A Polyfold Proof of the Arnold Conjecture
aa r X i v : . [ m a t h . S G ] J a n A POLYFOLD PROOF OF THE ARNOLD CONJECTURE
BENJAMIN FILIPPENKO, KATRIN WEHRHEIM
Abstract.
We give a detailed proof of the homological Arnold conjecture for nondegenerateperiodic Hamiltonians on general closed symplectic manifolds M via a direct Piunikhin-Salamon-Schwarz morphism. Our constructions are based on a coherent polyfold description for modulispaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from CP × M to C + × M and C − × M , as developed by Fish-Hofer-Wysocki-Zehnder as part of the SymplecticField Theory package. To make the paper self-contained we include all polyfold assumptions,describe the coherent perturbation iteration in detail, and prove an abstract regularization theoremfor moduli spaces with evaluation maps relative to a countable collection of submanifolds.The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish. Introduction
Let (
M, ω ) be a closed symplectic manifold and H : S × M → R a periodic Hamiltonianfunction. It induces a time-dependent Hamiltonian vector field X H : S × M → T M given by ω ( X H ( t, x ) , · ) = d H ( t, · ). We denote the set of contractible periodic orbits by(1) P ( H ) := (cid:8) γ : S → M (cid:12)(cid:12) ˙ γ ( t ) = X H ( t, γ ( t )) and γ is contractible (cid:9) and note that periodic orbits can be identified with the fixed points of the time 1 flow φ H : M → M of X H . We call this Hamiltonian system nondegenerate if φ H × id M is transverse to the diagonaland hence cuts out the fixed points transversely. In particular, this guarantees a finite set of periodicorbits. Arnold [A] conjectured in the 1960s that the minimal number of critical points of a Morsefunction on M is also a lower bound for the number of periodic orbits of a nondegenerate Hamiltoniansystem as above. In this strict form, the Arnold conjecture has been confirmed for Riemann surfaces[E] and tori [CZ]. A weaker form is accessible by Floer theory, introduced by Floer [F2, F3] in the1980s. It constructs a chain complex generated by P ( H ) that can be compared with the Morsecomplex generated by the critical points of a Morse function. When Floer homology is well-defined,it is usually independent of the Hamiltonian, and on a compact symplectic manifold can in factbe identified with Morse homology, which is also independent of the Morse function and computesthe singular homology. Using this approach, the following nondegenerate homological form of theArnold conjecture was first proven by Floer [F1, F4] in the absence of pseudoholomorphic spheres. Theorem 1.1.
Let ( M, ω ) be a closed symplectic manifold and H : S × M → R a nondegenerateperiodic Hamiltonian function. Then P ( H ) ≥ P dim Mi =0 dim H i ( M ; Q ) . Floer’s proof was later extended to general closed symplectic manifolds [HS, O, FO, LT], and in thepresence of pseudoholomorphic spheres of negative Chern number requires abstract regularizationsof the moduli spaces of Floer trajectories since perturbations of the geometric structures may notyield regular moduli spaces; see e.g. [MW]. Further generalizations and alternative proofs have beenpublished in the meantime, using a variety of regularization methods. The purpose of this note is toprovide a general and maximally accessible proof of Theorem 1.1 – using an abstract perturbationscheme provided by the polyfold theory of Hofer-Wysocki-Zehnder [HWZ], following an approach byPiunikhin-Salamon-Schwarz [PSS] based on [Sc2], and building on polyfold descriptions of Gromov-Witten moduli spaces [HWZ1] as well as their degenerations in Symplectic Field Theory [EGH, FH].To be more precise, let CF = ⊕ γ ∈P ( H ) Λ h γ i be the Floer chain group of the Hamiltonian H withcoefficients in the Novikov field Λ (see § CM, d) be the Morse complex with coefficients in Λ associated to a Morse function f : M → R and a suitable metric on M (see § Theorem 1.2.
There exist Λ -linear maps P SS : CM → CF , SSP : CF → CM , ι : CM → CM ,and h : CM → CM such that the following holds.(i) ι is a chain map, that is ι ◦ d = d ◦ ι .(ii) ι is a Λ -module isomorphism.(iii) h is a chain homotopy between SSP ◦ P SS and ι , that is ι − SSP ◦ P SS = d ◦ h + h ◦ d . Here we view the Floer chain group CF as a vector space over Λ – not as a chain complex,and in particular do not consider a Floer differential. Thus we are neither constructing a Floerhomology for H , nor identifying it with the Morse homology of f . However, the algebraic structuresin Theorem 1.2 suffice to deduce the homological Arnold conjecture for the Hamiltonian H as follows. Proof of Theorem 1.1.
Denote the sum of the Betti numbers k := P dim Mi =0 dim H i ( M ; Q ). Let( CM Q , d Q ) be the Morse complex over Q as defined in §
3. Then by the isomorphism of singular andMorse homology there exist c , . . . , c k ∈ CM Q that are cycles, d Q c i = 0, and linearly independentin the Morse homology over Q . Since the Morse differential d : CM → CM is given by Λ-linearextension of d Q from CM Q ⊂ CM the chains c , . . . , c k ∈ CM are also cycles d c i = d Q c i = 0 andlinearly independent in the Morse homology over Λ. By Theorem 1.2 (i),(ii), ι induces an isomor-phism Hι : HM → HM on homology. This in particular implies that [ ι ( c )] , . . . , [ ι ( c k )] ∈ HM arealso linearly independent in homology, that is for any λ , . . . , λ k ∈ Λ we have(2) P ki =1 λ i · ι ( c i ) ∈ im d = ⇒ λ = . . . = λ k = 0 . We now show that
P SS ( c ) , . . . , P SS ( c k ) ∈ CF are Λ-linearly independent, proving P ( H ) ≥ k since the elements of P ( H ) generate CF by definition. This proves the theorem.Let λ , . . . , λ k ∈ Λ be a tuple such that P ki =0 λ i · P SS ( c i ) = 0 . Then we deduce from Theorem 1.2 (iii) that P ki =0 λ i · ι ( c i ) = P ki =0 λ i · (cid:0) SSP (cid:0)
P SS ( c i ) (cid:1) + d h ( c i ) + h (d c i ) (cid:1) = SSP (cid:16)P ki =0 λ i · P SS ( c i ) (cid:17) + P ki =0 λ i · d h ( c i ) = d (cid:16)P ki =0 λ i · h ( c i ) (cid:17) , which implies λ = . . . = λ k = 0 by (2). (cid:3) This algebraically minimalistic approach of deducing the homological Arnold conjecture fromthe existence of maps
P SS and
SSP whose composition is chain homotopic to an isomorphismon the Morse complex was developed in 2011 discussions of the second author, Peter Albers, andJoel Fish with Mohammed Abouzaid and Thomas Kragh. These were prompted by our observationthat any rigorous proof of “Floer homology equals Morse homology” seemed to require equivarianttransversality; see Remark 1.3. Since equivariant transversality is generally obstructed – even forsimple equivariant sections of finite rank bundles – we were looking for a proof that requires the leastamount of specific geometric insights or new abstract tools. We ultimately expect the [PSS]-approachto yield an isomorphism between Floer and Morse homology, as well as generalizations of spectralinvariants [Sc3] to all closed symplectic manifolds, but the purpose of the present manuscript is toexemplify the use of existing polyfold results to obtain detailed, rigorous, and transparent proofs.For that purpose we include Appendix A, which summarizes all necessary polyfold notions andabstract tools. Here we moreover establish in Theorem A.9 a relative perturbation result that shouldbe of independent interest: It allows to bring moduli spaces with an evaluation map into generalposition to a countable collection of submanifolds. Besides these 10 pages of background, there are afew more technical complications due to the current lack of polyfold publications: Since the polyfolddescription of Floer-theoretic moduli spaces – while evident to experts – is not published apart froman outline in [W2], we reformulate all moduli spaces in our application into SFT moduli spaces,using the fact that graphs of perturbed pseudoholomorphic maps are J -curves for an appropriate J . Since the polyfold description of SFT moduli spaces [FH, FH1] is also not completely published POLYFOLD PROOF OF THE ARNOLD CONJECTURE 3 yet, we summarize the anticipated results in Assumptions 4.3, 5.5, 6.3. Finally, we give a detailedaccount of the iterative construction of coherent perturbations in the proofs of Lemma 6.4 and 6.6.To strike a balance between supplying technical details that are not easily available in the literatureand maximal accessibility, we have clearly labeled all such technical work. Readers willing to viewpolyfold theory as a black box can save 20 pages by skipping these parts.For accessibility we begin with reviews of the pertinent facts on the Novikov field, §
2, and Morsetrajectories, §
3. The proof of Theorem 1.2 then proceeds by constructing the
P SS and
SSP mapsin § C ± × M , constructing the isomorphism ι and chain homotopy h in § CP × M and its degeneration into C − × M and C + × M , and proving their algebraicrelations in § Remark 1.3. (i) There are essentially two approaches to the general Arnold conjecture as statedin Theorem 1.1. The first – developed by [F4] and used verbatim in [HS, O, FO, LT] – is toestablish the independence of Floer homology from the Hamiltonian function, and to identify theFloer complex for a C -small S -invariant Hamiltonian H : M → R with the Morse complex for H . This requires S -equivariant transversality to argue that isolated Floer trajectories must be S -invariant, hence Morse trajectories. A conceptually transparent construction of equivariant andtransverse perturbations – under transversality assumptions at the fixed point set which are met inthis setting – can be found in [Z], assuming a polyfold description of Floer trajectories.(ii) The second approach to Theorem 1.1 by [PSS] is to construct a direct isomorphism betweenthe Floer homology of the given Hamiltonian and the Morse homology for some unrelated Morsefunction. Two chain maps P SS : CM → CF , SSP : CF → CM between the Morse and Floercomplexes are constructed from moduli spaces of once punctured perturbed holomorphic spheres withone marking evaluating to the unstable resp. stable manifold of a Morse critical point, and with thegiven Hamiltonian perturbation of the Cauchy-Riemann operator on a cylindrical neighbourhood ofthe puncture. Then gluing and degeneration arguments are used to argue that both P SS ◦ SSP and
SSP ◦ P SS are chain homotopic to the identity, and hence
SSP is the inverse of
P SS onhomology. However, sphere bubbling can obstruct these arguments: In the first chain homotopy itcreates an ambiguity in the choice of nodal gluing when the intermediate Morse trajectory shrinksto zero length. (We expect to be able to avoid this by arguing that “index 1 solutions genericallyavoid codimension 2 strata” – another classical fact in differential geometry that should generalizeto polyfold theory.) The second chain homotopy is as claimed in Theorem 1.2 (iii) but with ι = id,which requires arguing that the only isolated holomorphic spheres with two marked points evaluatingto an unstable and stable manifold are constant. This again requires S -equivariant transversality(which we expect to be able to achieve with the techniques in [Z]).(iii) Theorem 1.2 is proven by following the [PSS]-approach as above but avoiding the use of newpolyfold technology such as equivariant or strata-avoiding perturbations. In particular, ι is the mapthat results from counting holomorphic spheres that intersect an unstable and stable manifold; itsinvertibility is deduced from an “upper triangular” argument.(iv) The techniques in this paper – combining existing perturbation technology with the polyfolddescriptions of SFT moduli spaces – would also allow to define the Floer differential, prove d = 0,establish independence of Floer homology from the Hamiltonian (and other geometric data), andprove that P SS and
SSP are chain maps. Then the chain homotopy between
SSP ◦ P SS and theisomorphism ι implies that P SS is injective and
SSP surjective on homology. However, provingthat
P SS and
SSP are isomorphisms on homology, or directly identifying the Floer complex of asmall S -invariant Hamiltonian with its Morse complex, requires the techniques discussed in (ii).Moreover, a proof of independence of Floer homology from the choice of abstract perturbationwould require a study of the algebraic consequences of self-gluing Floer trajectories in expecteddimension − A ∞ -setting in [LW]. We thank Peter Albers and Joel Fish for helping develop the outline of this project – and Edi Zehn-der for asking the initial question. The project was further supported by various discussions withMohammed Abouzaid, Helmut Hofer, Thomas Kragh, Kris Wysocki, and Zhengyi Zhou. Crucialfinancial support was provided by NSF grants DMS-1442345 and DMS-1708916.
BENJAMIN FILIPPENKO, KATRIN WEHRHEIM The Novikov field
We use the following Novikov field Λ associated to the symplectic manifold (
M, ω ). Let H ( M )denote integral homology and consider the map ω : H ( M ) → R given by the pairing ω ( A ) := h ω, A i for A ∈ H ( M ). The image of this pairing is a finitely generated additive subgroup of the realnumbers denoted Γ := im ω = ω ( H ( M )) ⊂ R . The
Novikov field
Λ is the set of formal sums λ = P r ∈ Γ λ r T r , where T is a formal variable, with rational coefficients λ r ∈ Q which satisfy the finiteness condition ∀ c ∈ R { r ∈ Γ | λ r = 0 , r ≤ c } < ∞ . The multiplication is given by λ · µ = (cid:0)P r ∈ Γ λ r T r (cid:1) · (cid:0)P s ∈ Γ µ s T s (cid:1) := P t ∈ Γ (cid:0)P r + s = t λ r µ s (cid:1) T t . This defines a field Λ by [HS, Thm.4.1] and the discussion preceding the theorem in [HS, § R .We will moreover make use of the following generalization of the invertibility of triangular matriceswith nonzero diagonal entries. Lemma 2.1.
Let M = ( λ ij ) ≤ i,j ≤ ℓ ∈ Λ ℓ × ℓ be a square matrix with entries λ ij ∈ Λ in the Novikovfield. Suppose that λ ij = P r ∈ Γ ,r ≥ λ ijr T r with λ ij = 0 for i = j and λ ii = 0 . Then M is invertible.Proof. Since Λ is a field, invertibility of M is equivalent to det( M ) = 0. Write det( M ) = P r ∈ Γ µ r T r ∈ Λ for some µ r ∈ Q . It suffices to show that µ = 0.We proceed by induction on the size of the matrix M . In the ℓ = 1 base case, when M is a 1 × M = [ λ ], we have det( M ) = λ = P r ∈ Γ µ r T r with µ r = λ r so µ = λ = 0.Now suppose that M is size ℓ × ℓ for some ℓ > ℓ − × ( ℓ −
1) matrix N satisfying the hypotheses of the Lemma, we have det( N ) = P r ∈ Γ µ Nr T r with µ N = 0. For 1 ≤ j ≤ ℓ , let C j denote the matrix obtained by deleting the first row and j -thcolumn of M . Then N := C is an ( ℓ − × ( ℓ −
1) matrix that satisfies the hypotheses of theLemma, and the cofactor expansion of the determinant yieldsdet( M ) = λ det( N ) + P ℓj =2 ( − j λ j det( C j ) . By hypothesis, all entries of M are of the form λ ij = P r ≥ λ ijr T r . Since the determinants det( N )and det( C j ) are polynomials of those entries, they are of the same form – with zero coefficientsfor T r with r <
0. Since we moreover have λ j = 0 for j ≥ T ) of λ j det( C j ) is 0. Hence the constant term ofdet( M ) = P µ r T r is µ = λ · µ N , where µ N = 0 by induction and λ = 0 by hypothesis. Thisimplies det( M ) = µ + . . . = 0 and thus finishes the proof. (cid:3) The Morse complex and half-infinite Morse trajectories
This section reviews the construction of the Morse complex as well as the compactified spaces ofhalf-infinite Morse trajectories which will appear in all our moduli spaces.3.1.
Euclidean Morse-Smale pairs.
