARIANT PERTURBATIONS AND MORSE-BOTT TRANSVERSALITY 137 connection A = A + a , there is a unique gauge transformation u A such that u A · A is inCoulomb-Neumann gauge with respect to A and u A ( p ) is 1. We also have u A = e Ha where H is a smoothing operator of order 1. Finally, let f be a compactly supported smoothreal valued function in B . We then define the map squared projection associated to thequadruple ( B , p , V, f ) as the mapˆ ω ( B ,p ,V,f ) : C σ ( Z, s Z ) → L ( X, i su (2))( A, s, φ ) f · ( π V ( φ · u A ) π V ( φ · u A ) ∗ ) . The main analytic result is the following.
Proposition . Given a triple ( B , p , V, f ) as above, the squared projection ˆ ω ( B ,p ,V,f ) : C σ ( Z, s Z ) → L ( X, i su (2)) is a tame Pin(2) -equivariant asd -perturbation in the sense of Definition 2.7.
Proof.
Gauge invariance is clear from the definition. To see the equivariance under the j action, one notices that u ¯ A = ¯ u A hence we have π V ( φ · j · u ¯ A ) = π V ( φ · u A · j ) = π ( φ · u A ) · j. In the last step we used the fact that the L projection to the quaternionic subspace V isquaternionic linear, which easily follows from the fact that h φ · j, φ · j i L ( X ) = h φ , φ i L ( X ) . We then prove condition (4). We have for a constant C independent of the configuration k ˆ ω ( B ,p ,V,f ) ( A, s, φ ) k L ( X ) = k π V ( φ · u A ) k L ( X ) ≤≤ C k π V ( φ · u A ) k L ( X ) ≤ C k φ · u A k L ( X ) = C. In the second step we used the fact that V is finite dimensional, so all norms are equivalent,while in the last step we used the fact that k φ k L ( X ) = 1. In particular, because the image ofthe map is contained in a finite dimensional subspace, the map ˆ ω ( B ,p ,V,f ) is precompact inthe L k +1 -topology.The proof of conditions (2) and (3) follows in the same way as the estimates in Chapter11 in the book. In fact, the proofs are much easier because our perturbation are defined in apurely four dimensional context, rather that being induced by three dimensional ones. Thekey estimate to be proven involves the derivatives of the function( A, s, φ ) φ · u A , and in fact we only need to consider the L norm of the target as again we are projectingto a finite dimensional subspace. Hence we are interested in bounding at a configuration( A + a , s , φ ) the L norms of the quantities( Hδa ) · · · ( Hδa n ) e Ha δφ and ( Hδa ) · · · ( Hδa n ) e Ha φ , where the δ s indicate the corresponding tangent vectors. Because k ≥
2, the required boundsreadily follow from the Sobolev multiplication theorem. (cid:3)
38 4. Pin(2)-MONOPOLE FLOER HOMOLOGY
We then turn in the construction of a Banach space of perturbations P ASD . Choose acountable collection of pointed balls ( B α , p α ) such that their union is the interior of X . Thenfor each n we choose a countable collection of n -dimensional quaternionic subspaces V α of C ∞ c ( B α , S + ) which is dense in the space of such subspaces. Finally choose a countable set ofcompactly supported smooth functions f α which is dense in the C ∞ topology of such space.From these we can construct a countable family { ˆ ω i } i ∈ N of squared projections. Using Floer’sconstruction as in Section 11 . . . Proposition . Given a family of squared projections { ˆ ω i } i ∈ N as above, there existsa separable Banach space P asd and a linear map D asd : P asd → C (cid:0) C σ ( X ) , L ( X, i su ( S + )) (cid:1) λ ˆ ω λ satisfying the following properties: • for each λ ∈ P asd , the element ˆ ω λ is a tame asd perturbation in the sense of Definition2.7; • the image of D asd contains all the perturbations in the collection { ˆ ω i } i ∈ N ; • for each k ≥ the map P asd × C σk ( X ) → L k ( X, i su ( S + ))( λ, γ ) ˆ ω λ ( γ ) is a smooth map of Banach manifolds. Finally we prove the following transversality result, see also Proposition 6.7 in Chapter 2.
Proposition . Suppose we have a fixed
Pin(2) -non-degenerate -equivariant q on Y .Consider the perturbation ˆ p + ˆ ω = β ( t )ˆ q + β ( t )(ˆ p ) + ˆ ω as above. Then there is a residual subset of P ( Y, s ) × P asd ( X ) such that for all pairs ( p , ˆ ω ) in this set the moduli spaces of solutions M ( X ∗ , [ C ]) are regular in the sense of Definition 6.6in Chapter . The statement for families of Proposition 6.7 in Chapter also holds. Proof.
The proof goes as the proof of Proposition 24 . . . . γ . As inthe usual proof we need to show that the differential of the map appearing in Lemma 24 . . M , ˜ R + ) : C σk ( X, s X ) × P ( Y ) × P asd ( X ) → V σk − × C σk − / ( Y, s )has dense image in the L × L / topology. We restrict the study to configurations in Coulomb-Neumann gauge. Consider a configuration ( V, v ) which is L orthogonal to the image of thedifferential at a point. By elliptic regularity V is in L and an integration by parts argumentshows that v is the restriction at ∂X of V . Because in our setting the perturbed Diracequation still has the unique continuation property (see the proof of Proposition 2.8), therestriction of γ to a slice { t } × Y on the collar is still irreducible, and we can show that V vanishes by just using Pin(2)-equivariant tame perturbations. When γ is reducible, the sameargument works to show that its spinor part is vanishing. On the other hand, if the other . INVARIANT CHAINS AND FLOER HOMOLOGY 139 component of V does not vanish one can find a Pin(2)-equivariant asd -perturbation ˆ ω in thedefining family { ˆ ω i } i ∈ N such that h m (0 , ˆ ω i ( γ )) , V i L ( X ) > , where m is the differential of the map (4.3) in the directions corresponding to the perturba-tions. Hence the result follows, and the case of a reducible configuration is analogous. (cid:3)
3. Invariant chains and Floer homology
In this section we construct the analogue of Manolescu’s recent invariants ([