The Morse complex can be constructed for any Morse-Smalepair of function and metric on a closed smooth manifold M (and more general spaces). However, wewill also work with half-infinite Morse trajectories, and to obtain natural manifold with boundaryand corner structures on these, we will restrict ourselves to the following special setting. Definition 3.1. A Euclidean Morse-Smale pair on a closed manifold M is a pair ( f, g ) con-sisting of a smooth function f ∈ C ∞ ( M, R ) and a Riemannian metric g on M satisfying a normalform and transversality condition as follows. POLYFOLD PROOF OF THE ARNOLD CONJECTURE 5 (i)
For every critical point p ∈ Crit( f ) of index | p | ∈ N there exists a local chart φ to aneighbourhood of ∈ R n such that φ ∗ f ( x , . . . , x n ) = f ( p ) − ( x + . . . + x | p | ) + ( x | p | +1 + . . . + x n ) ,φ ∗ g = d x ⊗ d x + . . . + d x n ⊗ d x n . (ii) For every pair of critical points p, q ∈ Crit( f ) the intersection of unstable and stable mani-folds is transverse, W − p ⋔ W + q . Remark 3.2.
One can check that given any Morse function f and metric g (e.g. one that isEuclidean in Morse normal coordinates around the critical points), any generic perturbation of g onannuli around the critical points yields a Euclidean Morse-Smale pair, see e.g. [BH, Prp.2] or [Sc1,Prp.2.24]. Hence Euclidean Morse-Smale pairs exist on every closed manifold, and for any givenMorse function.3.2. The Morse complex.
For distinct critical points p − = p + ∈ Crit( f ) the space of unbrokenMorse trajectories (which are necessarily nonconstant) is M ( p − , p + ) := (cid:8) τ : R → M (cid:12)(cid:12) ˙ τ = −∇ f ( τ ) , lim s →±∞ τ ( s ) = p ± (cid:9) / R (3) ∼ = (cid:0) W − p − ∩ W + p + (cid:1) / R ∼ = W − p − ∩ W + p + ∩ f − ( c ) . It is canonically identified with the intersection of unstable and stable manifold modulo the R -action given by the flow of −∇ f , or their intersection with a level set for any regular value c ∈ ( f ( p + ) , f ( p − )). Both formulations equip it with a canonical smooth structure of dimension | p − | −| p + | −
1, see e.g. [Sc1, § W − p forall p ∈ Crit( f ) induces orientations on the trajectory spaces M ( p − , p + ) by e.g. [Wb, § f, g ) is obtained by counting (with signs induced by the orientations)the zero dimensional spaces of unbroken trajectories,(4) CM Q := M p ∈ Crit( f ) Q h p i , d Q h p − i := X | p + | = | p − |− M ( p − , p + ) h p + i . It computes the singular homology of M ; see e.g. [Sc1, § CM Q = L i =0 ,..., dim M C i M by Morse indices C i M = L | p | = i Q h p i , and with d i := d Q | C i M we have H i ( M ; Q ) ∼ = ker d i / im d i +1 .The PSS and SSP morphisms will be constructed on the Morse complex with coefficients in theNovikov field Λ from Section 2,(5) CM = CM Λ := CM Q ⊗ Λ = L p ∈ Crit( f ) Λ h p i , with differential d = d Λ the Λ-linear extension of d Q (defined as above on generators). This complexis naturally graded with differential of degree 1,(6) C ∗ M = L dim Mi =0 C i M, C i M = L | p | = i Λ h p i , d : C i M → C i − M. Compactified spaces of Morse trajectories.
Our construction of moduli spaces will alsomake use of the following spaces of half-infinite unbroken Morse trajectories for p ± ∈ Crit( f ) M ( M, p + ) := (cid:8) τ : [0 , ∞ ) → M (cid:12)(cid:12) ˙ τ = −∇ f ( τ ) , lim s →∞ τ ( s ) = p + (cid:9) , M ( p − , M ) := (cid:8) τ : ( −∞ , → M (cid:12)(cid:12) ˙ τ = −∇ f ( τ ) , lim s →−∞ τ ( s ) = p − (cid:9) . These will be equipped with smooth structures of dimension dim M ( M, p + ) = dim M − | p + | resp.dim M ( p − , M ) = | p − | by the evaluation mapsev : M ( M, p + ) → M, τ τ (0) , ev : M ( p − , M ) → M, τ τ (0) , which identify the trajectory spaces with the unstable and stable manifolds M ( M, p + ) ∼ = W + p + resp. M ( p − , M ) ∼ = W − p − . Note that these spaces contain constant trajectories at a critical point, { τ ≡ p + } ∈ M ( M, p + ) and { τ ≡ p − } ∈ M ( p − , M ). To compactify these trajectory spaces in amanner compatible with Morse theory, we cannot simply take the closure of the unstable or stablemanifold W ± p ± ⊂ M , but must add broken trajectories involving the bi-infinite Morse trajectories. BENJAMIN FILIPPENKO, KATRIN WEHRHEIM
The bi-infinite trajectories from (3) which appear in such a compactification are always nonconstant,i.e. between distinct critical points p − = p + . So, unlike constant half-infinite length trajectories, ourconstructions will not involve constant bi-infinite trajectories, and we simplify subsequent notationby setting M ( p, p ) := ∅ for all p ∈ Crit( f ). With that we first introduce spaces of k -fold brokenhalf- or bi-infinite Morse trajectories for k ∈ N and p ± ∈ Crit( f ), M ( M, p + ) k := S p ,...,p k ∈ Crit( f ) M ( M, p ) × M ( p , p ) . . . × M ( p k , p + ) , M ( p − , M ) k := S p ,...,p k ∈ Crit( f ) M ( p − , p ) × M ( p , p ) . . . × M ( p k , M ) , (7) M ( p − , p + ) k := S p ,...,p k ∈ Crit( f ) M ( p − , p ) × M ( p , p ) . . . × M ( p k , p + ) . Now the compactifications of the spaces of half- or bi-infinite Morse trajectories are given by M ( M, p + ) := [ k ∈ N M ( M, p + ) k , M ( p − , M ) := [ k ∈ N M ( p − , M ) k , M ( p − , p + ) := [ k ∈ N M ( p − , p + ) k , with topology given by the Hausdorff distance between the images of the broken or unbroken trajec-tories. Compactness of these spaces is proven analogously to the bi-infinite Morse trajectory spacesin e.g. [BH, Prp.3], using [W1, Lemma 3.5]. Moreover, [W1, Lemma 3.3] shows that the evaluationmaps extend continuously toev : M ( M, p + ) → M, (cid:0) τ , [ τ ] , . . . , [ τ k ]) τ (0) , (8) ev : M ( p − , M ) → M, (cid:0) [ τ ] , . . . , [ τ k − ] , τ k ) τ k (0) . Smooth structures on these spaces are obtained by the following variation of a folk theorem, whichis proven in [W1], using techniques similar to those of [BH] for the bi-infinite trajectory spaces.
Theorem 3.3.
Let ( f, g ) be a Euclidean Morse-Smale pair and p ± ∈ Crit( f ) . Then M ( M, p + ) , M ( p − , M ) , and M ( p − , p + ) are compact, separable metric spaces and carry the structure of a smoothmanifold with corners of dimension dim M ( M, p + ) = dim M − | p + | , dim M ( p − , M ) = | p − | , and dim M ( p − , p + ) = | p − | − | p + | − . Their k -th boundary stratum is ∂ k M ( . . . ) = M ( . . . ) k . Moreover,the evaluation maps (8) are smooth. For reference, we recall the definition of a manifold with (boundary and) corners and its strata.
Definition 3.4. A smooth manifold with corners of dimension n ∈ N is a second countableHausdorff space M together with a maximal atlas of charts φ ι : M ⊃ U ι → V ι ⊂ [0 , ∞ ) n (i.e.homeomorphisms between open sets such that ∪ ι U ι = M ) whose transition maps are smooth.For k = 0 , . . . , n the k -th boundary stratum ∂ k M is the set of all x ∈ M such that for some (andhence every) chart the point φ ι ( x ) ∈ [0 , ∞ ) n has k components equal to . Remark 3.5. (i) To orient the Morse trajectory spaces in Theorem 3.3 we fix a choice of orien-tation on each unstable manifold W − p ∼ = M ( p, M ) for p ∈ Crit( f ), and orient W + p ∼ = M ( M, p )such that T p M = T p W − ⊕ T p W + induces the orientation on M given by the symplectic form.This also induces orientations on M ( p − , p + ) = W − p − ∩ W + p + / R that are coherent (by e.g. [Wb, § ∂ M ( · , · ) = S q ∈ Crit( f ) o ( · , q, · ) M ( · , q ) × M ( q, · ) with universal signs o ( · , q, · ) = ±
1. We compute the relevant cases: For M ( M, q ) × M ( q, p + ) ֒ → ∂ M ( M, p + ) withdim M ( q, p + ) = 0 the sign is o ( M, q, p + ) = ( − | p + | +1 . Indeed, a point in M ( q, p + ) is positivelyoriented if T W − q ∼ = h −∇ f i × N W + p + . Here we identify N p + W + p + ∼ = T p + W − p + , and the outer normaldirection is represented by ∇ f , so that the sign arises fromT W − p + × T W + p + ∼ = T W − q × T W + q ∼ = h −∇ f i × T W − p + × T W + q ∼ = T W − p + × h ( − | p + | ∇ f i × T W + q × T M ( q, p + ) . Similarly, for M ( p − , q ) ×M ( q, M ) ֒ → ∂ M ( p − , M ) with dim M ( p − , q ) = 0 the sign is o ( p − , q, M ) =+1 since −∇ f is an outer normal and T W − p − ∼ = h −∇ f i × T W − q when T M ( p − , q ) = + { } . POLYFOLD PROOF OF THE ARNOLD CONJECTURE 7 (ii) For computational purposes in § p − , p + ∈ Crit( f ) with the same Morse index | p − | = | p + | , M ( p − , M ) ev × ev M ( M, p + ) = (cid:8) ( τ − , τ + ) ∈ M ( p − , M ) × M ( M, p + ) (cid:12)(cid:12) ev( τ − ) = ev( τ + ) (cid:9) = ( ∅ ; p − = p + , ( τ − ≡ p − , τ + ≡ p + ) ; p − = p + . To verify this recall that the compactifications M ( p − , M ) and M ( M, p + ) are constructed in (7)via broken flow lines involving bi-infinite Morse trajectories in M ( p i , p i +1 ), which are (defined tobe) nonempty only for | p i | > | p i +1 | . So we have M ( p − , p ) × . . . × M ( p k , M ) ⊂ M ( p − , M ) only for | p k | < | p − | and M ( M, p ) × . . . × M ( p k , p + ) ⊂ M ( M, p + ) only for | p | > | p + | , and thus the imageof the evaluation maps are contained in unions of unstable/stable manifoldsev( M ( p − , M )) ⊂ W − p − ∪ S | q − | < | p − | W − q − , ev( M ( M, p + )) ⊂ W + p + ∪ S | q + | > | p + | W + q + . Since the intersections W − q − ∩ W + q + are transverse by the Morse-Smale condition, they can benonempty only for | q − | +dim M −| q + | ≥ dim M . So this intersection is empty whenever | q + | > | q − | .Thus for | q − | < | p − | = | p + | < | q + | in the above images we have empty intersections W − q − ∩ W + q + = ∅ as well as W − q − ∩ W + p + = ∅ and W − p − ∩ W + q + = ∅ . This proves ev( M ( p − , M )) ∩ ev( M ( M, p + )) = W − p − ∩ W + p + , and for p − = p + this intersection is empty by transversality in (3). Lastly, for p ± = p we have W − p ∩ W + p = { p } since gradient flows do not allow for nontrivial self-connecting trajectories.This proves M ( p, M ) ev × ev M ( M, p ) = { ( p, p ) } .4. The PSS and SSP maps
In this section we construct the PSS and SSP morphisms in Theorem 1.2 between Morse and Floercomplexes. As in the introduction, we fix a closed symplectic manifold (
M, ω ) and a smooth function H : S × M → R . This induces a time-dependent Hamiltonian vector field X H : S → Γ(T M ), whichwe assume to be nondegenerate. Thus it has a finite set of contractible periodic orbits, denoted by P ( H ) as in (1). We moreover pick a Morse function f : M → R and denote its – again finite – set ofcritical points by Crit( f ). Then we will work with the Floer and Morse complexes over the Novikovfield from Section 2, CF = ⊕ γ ∈P ( H ) Λ h γ i , CM = ⊕ p ∈ Crit( f ) Λ h p i , and construct the Λ-linear maps P SS : CM → CF , SSP : CF → CM from moduli spaces which weintroduce in § § § The Piunikhin-Salamon-Schwarz moduli spaces.
To construct the moduli spaces, weneed to make further choices as follows. • Let J be an ω -compatible almost complex structure on M .Then the Cauchy-Riemann operator on maps u : Σ → M parametrized by a Riemann surfaceΣ with complex structure j is ∂ J u := (cid:0) d u + J ( u ) ◦ d u ◦ j (cid:1) ∈ Ω , (Σ , u ∗ T M ). • Let g be a metric on M such that ( f, g ) is a Euclidean Morse-Smale pair as in Definition 3.1. Itexists by Remark 3.2. • Let β : [0 , ∞ ) → [0 ,
1] be a smooth cutoff function with β | [0 , ≡ β ′ ≥
0, and β | [ e, ∞ ) ≡ Y H ∈ Ω , ( C , Γ(T M )) in polarcoordinates Y H ( re iθ , x ) := β ( r ) (cid:0) JX H ( θ, x ) r − d r + X H ( θ, x ) d θ (cid:1) . In the notation of [MS, § Y H = − ( X H β ) , given by the anti-holomorphic part of the1-form with values in Hamiltonian vector fields X H β which arises from the 1-form with values insmooth functions H β ∈ Ω ( C , C ∞ ( M )) given by H β ( re iθ ) = β ( r ) H ( θ, · )d θ . BENJAMIN FILIPPENKO, KATRIN WEHRHEIM
The vector-field-valued 1-form Y H encodes the Floer equation on both the positive cylindricalend { z ∈ C | | z | ≥ e } ∼ = [1 , ∞ ) × S and the negative end {| z | ≥ e } ∼ = ( −∞ , − × S (where β ≡ v ( s, t ) := u ( e ± ( s + it ) ) of a map u : C → M satisfies the Floerequation ( ∂ s + J∂ t ) v ( s, t ) = JX H ( t, v ( s, t )) iff ∂ J u ( z ) = Y H ( z, u ( z )). • For each γ ∈ P ( H ), fix a smooth disk u γ : D → M with u γ | ∂D ( e it ) = γ ( t ).Then for u : C ± → M with lim R →∞ u ( Re ± it ) = γ ( t ), denote by u u γ : CP → M thecontinuous map given by gluing u to u ± γ (where the ± denotes the orientation of D ). By abuse oflanguage, we will call A := [ u u γ ] = ( u u γ ) ∗ [ CP ] ∈ H ( M ) the homology class represented by u .Moreover, we denote by e u γ : D → D × M the graph of u γ . Then the graph e u : C → C × M, z ( z, u ( z )) glues with e u ± γ to a continuous map representing [ e u e u γ ] = e A := [ CP ]+ A ∈ H ( CP × M ),or more precisely e A = [ CP ] × [pt] + [pt] × A . Now the condition [ v e u γ ] = e A makes sense forother maps v : C → C × M with the same asymptotic behaviour, and we say v represents e A . Infact, we will suppress the notation e A and label spaces with A – as this specifies the topologicaltype of v .Given such choices, the (choice-dependent) morphisms P SS : CM → CF and SSP : CF → CM will be constructed from the following moduli spaces for critical points p ∈ Crit( f ), periodic orbits γ ∈ P ( H ), and A ∈ H ( M ) M ( p, γ ; A ) := (cid:8) u : C + → M (cid:12)(cid:12) u (0) ∈ W − p , ∂ J u = Y H ( u ) , lim R →∞ u ( Re it ) = γ ( t ) , [ u u γ ] = A (cid:9) , M ( γ, p ; A ) := (cid:8) u : C − → M (cid:12)(cid:12) u (0) ∈ W + p , ∂ J u = Y H ( u ) , lim R →∞ u ( Re − it ) = γ ( t ) , [ u u γ ] = A (cid:9) . Each of these moduli spaces can be described as the zero set of a Fredholm section ∂ J − Y H : B ± → E ± .Here the Banach manifolds B ± are given by a weighted Sobolev closure of the set of smooth maps u : C ± → M representing the homology class A with point constraint u (0) ∈ W ∓ p and satisfyinga decay condition lim R →∞ u ( Re ± it ) = γ ( t ), but not necessarily satisfying the perturbed Cauchy-Riemann equation ∂ J u = Y H ( u ). Then ∂ J − Y H is a Fredholm section of index I ( p, γ ; A ) = CZ ( γ ) + 2 c ( A ) − dim M + | p | , (9) I ( γ, p ; A ) = − CZ ( γ ) + 2 c ( A ) + dim M − | p | , where CZ ( γ ) is the Conley-Zehnder index with respect to a trivialization of u ∗ γ T M as in e.g. [Sc2], c ( A ) is the first Chern class of ( T M, J ) paired with A , and | p | is the Morse index of p ∈ Crit( f ).If the moduli spaces were compact oriented manifolds, then we could define P SS (and analogously
SSP ) by a signed count of the index 0 solutions,
P SS h p i := M ( p, γ ; A ) · T ω ( A ) h γ i , where the sum is over γ ∈ P ( H ) and A ∈ H ( M ) with I ( p, γ ; A ) = 0. In many cases – if spherebubbles of negative Chern number can be excluded – this compactness and regularity can be achievedby a geometric perturbation of the equation, e.g. in the choice of almost complex structure. Ingeneral, obtaining well defined “counts” of the moduli spaces requires an abstract regularizationscheme. We will use polyfold theory to replace “ M ( p, γ ; A )” by a count of 0-dimensional perturbedmoduli spaces. In the presence of sphere bubbles with nontrivial isotropy, the perturbations will bemulti-valued, yielding rational counts. Remark 4.1.
Compactness, or rather Gromov-compactifications, of the moduli spaces M ( p, γ ; A )and M ( γ, p ; A ) will result from energy estimates [MS, Remark 8.1.7] for solutions of ∂ J u = Y H ( u ),(10) E ( u ) := R C | d u + X H β ( u ) | ≤ R C u ∗ ω + k R H β k ≤ ω ([ u u γ ]) + K. Here the curvature R H β dvol C = d H β + H β ∧ H β = β ′ H d r ∧ d θ has finite Hofer norm k R H β k = R C (max R H β − max R H β ) = R ∞ R S | β ′ ( r ) | (max x ∈ M H ( θ, x ) − min x ∈ M H ( θ, x )) d θ d r since β ′ has compact support in [1 , e ]. Since moreover P ( H ) is a finite set, we obtain the aboveestimate with a finite constant K := k R H β k + max γ ∈P ( H ) R D u ∗ γ ω . Thus the energy of the perturbedpseudoholomorphic maps in each of our moduli spaces will be bounded since we fix [ u u γ ] = A . POLYFOLD PROOF OF THE ARNOLD CONJECTURE 9
Now SFT-compactness [BEHWZ] asserts that for any
C > { u : C → M | ∂ J u = Y H ( u ) , lim R →∞ u ( Re ± it ) = γ ( t ) , E ( u ) ≤ C } is compact up to breakingand bubbling. This compactness will be stated rigorously in polyfold terms in Assumption 5.5 (ii).4.2. Polyfold description of moduli spaces.
We will obtain a polyfold description for the modulispaces in § M ( p, γ ; A ) ∼ = M ( p, M ) ev × ev M − ( γ ; A ) , M ( γ, p ; A ) ∼ = M + ( γ ; A ) ev × ev M ( M, p ) . This couples the half-infinite Morse trajectory spaces from § M ± ( γ ; A ) := (cid:8) u : C ± → M (cid:12)(cid:12) ∂ J u = Y H ( u ) , lim R →∞ u ( Re ± it ) = γ ( t ) , [ u u γ ] = A (cid:9) , (12)via the evaluation maps (8) and ev : M ± ( γ ; A ) → M, u u (0) . (13)More precisely, the general approach to obtaining counts or more general invariants from modulispaces such as (11) is to replace them by compact manifolds – or more general ‘regularizations’ whichstill carry ‘virtual fundamental classes’). Polyfold theory offers a universal regularization approachafter requiring a compactification M ( . . . ) ⊂ M ( . . . ) of the moduli space and a description of thecompact moduli space M ( . . . ) = σ − (0) as zero set of a sc-Fredholm section σ : B ( . . . ) → E ( . . . ) ofa strong polyfold bundle. For an introduction to the language [HWZ] used here see Appendix § A.The Morse trajectory spaces are compactified and given a smooth structure in Theorem 3.3. TheGromov compactification and perturbation theory for (12) will be achieved by identifying thesesspaces with moduli spaces that appear in Symplectic Field Theory (SFT) as introduced in [EGH],compactified in [BEHWZ, CM1], and given a polyfold description in [FH1]. Here we identify u : C → M with the map to its graph e u : C → C × M, z ( z, u ( z )) as in [MS, § M ± ( γ ; A ) ∼ = f M ± SFT ( e γ ; A ) / Aut( C ± ) to an SFT modulispace for the symplectic cobordism C ± × M between ∅ and S × M . Here S × M is equipped withthe stable Hamiltonian structure ( ± d t, ω + dH t ∧ dt ) whose Reeb field ± ∂ t + X H t has simply coveredReeb orbits given by the graphs ˜ γ : t ( ± t, γ ( t )) of the 1-periodic orbits γ ∈ P ( H ) . Moreover,Aut( C ± ) is the action of biholomorphisms φ : C → C by reparametrization v v ◦ φ on the SFTspace for an almost complex structure e J ± H on C ± × M induced by J , X H , and j = ± i on C ± , f M ± SFT ( e γ ; A ) := (cid:8) v : C ± → C ± × M (cid:12)(cid:12) ∂ e J ± H v = 0 , v ( Re ± it ) ∼ e γ R ( t ) , [ v e u γ ] = [ CP ] + A (cid:9) . More precisely, the asymptotic requirement is d C × M (cid:0) v ( Re ± i ( t + t ) ) , e γ R ( t ) (cid:1) → t ∈ S as R → ∞ for the graphs e γ R ( t ) = ( Re ± it , γ ( t )) of the orbit γ parametrized by S ∼ = {| z | = R } ⊂ C ± .To express the evaluation (13) in SFT terms note that a holomorphic map in the given homologyclass intersects the holomorphic submanifold { } × M in a unique point , so we can fix the point0 ∈ C ± in the domain where this intersection occurs and rewrite the moduli space M ± ( γ ; A ) ∼ = (cid:8) v ∈ f M ± SFT ( e γ ; A ) (cid:12)(cid:12) v (0) ∈ { } × M (cid:9) / Aut( C ± ,
0) with a slicing condition and quotient by thebiholomorphisms which fix 0 ∈ C ± . Thus we rewrite (11) into the fiber products over C ± × M M ( p, γ ; A ) ∼ = M ( p, M ) { }× ev × ev + M +SFT ( γ ; A ) , (14) M ( γ, p ; A ) ∼ = M − SFT ( γ ; A ) ev − × { }× ev M ( M, p )using evaluation maps on the SFT moduli space with one marked pointev ± : M ± SFT ( γ ; A ) := f M ± SFT ( e γ ; A ) (cid:14) Aut( C ± , → C ± × M, [ v ] v (0) . (15) For definitions of these notions see [CM1, § C × M the positive symplectization end is R + × S × M → C × M, ( r, θ, x ) ( e r + iθ , x ). After reversing orientation on C there is an analogous negative end R − × S × M ֒ → C − × M . Here we have implicitly chosen asymptotic markers that fix a parametrization of each Reeb orbit. For solutions in f M ± SFT ( e γ ; A ) this follows from pr C ± ◦ v : C ± → C ± being an entire function with a pole of order1 at infinity (prescribed by the asymptotics). For e J ± H -holomorphic curves in the compactification, it follows frompositivity of intersections, see e.g. [CM2, Prop.7.1]. Now we will obtain a polyfold description of the PSS/SSP moduli spaces (14) by the slicing construc-tion of [Fi1] applied to polyfold descriptions of the SFT-moduli spaces M ± SFT ( e γ ; A ) (compactified asspace of pseudoholomorphic buildings with one marked point). This result is outlined in [FH], butto enable a self-contained proof of our results, we formulate it as assumption, where we use C ± := C ± ∪ S ∼ = { z ∈ C ± | | z | ≤ } as target factor for a simplified evaluation map, as explained in the following remark. Remark 4.2.
Note that the compactified moduli space M ± SFT ( γ ; A ) – in view of the noncompacttarget C ± × M – contains broken curves v : Σ = C ± ⊔ R × S ⊔ . . . ⊔ R × S → Σ × M . We do notneed a precise description of this compactification (beyond the fact that it exists and is cut out bya sc-Fredholm section), but it affects the formulation of the evaluation maps [ v, z ] v ( z ) for amarked point z ∈ Σ that v might map to a cylinder factor R × S × M ⊂ Σ × M . We will simplify theresulting sc ∞ evaluation with varying target – being developed in [FH1] – to a continuous evaluationmap ev ± : M ± SFT ( γ ; A ) → C ± into the compactified target C ± .For that purpose we topologize C ± ∼ = {| z | ≤ } as a disk via a diffeomorphism C ± → {| z | < } , re iθ f ( r ) e iθ induced by a diffeomorphism f : [0 , ∞ ) → [0 ,
1) that is the identity near 0, andits extension to a homeomorphism C ± → {| z | ≤ } via S = R / π Z → {| z | = 1 } , θ e ± iθ . Thenfor any marked point z ∈ R × S on a cylinder we project the evaluation v ( z ) ∈ R × S × M to S × M = ∂ C ± × M by forgetting the R -factor. The resulting simplified evaluation map will beunchanged and thus still sc ∞ when restricted to the open subset (ev ± ) − ( C ± × M ) of the ambientpolyfold – as stated in (v) below. This open subset inherits a scale-smooth structure, and stillcontains some broken curves – just not those on which the marked point leaves the main component.This suffices for our purposes since the fiber product construction uses the evaluation map only inan open set of curves [ v, z ] with v ( z ) ≈ ∈ C ± . Assumption 4.3.
There is a collection of oriented sc-Fredholm sections of strong polyfold bundles σ SFT : B ± SFT ( γ ; A ) → E ± SFT ( γ ; A ) and continuous maps ev ± : B ± SFT ( γ ; A ) → C ± × M , indexed by γ ∈ P ( H ) and A ∈ H ( M ) , with the following properties.(i) The sections have Fredholm index ind( σ SFT ) = CZ ( γ ) + 2 c ( A ) + dim M + 2 on B +SFT ( γ ; A ) , resp. ind( σ SFT ) = − CZ ( γ ) + 2 c ( A ) + dim M + 2 on B +SFT ( γ ; A ) .(ii) Each zero set M ± SFT ( e γ ; A ) := σ − (0) is compact, and given any C ∈ R there are only finitelymany A ∈ H ( M ) with ω ( A ) ≤ C and nonempty zero set σ − (0) ∩ B ± SFT ( γ ; A ) = ∅ .(iii) Equivalence classes under reparametrization of Aut( C ± , of smooth maps v : C ± → C ± × M that satisfy v ( Re ± it ) = (cid:0) Re ± it , γ ( t ) (cid:1) for sufficiently large R > and represent the class [ v e u γ ] =[ CP ] + A form a dense subset B ± dense ( γ ; A ) ⊂ B ± SFT ( γ ; A ) contained in the interior. On this subset,the section is σ SFT ([ v ]) = [( v, ∂ e J ± H v )] and ev ± ([ v ]) is evaluation as in (15) .(iv) The intersection of the zero set with the dense subset σ − (0) ∩ B ± dense ( γ ; A ) ∼ = M ± SFT ( e γ ; A ) isnaturally identified with the SFT moduli space in (15) .(v) The sections σ SFT have tame sc-Fredholm representatives in the sense of [Fi1, Def.5.4] , and theevaluation maps ev ± restrict on the open subsets B ± , C SFT ( γ ; A ) := (ev ± ) − ( C ± × M ) ⊂ B ± SFT ( γ ; A ) to sc ∞ maps ev ± : B ± , C SFT ( γ ; A ) → C ± × M , which are σ SFT -compatibly submersive in the sense ofDefinition A.4. Finally, this open subset contains the interior, ∂ B ± SFT ( γ ; A ) ⊂ B ± , C SFT ( γ ; A ) . Remark 4.4.
The properties (iii),(iv) in this assumption are stated only to give readers an intuitivesense of what spaces we are working with. For the specific application in this paper it would besufficient to assume the existence of sc-Fredholm sections and submersive maps, along with thefurther sections, maps, and boundary stratifications stated in Assumptions 5.5 and 6.3. Property(v), which is used to construct fiber products in Lemma 4.5, should follow similarly to the explanationgiven in [Fi1, Ex.5.1] for the Gromov-Witten polyfolds [HWZ1].We also expect the existence of a direct polyfold description of the moduli space (12) in terms ofa collection of sc-Fredholm sections σ : B ± ( γ ; A ) → E ± ( γ ; A ) with the same indices, and submersivesc ∞ maps ev ± : B ± ( γ ; A ) → M with the following simplified properties. POLYFOLD PROOF OF THE ARNOLD CONJECTURE 11 (iii’) The smooth maps u : C → M which equal u ( Re ± it ) = γ ( t ) for sufficiently large R > A form a dense subset of B ± ( γ ; A ) that is contained in the interior. On thissubset, the section is σ ( u ) = ∂ J u − Y H ( u ), and the evaluation is ev ± ( u ) = u (0).(iv’) The intersection of M ± ( γ ; A ) := σ − (0) with the dense subset from (iii’) is naturally identifiedwith the moduli space M ± ( γ ; A ) in (12).While such a construction should follow from the same construction principles as in [FH], there ispresently no writeup beyond [W2], which proves the Fredholm property in a model case. Alterna-tively, one could abstractly obtain this construction from restricting the setup in Assumption 4.3 tosubsets consisting of maps of the form v ( z ) = ( z, u ( z )). Thus there is no harm in using (iii’) and(iv’) as intuitive guide for our work with the abstract setup.Given one or another polyfold description of the naturally identified moduli spaces (12) or (15)and corresponding evaluation maps, we will now extend the identifications (11) or (14) to a fiberproduct construction of polyfolds which will contain these PSS/SSP moduli spaces. For p ∈ Crit( f ), γ ∈ P ( H ), and A ∈ H ( M ) we define the topological spaces˜ B + ( p, γ ; A ) := (cid:8) ( τ , v ) ∈ M ( p, M ) × B +SFT ( γ ; A ) (cid:12)(cid:12) (0 , ev( τ )) = ev + ( v ) (cid:9) = (cid:8) ( τ , v ) ∈ M ( p, M ) × B + , C SFT ( γ ; A ) (cid:12)(cid:12) (0 , ev( τ )) = ev + ( v ) (cid:9) , (16) ˜ B − ( γ, p ; A ) := (cid:8) ( v, τ ) ∈ B − SFT ( γ ; A ) × M ( M, p ) (cid:12)(cid:12) (0 , ev( τ )) = ev − ( v ) (cid:9) = (cid:8) ( v, τ ) ∈ B − , C SFT ( γ ; A ) × M ( M, p ) (cid:12)(cid:12) (0 , ev( τ )) = ev − ( v ) (cid:9) . We will use [Fi1] to equip these spaces with natural polyfold structures and show that the pullbacksof the sections σ SFT by the projections to B ± SFT ( γ ; A ) yield sc-Fredholm sections whose zero setsare compactifications of the PSS/SSP moduli spaces. This will require a shift in levels which is oftechnical nature as each m -level B m ⊂ B contains the dense “smooth level” B ∞ ⊂ B m , which itselfcontains the moduli space M = σ − (0) ⊂ B ∞ ; see Remark A.3. Lemma 4.5.