Man13a ])from the Morse homology approach, the Pin(2) version of the Floer homology groups. Thisis done by exploiting the extra symmetry of the Floer chain complex that comes when thespin c structure is self-conjugate and we use a -equivariant perturbation.Suppose from now on that a Pin(2)-non-degenerate -equivariant tame perturbation q which is also regular in the Morse-Bott sense as in Theorem 2.6 has been fixed. Recall fromSection 1 that we have a natural identification of a critical submanifold [ C ] corresponding toan eigenspace of D q ,B for a connection B conjugate to a spin one with S , and the action of on the moduli space of configurations is identified with the antipodal map on it. This actioninduces an involution (still denoted by ) on the chain complex C F∗ ([ C ]) sending the δ -chain[ σ ] = [∆ , f ] to the δ -chain [ σ ] = [∆ , ◦ f ] . If the family countable family of δ -chains F is also invariant under this action then theinvolution is clearly a chain map on C F∗ ([ C ]), i.e. we have ◦ ∂ = ∂ ◦ . In particular the subspace of invariant δ -chains C F∗ ([ C ]) inv = { [ σ ] | [ σ ] = [ σ ] } ⊂ C F∗ ([ C ])is a subcomplex. There is also the subcomplex (id + ) (cid:0) C F∗ ([ C ]) (cid:1) , which is also -invariant.The proof of the following lemma is straightforward. Lemma . The inclusion (id + ) (cid:0) C F∗ ([ C ]) (cid:1) ֒ → C F∗ ([ C ]) inv is a quasi-isomorphism, andthe homology of both complexes is naturally H ∗ ( R P ) . Similarly, if [ b ] is an isolated critical point then also the configuration [ b ] is, hence thereis a natural involution on the chain complex C F∗ ([ b ] ∪ [ b ]) exchanging the two points. Wecan define again the subcomplexes C F∗ ([ b ] ∪ [ b ]) inv and (id + ) (cid:0) C F∗ ([ b ] ∪ [ b ]) (cid:1) (which coincide in this case) and the analogous of Lemma 3.1 is obvious in this case. Fur-thermore the involution also gives rise to a natural isomorphism (in the sense of Definition1.5 in Chapter 3) between the moduli spaces : ˘ M + z ([ C − ] , [ C + ]) → ˘ M + ( z ) ([ C − ] , [ C + ])˘ γ = ([˘ γ ] , . . . , [˘ γ m ]) → ([ ˘ γ ] , . . . , [ ˘ γ m ]) ,
40 4. Pin(2)-MONOPOLE FLOER HOMOLOGY with the additional property that ev ± ◦ = ◦ ev ± . This implies in particular that given any δ -chain [ σ ] in a critical submanifold [ C − ] we have the identity (cid:16) [ σ ] × ˘ M + z ([ C − ] , [ C + ]) (cid:17) = [ σ ] × ˘ M + ( z ) ( [ C − ] , [ C + ]) , hence the operators ∂ oo , ∂ os , ∂ uo , ∂ us and ¯ ∂ ss , ¯ ∂ su , ¯ ∂ us , ¯ ∂ uu all commute with the action of as theyinvolve these fiber products.The previous discussion implies that the chain complexes ( ˇ C ∗ , ˇ ∂ ), ( ˆ C ∗ , ˆ ∂ ) and ( ¯ C ∗ , ¯ ∂ ) areall equipped with an involutory chain map . We can then define the subcomplexes consistingof the invariant chains ( ˇ C inv ∗ , ˇ ∂ ) , ( ˆ C inv ∗ , ˆ ∂ ) , ( ¯ C inv ∗ , ¯ ∂ ) , and similarly the subcomplexes((id + ) ˇ C ∗ , ˇ ∂ ) , ((id + ) ˆ C ∗ , ˆ ∂ ) , ((id + ) ¯ C, ¯ ∂ ) . The following result readily follows along the same lines of Lemma 2.12 in Chapter 3 usingLemma 3.1.
Lemma . The inclusion (id + ) ˇ C ∗ ֒ → ˇ C inv ∗ is a quasi-isomorphism, and similarly forthe other variants. We are finally ready to introduce the main protagonist of the present chapter.
Definition . We define the Pin(2) -monopole Floer homology groups of Y equippedwith the self-conjugate spin c structure s denoted by c HS ∗ ( Y, s ) , c HS ∗ ( Y, s ) , HS ∗ ( Y, s )as the homology groups c HS ∗ ( Y, s ) = H ( ˇ C inv ∗ , ˇ ∂ ) c HS ∗ ( Y, s ) = H ( ˆ C inv ∗ , ˆ ∂ ) HS ∗ ( Y, s ) = H ( ¯ C inv ∗ , ¯ ∂ ) . Here again the choice of metric and perturbations is implicit in our notation.The objects we have just defined can be graded by the set J ( Y, s ) as in Section 2 of Chapter3, and we can define their negative completions c HS • ( Y, s ) , c HS • ( Y, s ) , HS • ( Y, s ) . As the spin c structure is torsion, they also admit an absolute rational grading, which we willdefine in more detail in the next Section. Before describing and proving the properties ofthese invariants as in Chapter 3, we perform an explicit calculation of these groups in thesimplest possible case, which will also be central in the construction of the maps induced bycobordisms. . INVARIANT CHAINS AND FLOER HOMOLOGY 141 Example . Consider S with the round metric and its unique spin c structure (whichis obviously self-conjugate). When we consider the unperturbed monopole equations, becausethe metric has scalar positive curvature and there is no homology, there only one solutionof the form [ B, B is gauge equivalent to the spin connection B , as [ B , D B does not have simple (quaternionic) spectrum, but we can choose a small -equivariantperturbation such that • q is Pin(2)-non-degenerate; • [ B,
0] is still the only critical point.Hence the critical submanifolds in B σk ( S , s ) consist of a doubly infinite sequence of spheres S , each corresponding to an eigenspace of D q ,B . Again Lemma 3.16 in Chapter 2 tells us thatthe perturbation is weakly self-indexing, so we can run the associated spectral sequence whichcollapses at the first page for dimensional reasons. This implies that the Pin(2)-monopoleFloer homology is just a direct sum of the homology groups of the invariant chains of thecritical submanifolds. Following Lemma 3.1 we have that: • c HS k ( S , s ) = F for k non negative and congruent to 0 , • c HS k ( S , s ) = F for k negative and congruent to 1 , • HS k ( S , s ) = F for k congruent to 0 , δ -chains in the first stable criticalsubmanifold (which coincides with the absolute rational grading). In particular, up to gradingshift of − c HS • ( S , s ) is naturally isomorphic as a graded group to the ring R inDefinition 1.3.The definition of the maps induced by cobordisms is essentially the same as Section 3 inChapter 3, with some slight modifications to be made in order to perform a -invariant con-struction. We first focus on the case of a cobordism equipped with a self-conjugate spin c struc-ture. In particular, given two three manifolds with self-conjugate spin c structures ( Y ± , s ± )and regular -equivariant perturbations q ± , a cobordism X endowed with a self-conjugatespin c structure s X between them and a invariant cohomology class of the form Q i V n on theconfiguration space B σk ( X, s ), we want to define the maps c HS • ( Q i V n | X, s X ) : c HS • ( Y − , s − ) → c HS • ( Y + , s + ) c HS • ( Q i V n | X, s X ) : c HS • ( Y − , s − ) → c HS • ( Y + , s + ) HS • ( Q i V n | X, s X ) : HS • ( Y − , s − ) → HS • ( Y + , s + ) . As in Section 3 of the previous Chapter, this is done by considering on the cobordism a finiteset of marked points p = { p , . . . , p m } . In this case, we also assume that the perturbation q i oneach end corresponding to each puncture is also -equivariant, regular and does not introduceirreducible critical points. We need to add extra Pin(2)-equivariant asd -perturbations on theblow up as defined in Section 2 in order to achieve regularity of the moduli spaces. In thiscase completed chain complexˇ C • ( Y − ) ⊗ C u • ( S ) ⊗ · · · ⊗ C u • ( S m )