For any p ∈ Crit( f ) , γ ∈ P ( H ) , and A ∈ H ( M ) there exist open subsets B + ( p, γ ; A ) ⊂ ˜ B + ( p, γ ; A ) and B − ( γ, p ; A ) ⊂ ˜ B − ( γ, p ; A ) which contain the smooth levels ˜ B ± ( . . . ; A ) ∞ of the fiberproducts (16) and inherit natural polyfold structures. The smooth level of their interior is ∂ B + ( p, γ ; A ) ∞ = M ( p, M ) { }× ev × ev + ∂ B + , C SFT ( γ ; A ) ∞ ,∂ B − ( γ, p ; A ) ∞ = ∂ B − , C SFT ( γ ; A ) ∞ ev − × { }× ev M ( M, p ) . Moreover, pullback of the sc-Fredholm sections of strong polyfold bundles σ ± SFT : B ± SFT ( γ ; A ) →E ± SFT ( γ ; A ) under the projection B ± ( . . . ; A ) → B ± SFT ( . . . ; A ) induces sc-Fredholm sections of strongpolyfold bundles σ +( γ,p ; A ) : B + ( γ, p ; A ) → E + ( γ, p ; A ) resp. σ − ( p,γ ; A ) : B − ( p, γ ; A ) → E − ( p, γ ; A ) ofindex I ( p, γ ; A ) resp. I ( γ, p ; A ) given in (9) . Their zero sets contain the moduli spaces from § σ +( p,γ ; A ) − (0) = M ( p, M ) { }× ev × ev + σ +SFT − (0) ⊃ M ( p, γ ; A ) ,σ − ( γ,p ; A ) − (0) = σ − SFT − (0) ev − × { }× ev M ( M, p ) ⊃ M ( γ, p ; A ) . Finally, each zero set σ ± ( ... ; A ) − (0) is compact, and given any p ∈ Crit( f ) , γ ∈ P ( H ) , and C ∈ R ,there are only finitely many A ∈ H ( M ) with ω ( A ) ≤ C and nonempty zero set σ ± ( ... ; A ) − (0) = ∅ .Proof. We will follow [Fi1, Cor.7.3] to construct the PSS polyfold, bundle, and sc-Fredholm section σ + p,γ ; A in detail, and note that the construction of the SSP section σ − γ,p ; A is analogous.Consider an ep-groupoid representative X = ( X, X ) of the polyfold B +SFT ( γ ; A ) with source andtarget maps denoted s, t : X → X together with a strong bundle P : W → X over the M -polyfold X and a structure map µ : X s × P W → X such that the pair ( P, µ ) is a strong bundle over X representing the polyfold bundle E +SFT ( γ ; A ) → B +SFT ( γ ; A ). In addition, consider a sc-Fredholm Here we can only make statements about the smooth level because we do not know what points of other levelsare included in the fiber products. This is sufficient for applications as the zero set of any sc-Fredholm section (andits admissible perturbations) is contained in the smooth level. As in Remark 4.4, this identification is stated for intuition and will ultimately not be used in our proofs. section functor S SFT : X → W of ( P, µ ) that represents σ +SFT . The ep-groupoid X and the bundle( P, µ ) are tame, since they represent a tame polyfold and a tame bundle, respectively. Moreover, S SFT is a tame sc-Fredholm section in the sense of [Fi1, Def.5.4] by Assumption 4.3(v).We view the Morse moduli space M ( p, M ) as the object space of an ep-groupoid with morphismspace another copy of M ( p, M ) and with unit map a diffeomorphism; that is, the only morphismsare the identity morphisms. The unique rank-0 bundle over M ( p, M ) is a strong bundle in the ep-groupoid sense, and the zero section of this bundle is a tame sc-Fredholm section functor. Next, notethat ˜ B + ( p, γ ; A ) ⊂ (cid:8) ( τ , v ) ∈ M ( p, M ) × | X | | ev + ( v ) ∈ { } × M (cid:9) ⊂ M ( p, M ) × | X ev | is representedwithin the open subset X ev := (ev + ) − ( C × M ) ⊂ X and the corresponding full ep-subgroupoid X ev of X , which represent the open subset B + , C SFT ( γ, A ) ⊂ | X | , and by Assumption 4.3(v) the restrictedevaluation ev + : X ev → C × M is sc ∞ and S SFT -compatibly submersive (see Definition A.4). Denoteby ev : M ( p, M ) → C × M, τ (0 , ev( τ )) the product of the trivial map to 0 ∈ C and the Morseevaluation map. We claim that the product map ev × ev + : M ( p, M ) × X ev → ( C × M ) × ( C × M )is S SFT -compatibly transverse to the diagonal ∆ ⊂ ( C × M ) × ( C × M ). Indeed, given ( τ , v ) ∈ (ev × ev + ) − (∆) let L ⊂ T Rv X ev be a sc-complement of the kernel of the linearization of ev + atsome v ∈ X ev ∞ that satisfies the conditions for S SF T -compatible submersivity in Definition A.4 w.r.t.a coordinate change ψ ev on a chart of X ev . Then the subspace { } × L ⊂ T Rτ M ( p, M ) × T Rv X ev satisfies the conditions for S SFT -compatible transversality of ev × ev + with ∆ at ( τ , v ) w.r.t. theproduct change of coordinates id × ψ ev in a product chart on the Cartesian product M ( p, M ) × X ev .(See [Fi1, Lem.7.1] for a discussion of the sc-Fredholm property on Cartesian products.)Next, note that M ( p, x ) ev × ev + X ev ∞ represents the smooth level of the fiber product topologi-cal space ˜ B + ( p, γ ; A ). So [Fi1, Cor.7.3] yields an open neighbourhood X ′ ⊂ M ( p, M ) ev × ev + X ev1 containing the smooth level M ( p, x ) ev × ev + X ev ∞ such that the full subcategory X ′ := ( X ′ , X ′ ) of M ( p, M ) × X ev1 is a tame ep-groupoid and the pullbacks of ( P, µ ) and S SFT to X ′ are a tame bundleand tame sc-Fredholm section. Here we used the fact that the smooth level M ( p, x ) ∞ = M ( p, x ) ofany finite dimensional manifold is the manifold itself; see Remark A.3.The tame ep-groupoid X ′ yields the claimed polyfold B + ( p, γ ; A ) := |X ′ | , and similarly the pull-backs of ( P, µ ) and S SFT through the projection X ′ → X define the claimed bundle and sc-Fredholmsection σ +( p,γ ; A ) : B + ( p, γ ; A ) → E + ( p, γ ; A ). The identification of the interior ∂ B + ( p, γ ; A ) ∞ followsfrom the degeneracy index formula d X ′ ( x , x ) = d M ( p,M ) ( x ) + d X ( x ) in [Fi1, Cor.7.3] and theinterior of the Morse trajectory spaces ∂ M ( p, M ) = M ( p, M ) from Theorem 3.3.The index formula in [Fi1, Cor.7.3] yields ind( σ +( p,γ ; A ) ) = ind( σ SFT )+ | p |− dim( C × M ) = I ( p, γ ; A )since dim M ( p, M ) = | p | and ind( σ SFT ) = CZ ( γ ) + 2 c ( A ) + dim M + 2.Finally, the zero set σ +( p,γ ; A ) − (0) is the fiber product of the zero sets as claimed, as these arecontained in the smooth level, and the restriction to ev − ( { } × M ) already restricts considerationsto the domain X ev from which the fiber product polyfold is constructed. Moreover, σ +( p,γ ; A ) − (0)is compact as in [Fi1, Cor.7.3], since both M ( p, M ) and σ +SFT − (0) are compact and both ev andev + are continuous. The final statement then follows from Assumption 4.3(ii). (cid:3) Construction of the morphisms.
To construct the Λ-linear maps PSS and SSP in Theo-rem 1.2 with relatively compact notation we index all moduli spaces from § I + := (cid:8) α = ( p, γ ; A ) (cid:12)(cid:12) p ∈ Crit( f ) , γ ∈ P ( H ) , A ∈ H ( M ) (cid:9) , I − := (cid:8) α = ( γ, p ; A ) (cid:12)(cid:12) p ∈ Crit( f ) , γ ∈ P ( H ) , A ∈ H ( M ) (cid:9) . To simplify notation we then denote I := I − ∪ I + and drop the superscripts from the polyfolds B ( α ) = B ± ( α ). Since Lemma 4.5 provides each moduli space M ( α ) for α ∈ I with a compactificationand polyfold description M ( α ) ⊂ σ − α (0), we can apply [HWZ, Theorems 18.2,18.3,18.8] to obtainadmissible regularizations of the moduli spaces, and counts of the 0-dimensional perturbed solutionspaces [HWZ, § Q + := Q ∩ [0 , ∞ ) the groupoid withonly identity morphisms. POLYFOLD PROOF OF THE ARNOLD CONJECTURE 13
Corollary 4.6. (i) For every α ∈ I , choice of neighbourhood of the zero sets σ − α (0) ⊂ V α ⊂ B ( α ) ,and choice of sc-Fredholm section functor S α : X α → W α representing σ α | V α , there exists a pair ( N α , U α ) controlling compactness in the sense of Definition A.5 with | S − α (0) | ⊂ |U α | ⊂ V α .For α ∈ I with σ − α (0) = ∅ we can choose U α = ∅ .(ii) For every collection ( N α , U α ) α ∈I of pairs controlling compactness, there exists a collection κ = (cid:0) κ α : W α → Q + (cid:1) α ∈I of ( N α , U α ) -admissible sc + -multisections in the sense of [HWZ, Defini-tions 13.4,15.5] that are in general position relative to ( S α ) α ∈I in the sense that each pair ( S α , κ α ) is in general position as per [HWZ, Def.15.6] .Here admissibility in particular implies κ α ◦ S α | X α r U α ≡ and thus κ α ◦ S α ≡ when σ − α (0) = ∅ .(iii) Every collection κ of admissible sc + -multisections in general position from (ii) induces a col-lection of compact, tame, branched ep + -groupoids (cid:0) κ α ◦ S α : X α → Q + (cid:1) α ∈I . In particular, eachperturbed zero set Z κ ( α ) := (cid:12)(cid:12) { x ∈ X α | κ α ( S α ( x )) > } (cid:12)(cid:12) ⊂ |U α | ∞ ⊂ |X α | ∞ ∼ = B ( α ) ∞ is compact, contained in the smooth level, and carries the structure of a weighted branched orbifoldof dimension I ( α ) as in (9) . Moreover, the inclusion in |U α | and general position of κ implies thatfor I ( α ) < or σ − α (0) = ∅ the perturbed zero set Z κ ( α ) = ∅ is empty.(iv) For α ∈ I with Fredholm index I ( α ) = 0 and κ α : W α → Q + as in (ii) the perturbed zero setis contained in the interior Z κ ( α ) ⊂ ∂ B ( α ) ∞ and yields a well defined count Z κ ( α ) := P | x |∈ Z κ ( α ) o σ α ( x ) κ α ( S α ( x )) ∈ Q . Here o σ α ( x ) ∈ {± } is determined by the orientation of σ α as in [HWZ, Thm.6.3] . If |U α |∩ ∂ B ( α ) = ∅ then this count is independent of the choice of admissible sc + -multisection κ α .(v) For every α ∈ I with Fredholm index I ( α ) = 1 and κ α : W α → Q + as in (ii) the boundary ofthe perturbed zero set is given by its intersection with the first boundary stratum of the polyfold, ∂Z κ ( α ) = Z κ ( α ) ∩ ∂ B ( α ) ∞ . With orientations o σ α | ∂ B ( α ) ( x ) ∈ {± } induced by the boundary restriction σ α | B ( α ) this implies ∂Z κ ( α ) = P | x |∈ ∂Z κ ( α ) o σ α | ∂ B ( α ) ( x ) κ α ( S α ( x )) = 0 . Remark 4.7. (i) The statements in (iv) and (v) of Corollary 4.6 require orientations of the sections σ α for α ∈ I . By the fiber product construction in Lemma 4.5 they do indeed inherit orientationsfrom the orientations of the Morse trajectory spaces in Remark 3.5, the orientations of σ ± SFT givenin Assumption 4.3, and an orientation convention for fiber products.In practice, we will construct the perturbations κ in Corollary 4.6 by pullback of perturbations λ = ( λ ± γ,A ) γ ∈P ,A ∈ H ( M ) of the oriented SFT-sections σ ± SFT . Thus it suffices to specify the orien-tations of the regularized zero sets, which is implicit in their identification with transverse fiberproducts of oriented spaces over the oriented manifold M , Z κ ( p, γ ; A ) = M ( p, M ) ev × ev + Z λ ( γ ; A ) , Z κ ( γ, p ; A ) = Z λ ( γ ; A ) ev − × ev M ( M, p ) . Orientations of the boundary restrictions in (v) are then induced by the orientations of Z κ ( α ), viaoriented isomorphisms of the tangent spaces R ν ( z ) × T z ∂Z κ ( α ) ∼ = T z Z κ ( α ), where ν ( z ) ∈ T z Z κ ( α )is an exterior normal vector at z ∈ ∂Z κ ( α ).(ii) Note that the counts in part (iv) of this Corollary may well depend on the choice of themulti-valued perturbations κ α – unless the ambient polyfold has no boundary, ∂ B ( α ) = ∅ . Indeed,although the moduli space M ( α ) is expected to have dimension 0, it may not be cut out transverselyfrom the ambient polyfold B ( α ), and moreover it may not be compact. Assumption 4.3 provides aninclusion in a compact set M ( α ) ⊂ σ − α (0), and the perturbation theory for sc-Fredholm sections ofstrong bundles then associates to σ − α (0) a perturbed zero set Z κ ( α ) ⊂ B ( α ) with weight function κ α ◦ S α : Z κ ( α ) → Q ∩ (0 , ∞ ). This process generally adds points on the boundary σ − α (0) r M ( α ) ⊂B ( α ) r ∂ B ( α ), which may or may not persist under variations of the perturbation κ α .The following construction of morphisms will depend on the choices of perturbations and orien-tation convention (see the previous remark) as well as geometric data fixed in § The algebraic properties in Theorem 1.2 will be achieved in § κ ± , and an overall sign adjustment. Definition 4.8.
Given collections κ ± = ( κ ± α ) α ∈I ± of admissible sc + -multisections in general posi-tion as in Corollary 4.6, we define the maps P SS κ + : CM → CF and SSP κ − : CF → CM to bethe Λ -linear extension of P SS κ + h p i := X γ,A I ( p,γ ; A )=0 Z κ + ( p, γ ; A ) · T ω ( A ) h γ i , SSP κ − h γ i := X p,A I ( γ,p ; A )=0 Z κ − ( γ, p ; A ) · T ω ( A ) h p i . Lemma 4.9.
The maps
P SS κ + : CM → CF and SSP κ − : CF → CM in Definition 4.8 are welldefined, i.e. the coefficients take values in the Novikov field Λ defined in § To prove that
P SS κ + is well defined we need to check finiteness of the following set for any p ∈ Crit( f ), γ ∈ P ( H ), and c ∈ R , n r ∈ ω ( H ( M )) ∩ ( −∞ , c ] (cid:12)(cid:12)(cid:12) P A ∈ H ( M ) ω ( A )= r Z κ + ( p, γ ; A ) = 0 o . Here ω : H ( M ) → R is given by pairing with the symplectic form on M , and recall from Lemma 4.5that there are only finitely many homology classes A ∈ H ( M ) with ω ( A ) ≤ c and σ − α (0) = ∅ . Onthe other hand, the perturbations κ + were chosen in Corollary 4.6 (iii),(iv) so that Z κ + ( . . . ; A ) = 0whenever σ − α (0) = ∅ . Thus there are in fact only finitely many A ∈ H ( M ) with ω ( A ) ≤ c and Z κ + ( . . . ; A ) = 0, which proves the required finiteness. The proof for SSP κ − is analogous. (cid:3) The chain homotopy maps
In this section we construct Λ-linear maps ι : CM → CM and h : CM → CM on the Morsecomplex over the Novikov field Λ given in (5), which appear in Theorem 1.2. For that purpose weagain fix a choice of geometric data as in § § § § ι, h for admissible regular choices ofperturbations in Definitions 5.7. To obtain the algebraic properties claimed in Theorem 1.2 (i)–(iii)we will then construct particular “coherent” choices of perturbations in § Moduli spaces for the isomorphism ι . We will construct ι : CM → CM from the followingmoduli spaces for critical points p − , p + ∈ Crit( f ), A ∈ H ( M ), using the almost complex structure J and the unstable/stable manifolds (see § f, g ) chosen in § M ι ( p − , p + ; A ) := (cid:8) u : CP → M (cid:12)(cid:12) u ([1 : 0]) ∈ W − p − , u ([0 : 1]) ∈ W + p + , ∂ J u = 0 , [ u ] = A (cid:9) . Note that a cylinder acts on this moduli space by reparametrization with biholomorphisms of CP that fix the two points [1 : 0] , [0 : 1]. However, we do not quotient out this symmetry so describethese moduli spaces as the zero set of a Fredholm section over a Sobolev closure of the set of smoothmaps u : CP → M in the homology class [ u ] = A satisfying the point constraints u ([1 : 0]) ∈ W − p − and u ([0 : 1]) ∈ W + p + . This determines the Fredholm index as(18) I ι ( p − , p + ; A ) = 2 c ( A ) + | p − | − | p + | . As in § J -holomorphic spheres with evaluation maps for z ∈ CP ,ev z : M ( A ) := (cid:8) u : CP → M (cid:12)(cid:12) ∂ J u = 0 , [ u ] = A (cid:9) → M, u u ( z ) . With this we can describe the moduli space (17) as fiber product with the half-infinite Morsetrajectory spaces from § z +0 := [1 : 0] and z − := [0 : 1](19) M ι ( p − , p + ; A ) ∼ = M ( p − , M ) ev × ev z +0 M ( A ) ev z − × ev M ( M, p + ) . Note here that we are not working with a Gromov-Witten moduli space, as we do not quotient byAut( CP ). This is due to the chain homotopy in Theorem 1.2 (iii), which will result from identifyinga compactification of M ( A ) with a boundary of the neck-stretching moduli space M SFT ( A ) in(26) that appears in Symplectic Field Theory [EGH]. For that purpose we identify a solution POLYFOLD PROOF OF THE ARNOLD CONJECTURE 15 u : CP → M with the map to its graph e u : CP → CP × M, z ( z, u ( z )) as in [MS, § M ( A ) ∼ = f M GW ([ CP ] + A ) := (cid:8) v : CP → CP × M (cid:12)(cid:12) ∂ e J v = 0 , [ v ] = [ CP ] + A (cid:9) Aut( CP )between the Cauchy-Riemann solution space for M and the Gromov-Witten moduli space for CP × M in class [ CP ] + A for the split almost complex structure e J := i × J on CP × M . To transfer theevaluation maps at z +0 = [1 : 0] and z − = [0 : 1] we keep track of these as (unique) marked pointsmapping to { z ± } × M and thus replace (19) by a fiber product over CP × M ,(20) M ι ( p − , p + ; A ) ∼ = M ( p − , M ) { z +0 }× ev × ev + M GW ( A ) ev − × { z − }× ev M ( M, p + ) . This uses the evaluation maps from a Gromov-Witten moduli space with two marked points,(21) ev ± : M GW ( A ) := f M GW ([ CP ] + A ) . Aut( CP , z − , z +0 ) → CP × M, [ v ] v ( z ± ) , where Aut( CP , z − , z +0 ) denotes the set of biholomorphisms φ : CP → CP which fix φ ( z ± ) = z ± . The polyfold setup in [HWZ1, Theorems 1.7,1.10,1.11] for Gromov-Witten moduli spaces nowprovides a strong polyfold bundle E GW ( A ) → B GW ( A ), and oriented sc-Fredholm section σ GW : B GW ( A ) → E GW ( A ) that cuts out a compactification M GW ( A ) = σ − (0) of M GW ( A ). Herea dense subset of the base polyfold B GW ( A ) consists of Aut( CP , z − , z +0 )-orbits of smooth maps v : CP → CP × M in the homology class [ v ] = [ CP ] + A , which implicitly carries the two markedpoints z ± ∈ CP . Nodal curves in M GW ( A ) then explicitly come with the data of two marked pointson their domain. On the dense subset the section is given by σ GW ([ v ]) = [( v, ∂ e J v )]. The setup in[HWZ1, Theorem 1.8] moreover provides sc ∞ evaluation maps ev ± : B GW ( A ) → CP × M at themarked points, which on the dense subset are given by ev ± ([ v ]) = v ( z ± ).Thus we have given each factor in the fiber product (20) a compactification that is either amanifold with corners given by the compactified Morse trajectory spaces in Theorem 3.3, or thecompact zero set M GW ( A ) = σ − (0) of a sc-Fredholm section. In § ι : CM → CM proceeds as in § ι inTheorem 1.2 (i) and (ii) will follow from the boundary stratifications of the Morse trajectory spaces M ( p − , M ) and M ( M, p + ) since the ambient polyfold B GW ( A ) has no boundary. However, thisrequires specific “coherent” choices of perturbations in § Remark 5.1.
Gromov-compactifications of the moduli spaces M ι ( p − , p + ; A ) will result from theenergy identity [MS, Lemma 2.2.1] for solutions of ∂ J u = 0,(22) E ( u ) := R C | d u | = R CP u ∗ ω = ω ([ u ]) . This fixes the energy of solutions on each solution space M ( A ), and Gromov compactness assertsthat { u : CP → M | ∂ J u = 0 , E ( u ) ≤ C } is compact up to bubbling for any C > ω ( A ) ≤ M ( A ) = ∅ except for A = 0 ∈ H ( M ) when the solution space is the space of constant maps M (0) = { u ≡ x | x ∈ M } ≃ M, which is compact and cut out transversely.Translated to graphs in CP × M with two marked points, this means M GW (0) ≃ CP × CP × M by adding two marked points in the domain. That is, ( z − , z + , x ) ∈ CP × CP × M corresponds tothe (equivalence class of) graphs e u x : z ( z, x ) with two marked points z − , z + ∈ CP . For z − = z + this tuple can be reparametrized to the fixed marked points z − , z +0 ∈ CP and then represents anAut( CP , z − , z +0 )-orbit. For z − = z + the tuple ( z − , z + , x ) corresponds to a stable map in M GW (0),given by the graph e u x with a node at z − = z + attached to a constant sphere with two distinctmarked points. This will be stated in polyfold terms in Assumption 5.5 (ii). The term ’compactification’ applied to spaces of pseudoholomorphic curves is always to be understood as Gromov-compactification, as M GW ( A ) ⊂ M GW ( A ) may not be dense. Moduli spaces for the chain homotopy h . To construct the moduli spaces from whichwe will obtain h : CM → CM , we again use the almost complex structure J and Morse-Smalepair ( f, g ) chosen in § Y H ∈ Ω , ( C , Γ(T M )) that arises from the fixed Hamiltonian function H : S × M → R and a choiceof smooth cutoff function β : [0 , ∞ ) → [0 ,
1] with β | [0 , ≡ β ′ ≥
0, and β | [ e, ∞ ) ≡
1. Gluing this1-form to another copy of Y H over C − with neck length R > Y RH ∈ Ω , ( CP , Γ(T M )) that vanishes near [1 : 0] , [0 : 1]and on CP r { [1 : 0] , [0 : 1] } = { [1 : re iθ ] | ( r, θ ) ∈ (0 , ∞ ) × S } is given by Y RH ([1 : re iθ ] , x ) := β R ( r ) (cid:0) JX H ( θ, x ) r − d r + X H ( θ, x ) d θ (cid:1) . Here β R ( r ) := β ( re R ) β ( r − e R ) is a smooth cutoff function β R : (0 , ∞ ) → [0 ,
1] that is identical to1 on [ e − R , e R − ] and identical to 0 on (0 , e − R ) ∪ ( e R , ∞ ). Now perturbing the Cauchy-Riemannoperator on CP by Y RH yields the following moduli spaces for critical points p − , p + ∈ Crit( f ), A ∈ H ( M ), and R ∈ [0 , ∞ ), M R ( p − , p + ; A ) := (cid:8) u : CP → M (cid:12)(cid:12) u ([1 : 0]) ∈ W − p − , u ([0 : 1]) ∈ W + p + , ∂ J u = Y RH ( u ) , [ u ] = A (cid:9) , and we will construct h from their union(23) M ( p − , p + ; A ) := F R ∈ [0 , ∞ ) M R ( p − , p + ; A ) . Remark 5.2.
Each vector-field-valued 1-form Y RH = − ( X H Rβ ) , is in the notation of [MS, § H Rβ ∈ Ω ( CP , C ∞ ( M )) given by H Rβ ( re iθ ) = β R ( r ) H ( θ, · )d θ . It is constructed so that it has the following properties:(i) For R = 0 we have Y H ≡ M ( p − , p + ; A ) = M ι ( p − , p + ; A ) is thesame moduli space (17) from which ι will be constructed.(ii) The restriction of any solution u ∈ M R ( p − , p + ; A ) to the middle portion { [1 : z ] ∈ CP | e − R < | z | < e R − } ∼ = (1 − R , R − × S satisfies the Floer equation ∂ s v + J∂ t v = JX H ( t, v ) afterreparametrization v ( s, t ) := u ([1 : e s + it ]).(iii) The shifts u − ( z ) := u ([1 : e − R z ]) and u + ( z ) := u ([ e R z : 1]) = u ([1 : e R z − ]) of any solution u ∈ M R ( p − , p + ; A ), restricted to { z ∈ C | | z | < e R − } , satisfy ∂ J u ± = Y H ( u ± ) as in the PSS/SSPmoduli spaces in § M ( p − , p + ; A ) is the zero set of a Fredholm section over a Banach manifold[0 , ∞ ) × B , where B is the same Sobolev closure as in § u : CP → M in the homology class [ u ] = A satisfying the point constraints u ([1 : 0]) ∈ W − p − and u ([0 : 1]) ∈ W + p + .Restricted to { } × B this is the Fredholm section that cuts out M ι ( p − , p + ; A ) in (17) with e J H = e J .This determines the Fredholm index as(24) I ( p − , p + ; A ) := I ι ( p − , p + ; A ) + 1 = 2 c ( A ) + | p − | − | p + | + 1 . Towards a compactification and polyfold description of these moduli spaces we again – as in § § § u : CP → M with the map to its graph. Moreover, we againfix marked points z +0 = [1 : 0], z − = [0 : 1] to implement evaluation maps to express the conditions u ( z ∓ ) ∈ W ± p ± . This yields a homeomorphism (in appropriate topologies) between the moduli space(23) and the fiber product over CP × M with the half-infinite Morse trajectory spaces from § M ( p − , p + ; A ) ∼ = M ( p − , M ) { z +0 }× ev × ev + M SFT ( A ) ev − × { z − }× ev M ( M, p + ) . Compared with (20) this replaces the Gromov-Witten moduli space in (21) with a family of modulispaces for almost complex structures e J RH on CP × M arising from Y RH for R ∈ [0 , ∞ ),(26) M SFT ( A ) := G R ∈ [0 , ∞ ) (cid:8) v : CP → CP × M (cid:12)(cid:12) ∂ e J RH v = 0 , [ v ] = [ CP ] + A (cid:9)(cid:14) Aut( CP , z − , z +0 ) . POLYFOLD PROOF OF THE ARNOLD CONJECTURE 17
Here, again, we implicitly include the two marked points z ± ∈ CP . Then, for R → ∞ , thedegeneration of the PDE ∂ e J RH v = 0 is the “neck stretching” considered more generally in SymplecticField Theory [EGH]. The evaluation maps from (21) directly generalize to(27) ev ± : M SFT ( A ) → CP × M, [ v ] v ( z ± ) . Now, as in § M SFT ( A ) = σ − (0)of a sc-Fredholm section that we will introduce in § h : CM → CM then again proceeds as in § h with ι and SSP ◦ P SS will moreoverrequire an in-depth discussion of the boundary stratification of the polyfold domains B SFT ( A ) ofthese sections, and “coherent” choices of perturbations in § Remark 5.3.
Gromov-compactifications of the moduli spaces M ( p − , p + ; A ) will result from energyestimates [MS, Remark 8.1.7] for solutions of ∂ J u = Y RH ( u ),(28) E R ( u ) := R CP | d u + X H Rβ ( u ) | ≤ R CP u ∗ ω + k R H Rβ k = ω ([ u ]) + 2 k H ( θ, · ) k . Here R H Rβ dvol CP = d H Rβ + H Rβ ∧ H Rβ = β ′ R H d r ∧ d θ has uniformly bounded Hofer norm k R H Rβ k = R CP (max R H Rβ − max R H Rβ ) = R ∞ R S | β ′ R ( r ) |k H ( θ, · ) k d θ d r = 2 k H ( θ, · ) k , where k H ( θ, · ) k := max x ∈ M H ( θ, x ) − min x ∈ M H ( θ, x ) and β R ∈ C ∞ ((0 , ∞ ) , [0 , β R | [ e − R ,e − R ] : r β ( re R ) with dd r β R ≥ R e − R e − R (cid:12)(cid:12) dd r β R (cid:12)(cid:12) d r = β ( e ) − β (1) = 1 ,β R | [ e R − ,e R ] : r β ( r − e R ) with dd r β R ≤ R e R e R − (cid:12)(cid:12) dd r β R (cid:12)(cid:12) d r = − (cid:0) β (1) − β ( e ) (cid:1) = 1 . This proves (28), and thus establishes energy bounds on the perturbed pseudoholomorphic maps ineach of our moduli spaces, where we fix [ u ] = A . Now SFT-compactness [BEHWZ] asserts that forany C > F R ∈ [0 , ∞ ) { u : CP → M | ∂ J u = Y RH ( u ) , E R ( u ) ≤ C } is compact up to breaking and bubbling. This compactness will be stated rigorously in polyfoldterms in Assumption 5.5 (ii).5.3. Construction of the morphisms.