42 4. Pin(2)-MONOPOLE FLOER HOMOLOGY contains the subcomplex complex(4.4) ˇ C inv • ( Y − ) ⊗ C u, inv • ( S ) ⊗ · · · ⊗ C u, inv • ( S m )generated by elements [ σ ] ⊗ [ σ ] ⊗ · · · ⊗ [ σ m ] such that each factor is invariant. Noticethat is is not the subspace of the invariants of the natural action of (because of the nonindecomposable elements). The homology of the second chain complex is naturally identifiedwith c HS • ( Y − ) ⊗ c HS • ( S ) ⊗ · · · ⊗ c HS • ( S m ) . Similarly there are natural isomorphisms : M + z ([ C − ] , C , X ∗ , [ C + ]) → M + z ([ C − ] , C , X ∗ , [ C + ])commuting with each evaluation map. This last observation implies that the subcomplexˇ C ∗ ( Y − , p ) defined by transversality conditions is also -invariant, and we can define as aboveits subcomplexˇ C • ( Y − , p ) inv = ˇ C • ( Y − , p ) ∩ (cid:0) ˇ C inv • ( Y − ) ⊗ C u, inv • ( S ) ⊗ · · · ⊗ C u, inv • ( S m ) (cid:1) . It follows as in Lemma 3.1 in Chapter 3 (via Lemma 3.1) that its inclusion in the tensorproduct of the invariant subcomplexes is a quasi-isomorphism. The chain map ˇ m satisfies theproperty ◦ ˇ m = ˇ m ◦ , hence it restricts to a chain mapˇ m inv : ˇ C • ( Y − , p ) inv → ˇ C • ( Y + ) inv , which induces the map in homology c HS • ( X, p ) : c HS • ( Y − ) ⊗ R ⊗ · · · ⊗ R → c HS • ( Y + ) . Here we used the identification from Example 3.4 for each small regular -equivariant pertur-bation d HM • ( S , q ) ∼ = R where R is the ring in definition 1.3. Here again there is a grading shift. Finally we definefor any element a ∈ R the element c HS • ( a | X, s X )( x ) = c HS • ( X, p )( x ⊗ a ⊗ · · · ⊗ a m ) . where the product of the a i is a .Having constructed these maps, we can discuss the general invariance and functorialityresult, which follows along the same lines of Section3 in Chapter 3. We first introduce anuseful definition. Definition . The category cob spin is the category whose objects are connected com-pact oriented 3-manifolds with a fixed spin structure and whose objects are isomorphismclasses of connected oriented cobordisms which admit a spin structure restricting to the givenones on the boundary.In the definition we consider spin structures and not self conjugate spin c structures becausethe composition of spin cobordisms is not necessarily spin. On the other hand it is clear fromthe definition that in our case morphisms compose well.As in the classical case, the key result in the proof of invariance and functoriality is thefollowing. . INVARIANT CHAINS AND FLOER HOMOLOGY 143 Theorem . The
Pin(2) -monopole Floer homology groups and the maps induced bycobordisms do not depend on the choice of metric and -equivariant perturbation. They definecovariant functors c HS • : cob spin → mod R c HS • : cob spin → mod R HS • : cob spin → mod R where mod R is the category of graded topological R -modules. More in general, if Y , Y and Y are -manifolds, X are X are cobordisms from Y to Y and from Y to Y such thatthe composite X ◦ X admits a self conjugate spin c structure s then c HS • ( a | X ◦ X , s ) = X s = s | X X s = s | X c HS • ( a | X , s ) ◦ c HS • ( a | X , s ) . where a a is a in R . Here the R -module structure is defined as in the classical case by considering the mapsinduced by the product cobordism I × Y , see Section 3 in Chapter 3. Proof.
The proof follows in the same way as the ones in Section 3 of Chapter 3, bytaking the maps induced on the tensor product of the invariant subcomplexes. The only nonobvious verification is the fact in the proof of Proposition 3.6 that the map ˆ m ( B , p ) definedby a disk with m punctures induces the multiplication R ⊗ · · · ⊗ R → R a ⊗ · · · ⊗ a m a · · · a m . To do this, one can first reduce to the case with only two punctures
R ⊗ R → R by using the associativity property (whose metric-stretching proof carries over without com-plications also in the case of more punctures). We then focus on the two properties Q ⊗ Q Q Q ⊗ Q , as the proof of all other relations follow in a very similar way. Let [ C − ] be the criticalsubmanifold corresponding to the first negative eigenvalue. Then the moduli space M + ([ C − ] , [ C − ] , ( B ) ∗{ p ,p } , [ C − ])is two dimensional and the evaluation map to each critical submanifold [ C − ] is surjectiveand has degree one. Because of the intersection structure of one dimensional chains in R P ,this implies that given a representative [ σ ] ⊗ [ σ ] of Q ⊗ Q its fibered product consists onan odd number of pairs consisting of a point and its image under , proving the first of theproperties. For the second property we know that the moduli space M + ([ C − ] , [ C − ] , ( B ) ∗{ p ,p } , [ C − ])is six dimensional. Any generic pair of points gives rise under the fibered product to agenerator of [ C − ], as it follows from the product structure in the classical case. In particular,as an invariant generator in each [ C − ] consists of a even number of points, the fibered productis zero in homology. (cid:3)
44 4. Pin(2)-MONOPOLE FLOER HOMOLOGY