In this section we construct the Λ-linear maps ι : CM → CM and h : CM → CM analogously to § § § § § CP × M via evaluation maps (21),(27). Polyfold descriptions of these moduli spaces and their properties are stated in the followingAssumption 5.5 for reference, with proofs in [HWZ1] resp. outlined in [FH]. Here we formulatethe evaluation map in the context of neck stretching, as explained in the following remark, using asplitting of the sphere as topological space with smooth structures on the complement of the equator CP ∞ := C + ∪ S C − ∼ = C + ⊔ S ⊔ C − , using the topologies and smooth structures on C ± = C ± ⊔ S ∼ = { z ∈ C ± | | z | ≤ } from Remark 4.2. Remark 5.4. (i) Recall from § B GW ( A ) a Gromov-Witten polyfold of curvesin class [ CP ] + A ∈ H ( CP × M ) with 2 marked points. These are determined by A ∈ H ( M )as we model graphs of maps CP → M , but should not be confused with a polyfold of curves in M . In particular, B GW ( A ) never contains constant maps and hence is well defined for A = 0. Theproperties of the Gromov-Witten moduli spaces for ω ( A ) ≤ Strictly speaking, R ∈ [0 ,
2] parametrizes a family of Gromov-Witten moduli spaces for varying almost complexstructure. At R = 2, the manifold S × M with its stable Hamiltonian structure (see § CP × M . Then R ∈ [2 , ∞ ) parametrizes the SFT neck-stretching. (ii) The SFT polyfolds B SFT ( A ) will similarly describe curves in class [ CP ] + A in a neck stretchingfamily of targets ( CP R × M ) R ∈ [0 , ∞ ] as in [BEHWZ, § CP R := D + ⊔ E R ⊔ D − . ∼ R with E R = ( [ − R, R ] × S ; R < ∞ , [0 , ∞ ) × S ⊔ ( −∞ , × S ; R = ∞ . Here we identify the boundaries of the closed unit disks D ± = { z ∈ C | | z | ≤ } with the boundarycomponents of the necks E R via ∂D ± ∈ e iθ ∼ R ( ( ± R, e ± iθ ) ; R < ∞ (0 ± , e ± iθ ) ; R = ∞ ) ∈ ∂E R , where we denote 0 + := 0 ∈ [0 , ∞ ) and 0 − := 0 ∈ ( −∞ ,
0] so that ∂E ∞ = { + } × S ⊔ { − } × S .To describe convergence and evaluation maps we also embed each CP R ⊂ CP ∞ = C + ⊔ S ⊔ C − by D + ⊔ [ − R, × S . ∼ R ∼ = D + ⊔ [0 , ∞ ) × S . ∼ ∞ =: C + ,D − ⊔ (0 , R ] × S . ∼ R ∼ = D − ⊔ ( −∞ , × S . ∼ ∞ =: C − , E R ⊃ { } × S ∼ = S ⊂ CP ∞ . For R = 0 this is to be understood as CP = D + ⊔ D − / ∂D + ∼ ∂D − with D ± r ∂D ± ∼ = C ± , and for all R < ∞ we view the resulting homeomorphism CP R ∼ = CP ∞ ∼ = CP as identifying the standardmarked points CP ∋ z +0 = [1 : 0] ∼ = 0 ∈ C + and CP ∋ z − = [0 : 1] ∼ = 0 ∈ C − . When theseembeddings are done via linear shifts [ − R, − ∼ = [0 , R −
1) and (1 , R ] ∼ = (1 − R,
0] extended by asmooth family of diffeomorphisms [ − , ∼ = [ R − , ∞ ) and (0 , ∼ = ( −∞ , − R ], then the pullbackof the almost complex structures e J RH on CP R × M converges for R → ∞ in C ∞ loc (cid:0) ( CP ∞ r S ) × M (cid:1) to the almost complex structures e J + H , e J − H on C + × M ⊔ C − × M = CP ∞ × M ⊂ CP ∞ × M ,which are used in the construction of the PSS and SSP moduli spaces in § ± : M SFT ( A ) → CP ∞ × M on the compactified SFT moduli space. At R = ∞ this involves pseudoholomorphic buildings in C + × M ⊔ R × S × M . . . ⊔ R × S × M ⊔ C − × M , and for any marked point with evaluation intoa cylinder R × S × M we project the result to S × M ⊂ CP ∞ × M by forgetting the R -component.Finally, this formulation with CP ∞ = C + ∪ S C − will allow us to compare the evaluation at R = ∞ with the product of the evaluations ev ± : M ± SFT ( γ ; A ) → C ± × M constructed in Remark 4.2.While this will be stated rigorously only in Assumption 6.3 (iii)(c), note here that we should expectthree top boundary strata of an ambient polyfold at R = ∞ , corresponding to the distribution ofmarked points on the curves in C + × M ⊔ C − × M . For the fiber product construction, only theboundary components with one marked point in each factor are relevant – in fact only those withmarked points near z +0 ∼ = 0 ∈ C + and z − ∼ = 0 ∈ C − . Thus we will work with the open subset(ev + ) − ( C + × M ) ∩ (ev − ) − ( C − × M ) where the two evaluations for any R ∈ [0 , ∞ ] are constrainedto take values in the open sets given by C ± ⊂ CP ∞ . Assumption 5.5.
There is a collection of oriented sc-Fredholm sections of strong polyfold bundles σ GW : B GW ( A ) → E GW ( A ) and σ SFT : B SFT ( A ) → E SFT ( A ) indexed by A ∈ H ( M ) , sc ∞ maps ev ± : B GW ( A ) → CP × M , and continuous maps ev ± : B SFT ( A ) → CP ∞ × M with the properties:(i) The sections have Fredholm indices ind( σ GW ) = 2 c ( A )+dim M +4 on B GW ( A ) resp. ind( σ SFT ) =2 c ( A ) + dim M + 5 on B SFT ( A ) .(ii) Each zero set M GW ( A ) := σ − (0) and M SFT ( A ) := σ − (0) is compact, and given any C ∈ R there are only finitely many A ∈ H ( M ) with nonempty zero set M GW ( A ) = ∅ resp. M SFT ( A ) = ∅ .Moreover, for ω ( A ) ≤ we have M GW ( A ) = ∅ except for A = 0 ∈ H ( M ) when σ GW | B GW (0) ⋔ is in general position with zero set M GW (0) ≃ CP × CP × M identified by B GW (0) ⊃ σ − (0) = M GW (0) ev + × ev − −→ (cid:8) ( z + , x, z − , x ) (cid:12)(cid:12) z − , z + ∈ CP , x ∈ M (cid:9) . (iii) The polyfolds B GW ( A ) have no boundary, ∂ B GW ( A ) = ∅ . For B SFT ( A ) there is a naturalinclusion [0 , ∞ ) × B GW ( A ) ⊂ B SFT ( A ) that covers the interior ∂ B SFT ( A ) = (0 , ∞ ) × B GW ( A ) and POLYFOLD PROOF OF THE ARNOLD CONJECTURE 19 identifies the boundary ∂ B SFT ( A ) to consist of the disjoint sets { } × B GW ( A ) and lim R →∞ { R } ×B GW ( A ) of B SFT ( A ) . Moreover, this inclusion identifies the section σ GW and evaluation maps ev ± with the restricted section σ SFT | { }×B GW ( A ) and evaluations ev ± | { }×B GW ( A ) . (A description of therelevant R = ∞ parts of the boundary ∂ B SFT ( A ) is given in Assumption 6.3.)(iv) The Aut( CP , z − , z +0 ) -orbits of smooth maps v : CP → CP × M which represent the class [ CP ] + A form a dense subset B dense ( A ) ⊂ B GW ( A ) . On this subset, the section is given by σ GW ([ v ]) = [( v, ∂ e J v )] . Moreover, [0 , ∞ ) × B dense ( A ) ⊂ B SFT ( A ) is a dense subset that intersects theboundary ∂ B SFT ( A ) exactly in { } × B dense ( A ) , and on which the section is given by σ SFT ( R, [ v ]) =[( v, ∂ e J RH v )] . On these dense subsets, ev ± ([ v ]) resp. ev ± ( R, [ v ]) is the evaluation as in (27) .(v) The intersection of the zero sets with the dense subsets σ − (0) ∩ B dense ( A ) ∼ = M GW ( A ) and σ − (0) ∩ [0 , ∞ ) × B dense ( A ) ∼ = M SFT ( A ) are naturally identified with the Gromov-Witten modulispace (21) and SFT moduli space in (26) .(vi) The sections σ GW and σ SFT have tame sc-Fredholm representatives in the sense of [Fi1, Def.5.4] .The product of evaluation maps ev + × ev − : B GW ( A ) → CP × M × CP × M is σ GW -compatiblysubmersive in the sense of Definition A.4. On the open subset B + , − SFT ( A ) := (ev + ) − ( C + × M ) ∩ (ev − ) − ( C − × M ) ⊂ B SFT ( A ) the evaluation maps ev ± : B SFT ( A ) → CP ∞ × M restrict to a σ SFT -compatibly submersive map (29) ev + × ev − : B + , − SFT ( A ) → C + × M × C − × M. On this domain intersected with { } × B GW ( A ) ⊂ ∂ B SFT ( A ) , this map coincides with the Gromov-Witten evaluations ev + × ev − viewed as maps ev + × ev − : B + , − GW ( A ) → C + × M × C − × M, where we identify C + ⊔ C − = CP r S and restrict to the domain { } × B + , − GW ( A ) := (cid:0) { } × B GW ( A ) (cid:1) ∩ B + , − SFT ( A ) = { } × (cid:0) (ev + ) − ( C + × M ) ∩ (ev − ) − ( C − × M ) (cid:1) . Here the properties (iv),(v) are stated to give an intuitive sense of what spaces we are workingwith. The polyfold description σ GW : B GW ( A ) → E GW ( A ) is developed for the homology classes[ CP ] + A ∈ H ( CP × M ) in [HWZ1], and assumption (v) should follow similarly to the explanationgiven in [Fi1, Ex.5.1] for the Gromov-Witten polyfolds [HWZ1]. Given any such polyfold descriptionsof the moduli spaces of pseudoholomorphic curves, we now extend the fiber product descriptions ofthe moduli spaces M ( ι ) ( p − , p + ; A ) ∼ = M ( p − , M ) { z +0 }× ev × ev + M GW / SFT ( A ) ev − × { z − }× ev M ( M, p + )in § § p − , p + ∈ Crit( f ) and A ∈ H ( M ) the topological spaces˜ B ι ( p − , p + ; A ) := (cid:8) ( τ − , v, τ + ) ∈ M ( p − , M ) × B GW ( A ) × M ( M, p + ) (cid:12)(cid:12) ( z ± , ev( τ ± )) = ev ± ( v ) (cid:9) := (cid:8) ( τ − , v, τ + ) ∈ M ( p − , M ) × B + , − GW ( A ) × M ( M, p + ) (cid:12)(cid:12) (0 , ev( τ ± )) = ev ± ( v ) (cid:9) , ˜ B ( p − , p + ; A ) := (cid:8) ( τ − , w, τ + ) ∈ M ( p − , M ) × B SFT ( A ) × M ( M, p + ) (cid:12)(cid:12) ( z ± , ev( τ ± )) = ev ± ( w ) (cid:9) = (cid:8) ( τ − , w, τ + ) ∈ M ( p − , M ) × B + , − SFT ( A ) × M ( M, p + ) (cid:12)(cid:12) (0 , ev( τ ± )) = ev ± ( w ) (cid:9) , where the last equality stems from the identification at the end of Remark 5.4 (ii). Then the abstractfiber product constructions in [Fi1] will be used as in Lemma 4.5 to obtain the following polyfolddescription for compactifications of the moduli spaces in § § Lemma 5.6.
Given any p − , p + ∈ Crit( f ) and A ∈ H ( M ) , there exist open subsets B ι ( p − , p + ; A ) ⊂ ˜ B ι ( p − , p + ; A ) and B ( p − , p + ; A ) ⊂ ˜ B ( p − , p + ; A ) which contain the smooth levels ˜ B ( ι ) ( p − , p + ; A ) ∞ of the fiber products and inherit natural polyfold structures with smooth level of the interior ∂ B ι ( p − , p + ; A ) ∞ = M ( p − , M ) { z +0 }× ev × ev + B + , − GW ( A ) ∞ ev − × { z − }× ev M ( M, p + ) ,∂ B ( p − , p + ; A ) ∞ = M ( p − , M ) { z +0 }× ev × ev + ∂ B + , − SFT ( A ) ∞ ev − × { z − }× ev M ( M, p + ) , and a scale-smooth inclusion φ ι : B ι ( p − , p + ; A ) ֒ → B ( p − , p + ; A ) , ( τ − , v, τ + ) ( τ − , , v, τ + ) . Moreover, pullback of the sections and bundles σ GW / SFT : B GW / SFT ( A ) → E GW / SFT ( A ) underthe projection B ( p − , p + ; A ) → B GW / SFT ( A ) induces sc-Fredholm sections of strong polyfold bun-dles σ ( p − ,p + ; A ) : B ( p − , p + ; A ) → E ( p − , p + ; A ) of index I ( p − , p + ; A ) as in (24) and σ ι ( p − ,p + ; A ) : B ι ( p − , p + ; A ) → E ι ( p − , p + ; A ) of index I ι ( p − , p + ; A ) = I ( p − , p + ; A ) − as in (18) . Further, these arerelated via the inclusion φ ι by natural orientation preserving identification σ ι ( p − ,p + ; A ) ∼ = φ ∗ ι σ ( p − ,p + ; A ) .The zero sets of these sc-Fredholm sections contain the moduli spaces from § § σ ( p − ,p + ; A ) − (0) = M ( p − , M ) { z +0 }× ev × ev + σ − (0) ev − × { z − }× ev M ( M, p + ) ⊃ M ( p − , p + ; A ) ,σ ι ( p − ,p + ; A ) − (0) = M ( p − , M ) { z +0 }× ev × ev + σ − (0) ev − × { z − }× ev M ( M, p + ) ⊃ M ι ( p − , p + ; A ) . Finally, each zero set σ ( ι )( p − ,p + ; A ) − (0) is compact, and given any p ± ∈ Crit( f ) and C ∈ R , there areonly finitely many A ∈ H ( M ) with ω ( A ) ≤ C and nonempty zero set σ ( ι )( p − ,p + ; A ) − (0) = ∅ .Proof. The inclusion φ ι is sc ∞ since the map B GW ( A ) ֒ → B SFT ( A ) , v (0 , v ) is a sc ∞ inclusionby Assumption 5.5 (iii). Apart from further relations involving φ ι , the proof is directly analogousto the fiber product construction in Lemma 4.5, using Assumption 5.5 – in particular the sc ∞ and σ SFT -compatibly submersive evaluation map (29) on the open subset B + , − SFT ( A ) ⊂ B SFT ( A ).This yields polyfold structures on open sets B ι ( p − , p + ; A ) ⊂ ˜ B ι ( p − , p + ; A ) and B ( p − , p + ; A ) ⊂ ˜ B ( p − , p + ; A ) as well as the pullback sc-Fredholm sections σ ( p − ,p + ; A ) = pr ∗ SFT σ SFT and σ ι ( p − ,p + ; A ) =pr ∗ GW σ GW under the projections pr GW / SFT : B ( ι ) ( p − , p + ; A ) → B GW / SFT ( A ). Here we have pr GW =pr SFT ◦ φ ι , so the bundle E ι ( p − , p + ; A ) = pr ∗ GW E GW ( A ) and section σ ι ( p − ,p + ; A ) = pr ∗ GW σ GW arenaturally identified with the pullback bundle φ ∗ ι E ( p − , p + ; A ) = pr ∗ GW E SFT ( A ) | { }×B GW ( A ) and section φ ∗ ι σ ( p − ,p + ; A ) = pr ∗ GW σ SFT | { }×B GW ( A ) using Assumption 5.5 (iii). Finally, the index of the inducedsection σ ( p − ,p + ; A ) , and similarly of σ ι ( p − ,p + ; A ) , is computed by [Fi1, Cor.7.3] asind( σ ( p − ,p + ; A ) ) = ind( σ SFT ) + dim M ( p − , M ) + dim M ( M, p + ) − CP × M )= 2 c ( A ) + dim M + 5 + | p − | + dim M − | p + | − − M = 2 c ( A ) + | p − | − | p + | + 1 = I ( p − , p + ; A ) . (cid:3) Given this compactification and polyfold description of the moduli spaces M ( α ) ⊂ σ − α (0) and M ι ( α ) ⊂ σ ια − (0) for all tuples in the indexing set I := (cid:8) α = ( p − , p + ; A ) (cid:12)(cid:12) p − , p + ∈ Crit( f ) , A ∈ H ( M ) (cid:9) , we can again apply [HWZ, Theorems 18.2,18.3,18.8] to the sc-Fredholm sections σ α and σ ια andobtain Corollary 4.6 verbatim for these collections of moduli spaces. In § σ ια = φ ∗ ι σ α arises from restriction of σ α , so admissible perturbations of σ α pullback to admissible perturbations of σ ια . For now, we choose perturbations independently and thusas in Definition 4.8 obtain perturbation-dependent, and not yet algebraically related, Λ-linear maps. Definition 5.7.
Given admissible sc + -multisections κ = ( κ ( p − ,p + ; A ) ) p ± ∈ Crit( f ) ,A ∈ H in general po-sition to ( σ ( p − ,p + ; A ) ) and κ ι = ( κ ι ( p − ,p + ; A ) ) p ± ∈ Crit( f ) ,A ∈ H in general position to ( σ ι ( p − ,p + ; A ) ) as inCorollary 4.6, we define the maps h κ : CM → CM and ι κ ι : CM → CM to be the Λ -linearextensions of h κ h p − i := X p + ,A I ( p − ,p +; A )=0 Z κ ( p − , p + ; A ) · T ω ( A ) h p + i , ι κ ι h p − i := X p + ,A Iι ( p − ,p +; A )=0 Z κ ι ( p − , p + ; A ) · T ω ( A ) h p + i . The proof that the coefficients of these maps lie in the Novikov field Λ is verbatim the same asLemma 4.9, based on the compactness properties in Lemma 5.6.
Remark 5.8.
The determination in Corollary 4.6 of Z κ ( p − , p + ; A ) , Z κ ι ( p − , p + ; A ) ∈ Q thatis used in Definition 5.7 requires an orientation of the sections σ ( p − ,p + ; A ) and σ ι ( p − ,p + ; A ) . As inRemark 4.7 this is determined via the fiber product construction in Lemma 5.6 from the orienta-tions of the Morse trajectory spaces in Remark 3.5 (i) and the orientations of σ GW , σ SFT given in As in Remark 4.4, this identification is stated for intuition and will ultimately not be used in our proofs.
POLYFOLD PROOF OF THE ARNOLD CONJECTURE 21
Assumption 5.5. In practice, we will construct the perturbations κ, κ ι by pullback of perturbations λ = ( λ A ) A ∈ H ( M ) of the SFT-sections σ SFT and their restriction λ ι to { } × B GW ( A ) ⊂ ∂ B SFT ( A ).So we can specify the orientations of the regularized zero sets by expressing them as transverse fiberproducts of oriented spaces over CP × M or C ± × M , Z κ ι ( p − , p + ; A ) = M ( p − , M ) ev +0 × ev + Z λ ι ( A ) ev − × ev − M ( M, p + ) , = M ( p − , M ) ev +0 × ev + (cid:0) Z λ ι ( A ) ∩ B + , − GW ( A ) (cid:1) ev − × ev − M ( M, p + ) ,Z κ ( p − , p + ; A ) = M ( p − , M ) ev +0 × ev + (cid:0) Z λ ( A ) ∩ B + , − SFT ( A ) (cid:1) ev − × ev − M ( M, p + ) , using ev ± : B GW ( A ) → CP × M resp. ev ± : B + , − GW / SFT ( A ) → C ± × M and the Morse evaluationsev ± : M ( . . . ) → CP × M, τ ( z ± , ev( τ )) resp. ev ± : M ( . . . ) → C ± × M, τ (0 , ev( τ )).6. Algebraic relations via coherent perturbations
In this section we prove parts (i)–(iii) of Theorem 1.2, that is the algebraic properties which relatethe maps
P SS : CM → CF , SSP : CF → CM constructed in §
4, and the maps ι : CM → CM , h : CM → CM constructed in §
5. More precisely, we will make so-called “coherent” choices ofperturbations in § § § ι is a chain map, (ii) ι is a Λ-moduleisomorphism, and (iii) h is a chain homotopy between the composition SSP ◦ P SS and ι .6.1. Coherent polyfold descriptions of moduli spaces.
The general approach to obtainingnot just counts as discussed in § M ( α ) = σ − α (0) as zero sets ofa “coherent collection” of sc-Fredholm sections (cid:0) σ α : B ( α ) → E ( α ) (cid:1) α ∈I of strong polyfold bundles.Here “coherence” indicates a well organized identification of the boundaries ∂ B ( α ) with unions ofCartesian products of other polyfolds in the collection I , which is compatible with the bundles andsections. A general axiomatic description of such coherent structures is being developed in [Fi2].As a first example, the moduli spaces M ι ( p − , p + ; A ) in § ι : CM → CM are given polyfold descriptions σ ι ( p − ,p + ; A ) : B ι ( p − , p + ; A ) → φ ∗ ι E ( p − , p + ; A ) in Lemma 5.6 that ariseas fiber products with polyfolds B GW ( A ) without boundary. Thus their coherence properties statedbelow follow from properties of the fiber product in [Fi1] and the boundary stratification of theMorse trajectory spaces in Theorem 3.3. We state this result to illustrate the notion of coherence.The full technical statement – on the level of ep-groupoids and including compatibility with bundlesand sections – can be found in the second bullet point of Lemma 6.4. Lemma 6.1.
For any p ± ∈ Crit( f ) and A ∈ H ( M ) the smooth level of the first boundary stratumof the fiber product B ι ( p − , p + ; A ) in Lemma 5.6 is naturally identified with ∂ B ι ( p − , p + ; A ) ∞ ∼ = [ q ∈ Crit( f ) M ( p − , q ) × ∂ B ι ( q, p + ; A ) ∞ ⊔ [ q ∈ Crit( f ) ∂ B ι ( p − , q ; A ) ∞ × M ( q, p + ) . Proof.
By the fiber product construction [Fi1, Cor.7.3] of B ι ( p − , p + ; A ) in Lemma 5.6, the degener-acy index satisfies d B ι ( p − ,p + ; A ) ( τ − , v, τ + ) = d M ( p − ,M ) ( τ − ) + d B GW ( A ) ( v ) + d M ( M,p + ) ( τ + ), and thesmooth level is B ι ( p − , p + ; A ) ∞ = M ( p − , M ) { z − }× ev × ev − B + , − GW ( A ) ∞ ev + × { z +0 }× ev M ( M, p + ). Thepolyfold B GW ( A ) and its open subset B + , − GW ( A ) are boundaryless by Assumption 5.5 (iii), which means d B GW ( A ) = d B + , − GW ( A ) ≡
0. Hence we have d B ι ( p − ,p + ; A ) ( τ − , v, τ + ) = 1 if and only if τ − ∈ ∂ M ( p − , M )and τ + ∈ ∂ M ( M, p + ) or the other way around. These two cases are disjoint but analogous, so it re-mains to show that the first case consists of points in the union S q ∈ Crit( f ) M ( p − , q ) × ∂ B ι ( q, p + ; A ) ∞ .For that purpose recall the identification ∂ M ( p − , M ) = S q ∈ Crit( f ) M ( p − , q ) × M ( q, M ) in Theo-rem 3.3, which is compatible with the evaluation ev : M ( p − , q ) × M ( q, M ) → M, ( τ , τ ) ev( τ ) by construction, and thus ∂ M ( p − , M ) { z − }× ev × ev − B + , − GW ( A ) ∞ ev + × { z +0 }× ev ∂ M ( M, p + )= (cid:16)S q ∈ Crit( f ) M ( p − , q ) × M ( q, M ) (cid:17) { z − }× ev × ev − B + , − GW ( A ) ∞ ev + × { z +0 }× ev M ( M, p + )= S q ∈ Crit( f ) M ( p − , q ) × (cid:16) M ( q, M ) { z − }× ev × ev − B + , − GW ( A ) ∞ ev + × { z +0 }× ev M ( M, p + ) (cid:17) = S q ∈ Crit( f ) M ( p − , q ) × ∂ B ι ( q, p + ; A ) ∞ Here we also used the identification of the interior smooth level in Lemma 5.6. (cid:3)
Next, the polyfold description in Lemma 5.6 for the moduli spaces M ( p − , p + ; A ) in § h : CM → CM , are obtained as fiber products of the Morse trajectory spaces withpolyfold descriptions σ SFT : B SFT ( A ) → E SFT ( A ) of SFT moduli spaces given in [FH, FH1]. We willstate as assumption only those parts of their coherence properties that are relevant to our argumentin § ι − SSP ◦ P SS = d ◦ h + h ◦ d. Here the contributions to d ◦ h + h ◦ dwill arise from boundary strata of the Morse trajectory spaces, whereas ι − SSP ◦ P SS arises fromthe following identification of the boundary of the polyfold B + , − SFT ( A ), which is given as open subsetof B SFT ( A ) in Assumption 5.5 (vi). Remark 6.2.
In the following we will use the word “face” loosely for Cartesian products of polyfoldssuch as F = B +SFT ( γ ; A + ) × B − SFT ( γ ; A − ) and their immersions into the boundary of another polyfoldsuch as ∂ B + , − SFT ( A ). We also refer to the image of the immersion F ֒ → ∂ B + , − SFT ( A ) as a face of B + , − SFT ( A ).Compared with the formal definition of faces in [HWZ, Definitions 2.21,11.1,16.13], ours are disjointunions of faces and carry more structure, as we will specify. Assumption 6.3.
The collection of oriented sc-Fredholm sections of strong polyfold bundles σ ± SFT : B ± SFT ( γ ; A ) → E ± SFT ( γ ; A ) , σ GW : B GW ( A ) → E GW ( A ) , σ SFT : B SFT ( A ) → E SFT ( A ) for γ ∈P ( H ) and A ∈ H ( M ) together with the evaluation maps ev ± : B ± SFT ( γ ; A ) → C ± × M , ev ± : B GW ( A ) → CP × M , ev ± : B SFT ( A ) → CP ∞ × M , and their sc ∞ restrictions on open subsets, ev ± : B ± , C SFT ( γ ; A ) → C ± × M , ev ± : B + , − GW / SFT ( A ) → C ± × M from Assumptions 4.3, 5.5 has thefollowing coherence properties.(i) For each γ ∈ P ( H ) and A − , A + ∈ H ( M ) such that A − + A + = A , there is a sc ∞ immersion l γ,A ± : B +SFT ( γ ; A + ) × B − SFT ( γ ; A − ) → ∂ B SFT ( A ) whose restriction to the interior ∂ B +SFT ( γ ; A + ) × ∂ B − SFT ( γ ; A − ) ⊂ B + , C SFT ( γ ; A + ) ×B − , C SFT ( γ ; A − ) is anembedding into the boundary of the open subset B + , − SFT ( A ) . They map into the limit set lim R →∞ { R }×B GW ( A ) from Assumption 5.5(iii), so cover most of the boundary ∂ B SFT ( A ) ⊃ { } × B GW ( A ) ⊔ S γ ∈P ( H ) A − + A += A l γ,A ± (cid:0) B +SFT ( γ ; A + ) × B − SFT ( γ ; A − ) (cid:1) . (ii) The union of the images l γ,A ± (cid:0) B + , C SFT ( γ ; A + ) × B − , C SFT ( γ ; A − ) (cid:1) ⊂ ∂ B + , − SFT ( A ) for all admissiblechoices of γ, A ± is the intersection of B + , − SFT ( A ) with lim R →∞ { R } × B GW ( A ) ⊂ ∂ B SFT ( A ) , i.e. ∂ B + , − SFT ( A ) = { } × B + , − GW ( A ) ⊔ ∂ R = ∞ B + , − SFT ( A ) , where ∂ R = ∞ B + , − SFT ( A ) = S γ ∈P ( H ) A − + A += A l γ,A ± (cid:0) B + , C SFT ( γ ; A + ) × B − , C SFT ( γ ; A − ) (cid:1) . When restricted to the interiors, this yields a disjoint cover of the top boundary stratum, ∂ B + , − SFT ( A ) = { } × B + , − GW ( A ) ⊔ F γ ∈P ( H ) A − + A += A l γ,A ± (cid:0) ∂ B +SFT ( γ ; A + ) × ∂ B − SFT ( γ ; A − ) (cid:1) . (iii) The immersions l γ,A ± are compatible with the evaluation maps, bundles, and sections – asrequired for the construction [FH1] of coherent perturbations for SFT, that is: See also the end of Remark 5.4 (ii) for the motivation of B + , − SFT ( A ) as open subset that intersects the boundarystrata lim R →∞ { R } × B GW ( A ) ⊂ ∂ B SFT ( A ) in the buildings which have one marked point in each of the componentsmapping to C ± × M , and no marked points mapping to intermediate cylinders R × S × M . The extra boundary faces of B SFT ( A ) arise from both marked points mapping to the same component in the R → ∞ neck stretching limit. These will not be relevant to our construction of coherent perturbations. POLYFOLD PROOF OF THE ARNOLD CONJECTURE 23 (a)
The boundary restriction of the evaluation maps ev ± | { }×B GW ( A ) ⊂ ∂ B SFT ( A ) coincides with theGromov-Witten evaluation maps ev ± : B GW ( A ) → CP ∞ , and the same holds for their sc ∞ restriction ev + × ev − | { }×B + , − GW ( A ) ⊂ ∂ B + , − SFT ( A ) = ev + × ev − : B + , − GW ( A ) → C + × M × C − × M with values in C ± ⊂ CP ∞ = C + ⊔ S ⊔ C − . The restriction of ev ± : B SFT ( A ) → CP ∞ toeach boundary face im l γ,A ± ⊂ ∂ B SFT ( A ) takes values in C ± ⊂ CP ∞ , and its pullback under l γ,A ± coincides with ev ± : B ± SFT ( γ ; A ± ) → C ± × M . Moreover, pullback of the restrictedsc ∞ evaluations ev + × ev − : B + , − SFT ( A ) → C + × M × C − × M under l γ,A ± coincides with ev + × ev − : B + , C SFT ( γ ; A + ) × B − , C SFT ( γ ; A − ) → C + × M × C − × M . (b) The restriction of σ SFT to F = { } × B GW ( A ) ⊂ ∂ B SFT ( A ) equals to σ GW via a naturalidentification E SFT ( A ) | F ∼ = E GW ( A ) . This identification reverses the orientation of sections. (c) The restriction of σ SFT to each face F = B +SFT ( γ ; A + ) × B − SFT ( γ ; A − ) ⊂ ∂ B SFT ( A ) is re-lated by pullback to σ +SFT × σ − SFT = σ SFT ◦ l γ,A ± via a natural identification l ∗ γ,A ± E SFT ( A ) ∼ = E +SFT ( γ ; A + ) × E − SFT ( γ ; A − ) . This identification preserves the orientation of sections. Coherent perturbations for chain map identity.
In this section we prove Theorem 1.2 (i),that is we construct ι κ ι in Definition 5.7 as a chain map on the Morse complex (5) with differential d : CM → CM given by (4). This requires the following construction of the perturbations κ ι thatis coherent in the sense that it is compatible with the boundary identifications of the polyfolds B ι ( p − , p + ; A ) in Lemma 6.1. Here we will indicate smooth levels by adding ∞ as superscript –denoting e.g. X ι, ∞ p − ,p + ; A as the smooth level of an ep-groupoid representing B ι ( p − , p + ; A ) ∞ . Lemma 6.4.