Example . The proof shows that c HS • ( S , q ) ∼ = F [[ V ]][ Q ] / ( Q ) {− } where deg Q = − Q = − R -module.Even thought in the rest of the present work we will only make use of the equivariantconstructions, it will be interesting in some developments of the theory to consider also theinteraction between these new invariants and the usual ones. Suppose we are given a pair s = ¯ s of conjugate but non self-conjugate spin c structures. Then the involution can bethought as a diffeomorphism : C ( Y, s ) → C ( Y, ¯ s )hence one can define a chain complex with an involution by taking the direct sumˇ C • ( Y, s ) ⊕ ˇ C • ( Y, ¯ s )where we pick a non-degenerate tame perturbation q on C ( Y, s ) and its image ∗ q on C ( Y, ¯ s ).Here the action of sends a critical point to its image under the diffeomorphism (which isstill a critical point). The homology of the invariant subcomplex, which we denote by c HS • ( Y, [ s ]) , is naturally isomorphic to the canonically isomorphic groups d HM • ( Y, s ) ∼ = d HM • ( Y, ¯ s )Here [ s ] denotes the equivalence class of s under the involution on the set of spin c struc-tures Spin c ( Y ) given by conjugation. This group can be thought as an R -module under theextension of coefficients F [[ V ]][ Q ] / ( Q ) → F [[ U ]] V V Q . This last map is the one induced in cohomology by the double cover BS → B Pin(2) . One can also define maps induced by cobordisms equipped with a pair of non self conjugatespin c structures. As an example, if ( X, s X ) is a cobordism between ( Y ± , s ± ) with s − self-conjugate and s + = ¯ s + , then ( X, ¯ s X ) is a cobordism between ( Y − , s − ) and ( Y + , ¯ s + ), and weobtain a map of R -modules c HS • ( X ; [ s X ]) : c HS • ( Y − , s − ) → c HS • ( Y + , [ s + ])from the -equivariant chain mapˇ m ( s ) ⊕ ˇ m (¯ s ) : ˇ C • ( Y − , s − ) → ˇ C • ( Y + , s + ) ⊕ ˇ C • ( Y + , ¯ s + )where the maps ˇ m ( s ) and ˇ m (¯ s ) are the ones defined in Chapter 3. Here, we chose theperturbations on the two collars to be -equivariant. We can define analogous maps in all theother cases. As discussed in the introduction, we can define the total group c HS • ( Y ) = M [ s ] ∈ Spin c ( Y ) / c HS • ( Y, [ s ]) , . INVARIANT CHAINS AND FLOER HOMOLOGY 145 for which the following result holds. Theorem . The total
Pin(2) -monopole Floer homology groups define a functors fromthe category cob of compact connected oriented three manifolds and isomorphism classes ofcobordism between them to the category of topological F [[ V ]] -modules. We now discuss a few additional properties that our invariants satisfy. We focus on thecase of a self-conjugate spin c structure s . The following result follows as Proposition 2.7 inChapter 3 by taking the invariant subcomplexes. Proposition . For any ( Y, s ) , there is an exact sequence of graded R -modules · · · i ∗ −→ c HS ∗ ( Y, s ) j ∗ −→ c HS ∗ ( Y, s ) p ∗ −→ HS ∗ ( Y, s ) i ∗ −→ c HS ∗ ( Y, s ) j ∗ −→ . . . where the maps i ∗ , j ∗ and p ∗ have degree , , − respectively. The following result should be thought as a version of the Gysin exact sequence (in thesimple case of S bundles, i.e. double coverings). Proposition . For every three manifold Y and self conjugate spin c structure s , thereis an exact sequence · · · → c HS k +1 ( Y, s ) e ∗ −→ c HS k ( Y, s ) ι ∗ −→ d HM k ( Y, s ) π ∗ −→ c HS k ( Y, s ) → c HS k − ( Y, s ) → · · · The maps e ∗ , ι ∗ and π ∗ are maps of R -modules. The similar result holds for the other versionsof Pin(2) -monopole Floer homology.
Proof.
There is a short exact sequence of chain complexes0 → ˇ C inv ∗ ֒ → ˇ C ∗ id+ −→ (id + ) ˇ C ∗ → . The sequence in the statement is the associated long exact sequence, where we use Lemma3.1 to identify the homology groups. (cid:3)
The cohomology groups c HS ∗ ( Y, s ) , c HS ∗ ( Y, s ) and HS ∗ ( Y, s ) are defined as in Section 2 ofChapter 3 as the groups of − Y with the appropriate change in the gradings. These comewith canonical intersection pairings with the corresponding homology groups, for example c HS k ( Y, s ) ⊗ c HS k ( Y, s ) → F , where we consider the intersection of -invariant as the intersection of the corresponding (aftersubdivision) cycles in the quotient by the action of . We cannot prove that such pairing isperfect for a general three manifold, but we expect that the proof of the next result could beadapted once has a better knowledge of the reducible solutions analogous to that of Chapter35 in the book. Proposition . Suppose Y is a rational homology sphere. Then the intersection pair-ing is perfect. Proof.
We focus on the to version of the invariants. Because the space is a rational ho-mology sphere, we can choose a perturbation with a single reducible critical point downstairs.For the grading high enough, the result follows from the fact that the intersection pairing on R P is perfect (see Proposition 1.17 in Chapter 3), as there are only finitely many irreduciblesolutions. We can then proceed with the following inductive proof. We have the commutativediagram
46 4. Pin(2)-MONOPOLE FLOER HOMOLOGY d HM k +1 ( Y ) c HS k +1 ( Y ) c HS k ( Y ) d HM k ( Y ) c HS k ( Y )Hom d HM k +1 ( Y ) Hom c HS k +1 ( Y ) Hom c HS k ( Y ) Hom d HM k ( Y ) Hom c HS k ( Y ) a b F c F where the upper row is the Gysin sequence form Proposition 3.10 and the lower row is thedual of the corresponding sequence for cohomology (or equivalently for − Y ), and the verticalmaps are the ones induced by the intersection pairing. In particular we know that a and c areisomorphism. Suppose now b is also an isomorphism. Then applying the four lemma to theleft part of the diagram, we obtain that F is injective. Applying then the four lemma to theright part of the diagram we obtain that F is also surjective, hence it is an isomorphism. (cid:3)
4. Some computations
In this section we provide some basic computations of the invariants which can be per-formed by explicitly solving the equations as in Chapters 36 and 37 of the book. Before doingthis, we discuss a non-vanishing result and quickly review the definition of absolute gradings.
Proposition . For every three manifold Y and self conjugate spin c structure s , thegroups c HS ∗ ( Y, s ) , c HS ∗ ( Y, s ) , HS ∗ ( Y, s ) are non-zero in infinitely many gradings. Proof.
This readily follows from the Gysin exact sequence in Proposition 3.10 and thedeep non-vanishing result for the classical monopole Floer groups when the spin c structurehas torsion first Chern class, see Corollaries 35 . . . . (cid:3) We quickly review the discussion of the absolute rational gradings for torsion spin c struc-tures in Section 28 . c structures our definition of absolute grading (as the three dimensional spincobordism group is trivial) we will consider the more general case of a non-self conjugatespin c structure, as it sometimes makes computations easier. Furthermore, this will allow toconsider also the interaction with the non-equivariant counterparts of the theory. Given anintegral 2-dimensional cohomology class on a cobordism W that restricts to a torsion classon the boundary, we define h c, c i = (˜ c ∪ ˜ c )[ W, ∂W ] ∈ Q where ˜ c is any lift the image of c in the rational cohomology H ( W ; Q ) to H ( W, ∂W ; Q ).Given now a self-conjugate spin c structure s on Y , choose any four manifold ˜ X whose bound-ary is Y and over which the s extends. We will think of this manifold as a cobordism X from S to Y obtained by removing a ball. In the following definition, we have a fixed equi-variant perturbation q on Y (see also Definition 28 . . . SOME COMPUTATIONS 147 Morse-Bott perturbation on S . Recall the definition of relative grading on a cobordism fromSection 6 in Chapter 2. Definition . Given a self-conjugate spin c structure s on Y , let X be any cobordismfrom S to Y as above. For a δ -chain [ σ ] with value in a critical submanifold [ C ], we defineits absolute grading gr Q ([ σ ]) as the rational numbergr Q ([ σ ]) = − gr z ([ C ] , X, [ C ]) + dim[ σ ] + 14 h c ( S + ) , c ( S + ) i − ι ( X ) − σ ( X ) ∈ Q where • [ C ] is the the stable reducible critical manifold of S corresponding to the smallestpositive eigenvalue of the Dirac operator, and z is any relative homotopy class; • S + is the spinor bundle for the corresponding spin c structure on X ; • ι ( X ) is the characteristic number12 ( χ ( X ) + σ ( X ) − b ( Y ))from Definition 25 . . Q ([ a ]) = ( gr Q ([ a ]) , [ a ] boundary stablegr Q ([ a ]) − , [ a ] boundary unstable . Recall that the characteristic number ι ( X ) is an integer as it is the index of an operator,see Lemma 25 . . . Proposition . The absolute grading gr Q ([ σ ]) is well defined, i.e. independent of thechoice of X and homotopy class z . Its fractional part is the same as the fractional part of ( h c , c i − σ ( X )) / . If Y is a homology three sphere and [ σ ] is a δ -chain in a boundary-stablereducible critical submanifold, its grading gr([ σ ]) is an integer with the same parity as itsdimension. Finally, in any case the duality isomorphism ˇ ω : c HS • ( − Y, s ) → c HS • ( Y, s ) maps elements of grading k to elements of grading − − b ( Y ) − k . Remark . This absolute grading convention differs from the one adopted in HeegaardFloer homology (see [
OS03 ]), which can be recovered by subtracting − b ( Y ) / Definition . The standard R -module M is the graded R -module M = F [ V − , V ]][ Q ] / ( Q ) {− } , where F [ V − , V ]] denotes the ring of Laurent series in V . In particular the element generatingthe homogeneos component of degree zero is V − Q .
48 4. Pin(2)-MONOPOLE FLOER HOMOLOGY
The grading shift is performed so that M agrees as a graded module with HS • ( S ), seeCorollary 4.9. Proposition . Let Y be a rational homology sphere, and s a self-conjugate spin c structure on Y , then after grading shifts we have the isomorphism of graded R -modulesHS • ( Y, s ) ∼ = M If Y is a homology sphere, the generator V has even degree. The result tells us that in this case the only interesting content of the invariant are theabsolute rational gradings.
Proof.
The bar version of the Floer invariants only involves reducible critical points, andin the case of a rational homology sphere Y there is only [ B ,
0] before blowing up. Hence,we are in the same situation as Example 3.4. The determination of module structure followswith the same proof as in Theorem 3.6, see also Example 3.7. The fact that in the case of ahomology sphere v has even degree follows the fact that it is represented by a δ -chain of evendimension and Proposition 4.3. (cid:3) The result we have just stated only deals with the reducible solutions, and we do not havein general any knowledge about the irreducible ones. In particular, the only computation wecan perform by hand is the case in which the manifold has positive scalar curvature or is flat.We first have the following result.
Proposition . If the manifold Y has positive scalar curvature, then the map j ∗ iszero, hence the long exact sequence splits into a canonical direct sum decompositionHS • ( Y, s ) = c HS • ( Y, s ) ⊕ c HS • ( Y, s ) {− } . where the braces indicate the grading shift. Proof.
This follows in the same way as in Section 36 . -invariantMorse function on the torus T = H ( Y ; R ) /H ( Y ; Z )parametrizing the reducible solutions to the monopole equations, one can construct a Pin(2)-equivariant tame perturbation such that the boundary map ¯ ∂ us is zero (see Lemma 36 . . ∂ su is also zero at the chain level. Indeed the image is always negligiblefor dimensional reasons unless we are considering two critical manifolds [ C ] and [ C − ] cor-responding to the first and positive and negative eigenvalues respectively, and zero chains inthe former. On the other hand, the subset of trajectories in the unparametrized moduli space˘ M ([ C ] , [ C − ]) converging to a given point is already compact simply because there are nopossible intermediate resting points. So the δ -chain defined in [ C − ] by a zero chain in [ C ]is three dimensional and has no boundary, so is negligible. The vanishing of this two mapsimplies that the chain complex ¯ C ∗ splits as the direct sum of ˇ C ∗ and ˆ C ∗ . (cid:3) Remark . By the characterization of the moduli spaces in Lemma 3.16 in Chapter 2we can see that the chain defined above is indeed diffeomorphic to a three sphere. . SOME COMPUTATIONS 149
Corollary . Suppose Y is a rational homology sphere with positive scalar curvature,and s a self-conjugate spin c structure. Then we have up to grading shifts the isomorphism of R -modules HS • ( Y, s ) ∼ = M c HS • ( Y, s ) ∼ = M / R · c HS • ( Y, s ) ∼ = R{− } . Furthermore, if Y is S , the absolute grading of the minimum non zero element in c HS • is ,and if Y is the Poincar´e homology sphere (oriented as the boundary of the plumbing along thegraph − E ) the absolute grading is − . Proof.
The computation of the gradings follows from the usual ones (see for example[
KMOS07 ]), as the homology of the chain complex computes the usual invariants. (cid:3)
We then turn to the case of S × S . This case is also tractable because of the positivescalar curvature, and in particular because of Proposition 4.7 we only need to compute thebar version of the theory. Proposition . Letting s is the only self-conjugate spin c structure on S × S , up tograding shifts we have the isomorphisms of R -modulesHS • ( S × S , s ) ∼ = L {− } ⊗ H ∗ ( S ; F ) c HS • ( S × S , s ) ∼ = L {− } / R · ⊗ H ∗ ( S ; F ) c HS • ( S × S , s ) ∼ = R{− } ⊗ H ∗ ( S ; F ) . In particular, the minimum absolute grading of a non zero element in c HS • ( S × S , s ) is − . Proof.
This computation can be performed directly by choosing an appropriate pertur-bation. Because of Proposition 4.7, we can just focus on the bar version of the chain complex,and can choose as in Section 36 . f ◦ p where p : B ( Y, s ) → T is a smooth gauge-invariant retraction equivariant for the action of and f is a -invariantMorse function with exactly two critical points, which are the reducible critical points cor-responding to the two spin connections on S × S , namely [ B ,
0] with index 1 and [ B , B ,
0] and ends at [ B , T connecting such two connections, and they are conjugate under the actionof . Consider the critical submanifolds [ C i ] and [ C i ] the critical submanifolds correspondingto the i th eigenvalue of D q ,B and D q ,B . Because of dimensional reasons (see Proposition4.7), the only interesting moduli spaces when computing differentials are those of the form˘ M + ([ C i ] , [ C i ]). These are to two copies of C P identified via the action of , and each of theevaluation maps are either constant or a diffeomorphism onto the image. This implies thattheir total contribution on the invariant chains is trivial, hence the only non-zero differentialsare the ones in the chain complexes of the critical submanifolds, which implies the result.
50 4. Pin(2)-MONOPOLE FLOER HOMOLOGY
The R -module structure is determined in a similar way as in the proof of Theorem 3.6. Inparticular, the only additional thing to check in this case is that the action of Q and V doesnot send a δ -chain over one reducible over the other reducible. This follows as above fromthe fact that there are the corresponding moduli space has two components related by theaction of hence their contribution on the invariant chains is zero. Finally the fact that thelowest grading of a non zero element is − S × S . (cid:3) Remark . Given this computation, we can define the map induced by a cobordismand an element in the exterior algebra Λ ∗ ( H ( W ; Z ) / Tor ⊗ F ) as in Remark 3.10 in Chapter3. We now study the case of the three torus T , which is more involved. Proposition . Let s be the only self conjugate spin c structure on the three torus. Wehave the isomorphisms of graded R -modulesHS • ( T , s ) ∼ = M{− } ⊗ (cid:0) H ( T ; F ) ⊕ H ( T ; F ) (cid:1)c HS • ( T , s ) ∼ = M{− } ⊗ (cid:0) H ( T ; F ) ⊕ H ( T ; F ) (cid:1)c HS • ( T , s ) ∼ = R{− } ⊗ (cid:0) H ( T ; F ) ⊕ H ( T ; F ) (cid:1) . In particular the absolute grading of the minimum non zero element in c HS • ( T , s ) is − . Proof.
The proof follows the same line of Proposition 4.10, with some complications dueto the fact that the chain complexes we are dealing with are much bigger than that case. Asin Section 34 . ǫ and δ very small (see Proposition 37 . .
1) so thatthe perturbed functional(4.5) −L = L − ( δ/ k Φ k + ǫf has no irreducible critical points. Here f is a -invariant function obtained via a retraction asin Proposition 4.10 by Morse function on the space of reducible solutions of the unperturbedequations T (which is a three torus itself) which has a standard form with eight critical points(the gauge equivalence classes of spin connections): x of index 0 (corresponding to the onlyclass of flat connections for which the Dirac operator has kernel), three points y α of index1, three points w α of index 2 and z of index 3. Then for each i ∈ Z we have eight criticalsubmanifolds [ C wi ][ C z i ] [ C z i ] [ C z i ][ C y i ] [ C y i ] [ C y i ][ C xi ]corresponding to the i th eigenvalue of the corresponding Dirac operators. The interestingfeature is that the grading on [ C xi ] is shifted up by two because of the spectral flow. First ofall, we make some simple considerations on the moduli spaces that arise.(1) The moduli spaces ˇ M + ([ C wi ] , [ C z α i ]) and ˇ M + ([ C z α i ] , [ C y β i ]) all consist of pairs of C P related by the action of as in Proposition 4.10, each evaluation map being constantor a diffeomorphism. . SOME COMPUTATIONS 151 (2) The moduli spaces ˇ M + ([ C y β i ] , [ C xi ]) consist of pairs of points related by the action of .(3) the moduli spaces ˇ M + ([ C wi ] , [ C y β i ]) is naturally decomposed as the union of two 3-dimensional δ -cycles in [ C y β i ] related by the action of . To see this, notice that thespace of unparametrized flow lines for f between w and y β can be identified withthe union of four intervals I, J, I and J , and the action of in exchanges them inpairs. The observation in point (1) implies that the stratum of the moduli space overeach boundary point of such interval (which corresponds to a broken trajectory) iseither empty or consists of a copy of C P . The claim follows by taking the cyclesparametrized by the unions I ∪ J and I ∪ J . Notice that the possible complicationthat the boundary consists of pairs of conjugate C P mapping to conjugate points isnot an issue because these δ -chains are negligible, hence zero in our context.(4) For the same reason, the moduli spaces ˇ M + ([ C z α i ] , [ C xi ]) can be written as the disjointunion of two one dimensional δ -cycles in [ C xi ] related by the action of .(5) Finally, analogous considerations imply that the moduli space ˇ M + ([ C wi ] , [ C xi ]) is a twodimensional δ -cycle [ σ i ] in [ C xi ].Our claim is that the δ -cycle [ σ i ] represents a generator of the top homology of [ C xi ]. Oncethis is settled, the computation can be performed by running the spectral sequence associatedto the energy filtration for the perturbed functional −L (see Section 2 in Chapter 3). Indeedfirst page of this spectral is the following H inv2 ([ C wi ]) H inv1 ([ C wi ]) L H inv2 ([ C z α i ]) H inv2 ([ C xi ]) H inv0 ([ C wi ]) L H inv1 ([ C z α i ]) L H inv2 ([ C y β i ]) H inv1 ([ C xi ]) L H inv0 ([ C z α i ] L H inv2 ([ C y β i ]) H inv0 ([ C xi ]) L H inv2 ([ C y β i ])Of course the inv indicates that we are considering the corresponding -invariant subcomplexof the critical submanifolds, and each of the groups is a copy of F . Here we focus only on thepart involving the eight critical manifolds with index i because the map on the first page d : H inv0 (([ C y i +1 ]) ⊕ H inv0 ([ C y i +1 ]) ⊕ H inv0 ([ C y i +1 ]) → H inv2 ([ C xi ])is zero by symmetry considerations. Exploiting the discussion on the moduli spaces above wecan conclude that there are only two non trivial differentials: • on the first page, there is a differential d : H inv2 (([ C y i ]) ⊕ H inv2 ([ C y i ]) ⊕ H inv2 ([ C y i ]) → H inv0 ([ C xi ])which is non zero on each summand coming from the moduli spaces in case (2); • on the third page, there is a differential d : H inv2 ([ C wi ]) → H inv2 ([ C xi ])coming from chain [ σ i ] in case (5), which we know is a generator of the latter group.
52 4. Pin(2)-MONOPOLE FLOER HOMOLOGY
The module structure follows in a way similar to that of Proposition 4.10. The only additionaldetail is that we can find for β = 1 , , H inv2 ([ C y β i ]) involving a top generator of exactly one of the [ C y β i ], which again follows from theclaim on [ σ i ].Finally, to prove our claim about the cycle [ σ i ], we consider the energy filtration on thetotal complex, not just the invariant subcomplex. In particular, symmetry considerationsimply that H ([ C wi ]) , H ([ C x α i ]) for α = 1 , , , H ([ C xi ])all survive until the E page. On the other hand, the usual monopole Floer group in thegrading corresponding to the last four groups has rank three, so that the differential d : H inv0 ([ C wi ]) → H inv0 ([ C xi ])is non trivial. Hence [ σ i ] is a generator of the top homology of [ C xi ]. (cid:3) Remark . The perturbation in the last computation is not weakly self-indexing.We finally provide some computations for the other flat three manifolds. The computationfor the Hantzsche-Wendt manifold (which is a rational homology sphere) is analogous to thatof Corollary 4.9 because one can achieve transversality using a perturbation as in the case ofthe three torus without introducing irreducible critical points. The remaining case are all flatthree manifolds which are torus bundles over S with b ( Y ) = 1. Proposition . Let Y be flat torus bundle over the circle, and let s be the self-conjugate spin c structure corresponding to the -plane field ξ tangent to the fiber. Then wehave the isomorphisms of R -modulesHS • ( Y, s ) ∼ = Q · M ⊗ ( F ⊕ F { } ) c HS • ( Y, s ) ∼ = Q · M / R · Q ⊗ ( F ⊕ F { } ) c HS • ( Y, s ) ∼ = R · Q ⊗ ( F {− } ⊕ F { } ) . Remark . Notice that some of these manifolds have other self-conjugate spin c struc-tures other than the one we are considering. Proof.
As in the case of the three torus, we can choose a perturbation as in equation(4.5) where the Morse function on the circle of flat connections T ( s ) is such that • f is -invariant; • f has two maxima corresponding to the spin connections [ B ] and [ B ] and two minima[ A ] and [ A ] corresponding to the flat connections such that the corresponding Diracoperator has kernel.The two minima are conjugate via the involution . Here we set [ B ] to be the spin connectionsuch that on a path connecting it to a minimum the family of Dirac operators D A − ǫ for ε > −
1. Call [ C i ] , [ C i ] and [ b i ] , [ b i ] the critical submanifoldscorresponding to the i th eigenvalue of the respective Dirac operators. In particular the firsttwo are copies of C P while the latter consist of single points. We have that for dimensionalreasons (as in Proposition 4.7) the complex is a direct sum of pieces as in the followingdiagram: . MANOLESCU’S β INVARIANT AND THE TRIANGULATION CONJECTURE 153 C ([ C i ]) C ([ C i ]) C ([ b i +1 ]) ⊕ C ([ b i +1 ]) C ([ C i ]) C ([ C i ]) C ([ b i ]) ⊕ C ([ b i ]) C ([ C i ]) C ([ C i ]) ∂ F ∂ ss ∂ F ∂ F ∂ ss ∂ F Here we assume i ≥
0, and the chain complex of the critical submanifold [ C i ] is shifteddown by two because of the spectral flow. Each of the moduli spaces M + ([ C i ] , [ b i +1 ]) and M + ([ C i ] , [ b i +1 ]) consists of a single point, and these are conjugated under the action of .The same holds for M + ([ C i ] , [ b i ]) and M + ([ C i ] , [ b i ]), and the result readily follows. (cid:3)
5. Manolescu’s β invariant and the Triangulation conjecture In this final section we define the counterpart of Manolescu’s β invariant introduced in[ Man13a ] in our Morse-theoretic approach, and prove the main properties which are used inthe disproof of the Triangulation conjecture as discussed in the Introduction. The essentialingredient is the absolute grading discussed in the previous section. The invariant β and itscompanions α and γ are the Pin(2) analogue of the Froyshøv invariant h in monopole Floerhomology (see [ Frø10 ] and Chapter 39 in the book) and the Heegaard Floer correction terms([
OS03 ]). Let Y be a rational homology sphere, and consider a self-conjugate spin c structure s on it. Clearly, if Y is actually a homology sphere, or more in general if it the first homologyhas no two-torsion, there is only one such spin c -structure. As stated in Proposition 4.6, upto grading shifts HS • ( Y, s ) is isomorphic as a R -module to the standard Pin(2)-module M from Definition 4.5. A rational homology sphere Y with a self-conjugate spin c structure s gives rise to a preferred proper homogeneous quotient of M , namely the image of the map(4.6) i ∗ : HS • ( Y, s ) → c HS • ( Y, s ) . We use these submodules to define three numerical invariants of the rational homology sphere.
Definition . For α ≥ β ≥ γ rational numbers all congruent to ǫ modulo 1, the standard Pin(2) -module M α,β,γ is defined to be the R -module obtained as the quotient of (asuitably shifted) standard module M by a graded submodule such that • the element of lowest degree of the form Q V a has degree 2 α ; • the element of lowest degree of the form QV b has degree 2 β + 1; • the element of lowest degree of the form V c has degree 2 γ + 2.
54 4. Pin(2)-MONOPOLE FLOER HOMOLOGY
It follows from the structure as a R -module that the image of the map i ∗ in equation(4.6) is a module of the form M α,β,γ . Definition . For a rational homology sphere Y equipped with a self-conjugate spin c structure s , the Manolescu’s invariants α ( Y, s ), β ( Y, s ) and γ ( Y, s ) are defined as the rationalnumbers α ≥ β ≥ γ such that i ∗ (cid:0) HS • ( Y, s ) (cid:1) ∼ = M α,β,γ as graded R -modules. Example . Following Corollary 4.9, in the case of S the grading preserving identifi-cation HS • ( S ) ∼ = M . Moreover i ∗ is surjective and c HS • ( S ) consists of the part in non-negative. Hence we have α ( S ) = β ( S ) = γ ( S ) = 0 . Similarly, the case of the Poncar´e homology sphere Y is identical up to a shift in degree by −
2, hence α ( Y ) = β ( Y ) = γ ( Y ) = − . More in general, whenever Y is a homology sphere these three numbers are integers thanksto Proposition 4.3. Remark . The following interpretation is closer to Manolescu’s original definition. Wehave that i ∗ defines an injective map on HS • ( Y, s ) / im( p ∗ ) so we can also define α ( Y, s ) = 12 (cid:16) min { gr Q ( x ) | x ∈ i ∗ F [ V − , V ]] , x = 0 } (cid:17) β ( Y, s ) = 12 (cid:16) min { gr Q ( x ) | x ∈ i ∗ Q F [ V − , V ]] , x = 0 } − (cid:17) γ ( Y, s ) = 12 (cid:16) min { gr Q ( x ) | x ∈ i ∗ Q F [ V − , V ]] , x = 0 } − (cid:17) under the usual identification of HS • ( Y, s ) with M .The next result is the central theorem regarding Manolescu’s invariants. Recall that theRohklin invariant of a rational homology three sphere Y equipped with a spin structure s isdefined as µ ( Y, s ) = 18 σ ( X ) mod 2 Z for any smooth oriented four manifold X with boundary Y which admits a spin structurerestricting to the given one on the boundary. This is well defined because of the celebratedRokhlin’s theorem, and is an integer modulo two in the case Y is a homology sphere (see forexample [ Sav02 ] for an introduction to the subject).
Theorem . Let Y be a rational homology sphere, and s a self-conjugate spin c structure.Then the Manolescu’s invariants α ( Y, s ) , β ( Y, s ) and γ ( Y, s ) satisfy the inequalities α ( Y, s ) ≥ β ( Y, s ) ≥ γ ( Y, s ) . Furthermore, they satisfy the following properties: . MANOLESCU’S β INVARIANT AND THE TRIANGULATION CONJECTURE 155 (1) if − Y denotes Y with the opposite orientation, then α ( − Y, s ) = − γ ( Y, s ) β ( − Y, s ) = − β ( Y, s ) γ ( − Y, s ) = − α ( Y, s );(2) the reduction modulo two of α ( Y, s ) , β ( Y, s ) and γ ( Y, s ) are the opposite of the Rokhlininvariant − µ ( Y, s ) ; (3) given a smooth spin cobordism W with negative definite intersection form from ( Y , s ) to ( Y , s ) , then α ( Y , s ) ≥ α ( Y , s ) + 18 b ( W ) β ( Y , s ) ≥ β ( Y , s ) + 18 b ( W ) γ ( Y , s ) ≥ γ ( Y , s ) + 18 b ( W ) . In particular the invariants α ( Y, s ) , β ( Y, s ) and γ ( Y, s ) are invariant under homology cobor-dism. Remark . We expect our Pin(2)-monopole Floer homology invariants (and in partic-ular these numerical invariants) to coincide with those defined in [
Man13a ].Restricting to the case in which Y is a homology sphere, we obtain the following result.This implies Theorem 0.1 in the Introduction, see the discussion after the result. Corollary
Man13a ]) . There is an integer valued invariant β associatedto each homology sphere Y which satisfies the following properties: (1) if − Y is Y with the opposite orientation, then β ( − Y ) = − β ( Y ) ; (2) the reduction modulo two of β ( Y ) is the Rokhlin invariant µ ( Y ) ; (3) if W is a spin cobordism between two homology spheres Y and Y with negative definiteintersection form, then β ( Y ) ≥ β ( Y ) + 18 b ( W ) . In particular, β is invariant under homology cobordism. Proof of Theorem 5.5.
The fact that α ( Y, s ) ≥ β ( Y, s ) ≥ γ ( Y, s )is implicit in the definition of the invariants and follows from the R -module structure. Re-garding property (2), if X is a spin four manifold bounding Y we have (as in Lemma 28 . . σ ] is a zero dimensional δ -chain is some criticalsubmanifold, we have ¯gr Q ([ σ ]) ≡ − σ ( X ) mod 4 Z . This follows from the fact that we can compute gr z ([ σ ] , X, [ σ ]) by means of the reducibleconfiguration corresponding to a spin connection on the cobordism. If we choose a Pin(2)-equivariant perturbation on S this relative grading turns out to be the sum of the (real)
56 4. Pin(2)-MONOPOLE FLOER HOMOLOGY index of a perturbed Dirac operator (which is divisible by four as it is quaternionic) and ι ( X ). This readily implies that α ( Y, s ) is congruent to the opposite of the Rokhlin invariantmodulo two, and the same proof applies to the invariants β ( Y, s ) and γ ( Y, s ).Property (1) follows from the definitions of the invariant using the fact that if HS k ( Y, s ) i ∗ −→ c HS k ( Y, s )is injective, then by Poincar´e duality also HS − − k ( − Y, s ) p ∗ −→ c HS − − k ( − Y, s )is injective (see Proposition 4.3 for the gradings), hence by the universal coefficient theorem(Proposition 3.11) the dual map c HS − − k ( − Y, s ) p ∗ −→ HS − − k ( − Y, s )is surjective. Here we use the alternative point of view from Remark 5.4 and the fact thatthe kernel of i ∗ is the image of p ∗ .To prove property (3), we first modify the cobordism by surgery paying attention to theframings so that we obtain a spin four manifold with b ( W ) zero and the same intersectionform. We still call this cobordism W . For a fixed self-conjugate spin c structure s on W restriction to the given ones on the boundary, we can then consider the induced map HS • ( W, s ) : HS • ( Y , s ) → HS • ( Y , s ) . The claim is that this is an isomorphism of absolute degree b ( X ). The topological as-sumptions imply that on the cylindrical-end manifold W ∗ there is a unique anti-self-dualspin c connection (up to gauge equivalence), the spin connection A . If we chose the pertur-bations on the two ends to be small enough, the hypothesis that the cobordism is negativedefinite implies that we can achieve transversality for the moduli spaces without appealingto Pin(2)-equivariant asd -perturbations, so that we are in the same setting of the proof ofTheorem 39 . . C ] and [ C ] lyingover the unique reducible solution, it follows from the grading formulas and the fact that σ ( X ) = − b ( X ) that the moduli space M + ([ C ] , W ∗ , [ C ]) is two dimensional if and only ifthey have relative grading gr z ([ C ] , W, [ C ]) = 14 b ( X ) − . It is important to remark here that given any critical manifold [ C ] there is exactly one critical manifold [ C ] such that the above relation holds. This follows from the fact that thegrading shifts of the reducible critical submanifolds is determined by the Rokhlin invariants,as discussed above. In the case this holds, the moduli space M + ([ C ] , W ∗ , [ C ]) is a copyof C P on which acts as the antipodal map. Furthermore both evaluation maps ev ± arediffeomorphisms because they are equivariant hence they cannot be constant (cfr. the proofof Proposition 4.7). This proves our claim as all other maps are zero because of the usualdimensional argument. Finally, this implies that the commutative diagram of R -modules . MANOLESCU’S β INVARIANT AND THE TRIANGULATION CONJECTURE 157 HS • ( Y , s ) HS • ( Y , s ) c HS • ( Y , s ) c HS • ( Y , s ) HS • ( W, s ) i ∗ c HS • ( W, s ) i ∗ is identified with the commutative diagram of R -modules M{ ǫ } M{ ǫ }M α ,β ,γ M α ,β ,γ xx where the ǫ i indicate suitable grading shifts, the x on the top row is an isomorphism of degree b ( X ), and the groups in the bottom rows are the images of the respective maps i ∗ . Inparticular the vertical arrows are surjective, and Property (3) follows. Finally, the invarianceunder homology cobordism invariance follows because if W is a homology cobordism from Y to Y then ¯ W is a homology cobordism from Y to Y . (cid:3) ibliography [AB95] D. M. Austin and P. J. Braam. Morse-Bott theory and equivariant cohomology. In The Floermemorial volume , volume 133 of
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