There is a choice of ( κ ια ) α ∈I in Corollary 4.6 for I = { ( p − , p + ; A ) | p ± ∈ Crit( f ) , A ∈ H ( M ) } that is coherent w.r.t. the identifications in Lemma 6.1 in the following sense. • Each κ ια : W ια → Q + for α ∈ I is an admissible sc + -multisection of a strong bundle P α : W ια →X ια that is in general position to a sc-Fredholm section functor S ια : X ια → W ια which represents σ ια | V α on an open neighbourhood V α ⊂ B ι ( α ) of the zero set σ − α (0) . • The identification of top boundary strata in Lemma 6.1 holds for the representing ep-groupoids, ∂ X ι, ∞ p − ,p + ; A ∼ = S q ∈ Crit( f ) M ( p − , q ) × ∂ X ι, ∞ q,p + ; A ⊔ S q ∈ Crit( f ) ∂ X ι, ∞ p − ,q ; A × M ( q, p + ) , and the oriented section functors S ια : X ια → W ια are compatible with these identifications in thesense that the restriction of S ιp − ,p + ; A to any face F ∞ ( p − ,q − ) ,α ′ := M ( p − , q − ) × ∂ X ι, ∞ α ′ ⊂ ∂ X ι, ∞ p − ,p + ; A resp. F ∞ α ′ , ( q + ,p + ) := ∂ X ι, ∞ α ′ × M ( q + , p + ) ⊂ ∂ X ι, ∞ p − ,p + ; A for another α ′ ∈ I coincides on the smoothlevel with the pullback S ια | F ∞ = pr ∗F S ια ′ | F ∞ of S ια ′ via the projection pr F : F = F ( p − ,q − ) ,α ′ := M ( p − , q − ) × ∂ X ια ′ → X ια ′ resp. pr F : F = F α ′ , ( q + ,p + ) := ∂ X ια ′ × M ( q + , p + ) → X ια ′ . • Each restriction κ ια | P − α ( F ∞ ) to a face F ∞ = F ∞ ( p − ,q − ) ,α ′ resp. F ∞ = F ∞ α ′ , ( q + ,p + ) is given bypullback κ ια | P − α ( F ∞ ) = κ ια ′ ◦ pr ∗F via the identification P − α ( F ∞ ) ∼ = pr ∗F W ια ′ | ∂ X ι, ∞ α ′ and naturalmap pr ∗F : pr ∗F W ια ′ → W ια ′ .For any such choice of κ ι = ( κ ια ) α ∈I , the resulting map ι κ ι : CM → CM in Definition 5.7 satisfies ι κ ι ◦ d + d ◦ ι κ ι = 0 . By setting ι h p i := ( − | p | ι κ ι h p i we then obtain a chain map ι : C ∗ M → C ∗ M ,that is ι ◦ d = d ◦ ι .Proof. We will first assume the claimed coherence and discuss the algebraic consequences up to signs,then construct the coherent data, and finally use this construction to compute the orientations.
Construction of chain map:
Assuming ι κ ι ◦ d + d ◦ ι κ ι = 0, recall that d decreases the degree onthe Morse complex (6) by 1. Thus ι : C ∗ M → C ∗ M defined as above satisfies for any q ∈ Crit( f )( ι ◦ d − d ◦ ι ) h q i = ( − | q |− ι κ ι (d h q i ) − d(( − | q | ι κ ι h q i ) = ( − | q |− (cid:0) ι κ ι ◦ d + d ◦ ι κ ι (cid:1) h q i = 0 . By Λ-linearity this proves ι ◦ d = d ◦ ι on C ∗ M . Proof of identity:
To prove ι κ ι ◦ d + d ◦ ι κ ι = 0 note that both ι κ ι and d are Λ-linear, so theclaimed identity is equivalent to the collection of identities ( ι κ ι ◦ d) h p − i + (d ◦ ι κ ι ) h p − i = 0 for all generators p − ∈ Crit( f ). That is we wish to verify X q,p + ,A Iι ( q,p +; A )=0 | q | = | p −|− M ( p − , q ) · Z κ ι ( q, p + ; A ) · T ω ( A ) h p + i + X q,p + ,A Iι ( p − ,q ; A )=0 | p + | = | q |− Z κ ι ( p − , q ; A ) · M ( q, p + ) · T ω ( A ) h p + i = 0 . Here, by the index formula (18), both sides can be written as sums over p + ∈ Crit( f ) and A ∈ H ( M )for which I ι ( p − , p + ; A ) = 1. Then it suffices to prove for any such pair α = ( p − , p + ; A ) with I ι ( α ) = 1(30) P | q | = | p − |− M ( p − , q ) · Z κ ι ( q, p + ; A ) + P | q | = | p + | +1 Z κ ι ( p − , q ; A ) · M ( q, p + ) = 0 . This identity will follow by applying Corollary 4.6 (v) to the sc + -multisection κ α : W ια → Q + . Itsperturbed zero set is a weighted branched 1-dimensional orbifold Z κ ι ( α ), whose boundary is givenby the intersection with the smooth level of the top boundary stratum ∂ B ι ( α ) ∩ V α = | ∂ X ια | . Bycoherence (and with orientations discussed below) this boundary is ∂Z κ ι ( α ) = Z κ ι ( α ) ∩ | ∂ X ια | = S q ∈ Crit( f ) Z κ ι ( α ) ∩ (cid:0) M ( p − , q ) × | ∂ X ιq,p + ; A | (cid:1) ⊔ S q ∈ Crit( f ) Z κ ι ( α ) ∩ (cid:0) | ∂ X ιp − ,q ; A | × M ( q, p + ) (cid:1) = S q ∈ Crit( f ) M ( p − , q ) × (cid:0) Z κ ι ( q, p + ; A ) ∩ | ∂ X ιq,p + ; A | (cid:1) ⊔ S q ∈ Crit( f ) (cid:0) Z κ ι ( p − , q ; A ) ∩ | ∂ X ιp − ,q ; A | (cid:1) × M ( q, p + ) , = S | q | = | p − |− M ( p − , q ) × Z κ ι ( q, p + ; A ) ⊔ S | q | = | p + | +1 Z κ ι ( p − , q ; A ) × M ( q, p + ) . Here the first summand of the third identification on the level of object spaces, (cid:8) ([ τ ] , x ) ∈ M ( p − , q ) × ∂ X q,p + ; A ⊂ ∂ X α (cid:12)(cid:12) κ α ( S ια ([ τ ] , x )) > (cid:9) ∼ = (cid:8) ([ τ ] , x ) ∈ M ( p − , q ) × ∂ X q,p + ; A (cid:12)(cid:12) κ q,p + ; A ( S ιq,p + ; A ( x )) > (cid:9) = M ( p − , q ) × (cid:8) x ∈ ∂ X q,p + ; A (cid:12)(cid:12) κ q,p + ; A ( S ιq,p + ; A ( x )) > (cid:9) , follows if we assume coherence of sections and multisections on the faces F ( p − ,q ) ,α ′ ⊂ ∂ X ια , κ α ( S ια ([ τ ] , x )) = κ α ( S ιq,p + ; A ( x )) = κ q,p + ; A ( S ιq,p + ; A ( x )) . The second summand is identified similarly by assuming coherence on the faces F α ′ , ( q − ,p + ) ⊂ ∂ X ια .Finally, the fourth identification in ∂Z κ ι ( α ) for α = ( p − , p + ; A ) with I ι ( α ) = 1 follows fromindex and regularity considerations as follows. Corollary 4.6 (iii),(iv) guarantees that the perturbedsolution spaces Z κ ι ( α ′ ) are nonempty only for Fredholm index I ι ( α ′ ) ≥
0, and for I ι ( α ′ ) = 0 arecontained in the interior, Z κ ι ( α ′ ) ⊂ ∂ B ( α ′ ). The Morse trajectory spaces M ( p − , q ) resp. M ( q, p + )are nonempty only for | p − | − | q | ≥ | q | − | p + | ≥
1, so the perturbed solution spaces in theCartesian products have Fredholm index (18) I ι ( q, p + ; A ) = 2 c ( A ) + | q | − | p + | = I ι ( p − , p + ; A ) + | q | − | p − | = 1 + | q | − | p − | ≤ , and analogously I ι ( p − , q ; A ) = I ι ( p − , p + ; A )+ | p + |−| q | ≤
0. By the above regularity of the perturbedsolution spaces this implies that the unions on the left hand side of the fourth identification are over | q | = | p − | − | q | = | p + | + 1 as in (30), and for these critical points we have the inclusions Z κ ι ( q, p + ; A ) ⊂ ∂ B ( q, p + ; A ) and Z κ ι ( p − , q ; A ) ⊂ ∂ B ( p − , q ; A ) that verify the equality.This finishes the identification of the boundary ∂Z κ ι ( α ). Now Corollary 4.6 (v) asserts thatthe sum of weights over this boundary is zero – when counted with signs that are induced by theorientation of Z κ ι ( α ). So in order to prove the identity (30) we need to compare the boundaryorientation of ∂Z κ ι ( α ) with the orientations on the faces. We will compute the relevant signs in (31)below, after first making coherent choices of representatives S ια : X ια → W ια of the oriented sections σ ια , and constructing coherent sc + -multisections κ ια : W ια → Q + for α ∈ I . Coherent ep-groupoids, sections, and perturbations:
Recall that the fiber product construc-tion in Lemma 5.6 defines each bundle W ια = pr ∗ α W GW A for α = ( p − , p + ; A ) ∈ I as the pullback Here and in the following we suppress indications of the smooth level, as the perturbed zero sets automaticallylie in the smooth level; see Remark A.3.
POLYFOLD PROOF OF THE ARNOLD CONJECTURE 25 of a strong bundle W GW A → X GW A under a projection of ep-groupoids – with abbreviated notationev ± := { z ± } × ev : M ( . . . ) → CP × M –pr p − ,p + ; A : X ιp − ,p + ; A = M ( p − , M ) ev − × ev − X GW A ev + × ev +0 M ( M, p + ) −→ X GW A . Moreover, the section S ια = S GW A ◦ pr α is induced by the section S GW A : X GW A → W GW A which cutsout the Gromov-Witten moduli space M GW ( A ) = | ( S GW A ) − (0) | . Then the identification of the topboundary stratum proceeds exactly as the proof of Lemma 6.1. Coherence of the bundles and sectionsfollows from coherence of the projections pr α : X ια → X GW A in the sense that pr α | F ∞ = pr α ′ ◦ pr F forall smooth levels of faces F ⊃ F ∞ ⊂ ∂ X ια and their projections pr F : F = F ( p − ,q − ) ,α ′ → X ια ′ resp.pr F : F = F α ′ , ( q − ,p + ) → X ια ′ . For example, the face F = F ( p − ,q − ) , ( q − ,p + ; A ) with F ∞ ⊂ ∂ X ιp − ,p + ; A identifies (cid:0) [ τ ] , ( τ − , [ v ] , τ + ) (cid:1) ∈ F ∞ ( p − ,q − ) , ( q − ,p + ; A ) = M ( p − , q − ) × ∂ X ι, ∞ q − ,p + ; A = M ( p − , q − ) × M ( q − , M ) ev − × ev − X GW , ∞ A ev + × ev +0 M ( M, p + )with (cid:0) ([ τ ] , τ − ) , [ v ] , τ + (cid:1) ∈ M ( p − , M ) − × ev − X GW , ∞ A ev + × ev +0 M ( M, p + ) ⊂ ∂ X ι, ∞ p − ,p + ; A , and pr p − ,p + ; A (cid:0) ([ τ ] , τ − ) , [ v ] , τ + (cid:1) = [ v ] ∈ X GW A coincides with (pr q − ,p + ; A ◦ pr F ) (cid:0) [ τ ] , ( τ − , [ v ] , τ + ) (cid:1) =pr q − ,p + ; A ( τ − , [ v ] , τ + ) = [ v ] ∈ X GW A . Now any choice of sc + -multisections ( λ GW A : W GW A → Q + ) A ∈ H ( M ) induces a coherent collection of sc + -multisections (cid:0) κ ια := λ GW A ◦ pr ∗ α : pr ∗ α W GW A → Q + (cid:1) α ∈I bycomposition with the natural maps pr ∗ α : pr ∗ α W GW A → W GW A covering pr α : X ια → X GW A . Indeed,pr α | F = pr α ′ ◦ pr F lifts to pr ∗ α | P − α ( F ∞ ) = pr ∗ α ′ ◦ pr ∗F so that κ ια | P − α ( F ∞ ) = λ GW A ◦ pr ∗ α | P − α ( F ∞ ) = λ GW A ◦ pr ∗ α ′ ◦ pr ∗F = κ ια ′ ◦ pr ∗F . Construction of admissible Gromov-Witten perturbations:
It remains to choose the sc + -multisections ( λ GW A : W GW A → Q + ) A ∈ H ( M ) so that the induced coherent collection κ ι = (cid:0) λ GW A ◦ pr ∗ α (cid:1) α ∈I is admissible and in general position. To do so, for each A ∈ H ( M ) we apply Theorem A.9to the sc-Fredholm section functor S GW A : X GW A → W GW A , the sc ∞ submersion ev − × ev + : X GW A → CP × M × CP × M , and the collection of Cartesian products of stable and unstable manifolds { z − } × W − p − × { z +0 } × W + p + for all pairs of critical points p − , p + ∈ Crit( f ).After fixing a pair controlling compactness ( N A , U A ) for each A ∈ H ( M ), Theorem A.9 yields( N A , U A )-admissible sc + -multisections λ GW A : W GW A → Q + in general position to S GW A for each A ∈ H ( M ). Moreover, they can be chosen such that restriction of evaluations to the perturbed zeroset ev − × ev + : Z λ GW A → CP × M × CP × M is transverse to all of the products of unstable andstable submanifolds { z − } × W − p − × { z +0 } × W + p + for p − , p + ∈ Crit( f ). Note that these embeddedsubmanifolds cover the images of all evaluation maps on the compactified Morse trajectory spacesev − × ev +0 : M ( p − , M ) × M ( M, p + ) → CP × M × CP × M , by construction of the evaluationsev : M ( . . . ) → M in (8), which determine ev ± ( τ ) = ( z ± , ev( τ )). Thus we obtain transverse fiberproducts M ( p − , M ) ev − × ev − Z λ GW A ev + × ev +0 M ( M, p + ) for every α ∈ I . This translates into thepullbacks κ ια = λ GW A ◦ pr ∗ α being in general position to the pullback sections S ια for α ∈ I . Moreover, κ ια is admissible with respect to a pullback of ( N A , U A ), so the perturbed zero set is a compactweighted branched orbifold for each α = ( p − , p + ; A ),“ (cid:12)(cid:12) ( S ια + κ ια ) − (0) (cid:12)(cid:12) ” = Z κ ι ( α ) = M ( p − , M ) ev − × ev − Z λ GW A ev + × ev +0 M ( M, p + ) . This finishes the construction of coherent perturbations.
Computation of orientations:
To prove the identity (30) it remains to compute the effect ofthe orientations in Remark 5.8 on the algebraic identity in Corollary 4.6 (v) that arises from theboundary ∂Z κ ι ( α ) of the 1-dimensional weighted branched orbifolds arising from regularization ofthe moduli spaces with index I ι ( α ) = I ι ( p − , p + ; A ) = 1. Here Z λ GW A is of even dimension and has noboundary since the Gromov-Witten polyfolds in Assumption 5.5 have no boundary, and the index of σ GW is even. For the Morse trajectory spaces, the boundary strata are determined in Theorem 3.3,with relevant orientations computed in Remark 3.5. Thus for I ι ( α ) = | p − | − | p + | + 2 c ( A ) = 1 we can compute orientations – at the level of well defined finite dimensional tangent spaces at asolution; in whose neighbourhood the evaluation maps are guaranteed to be scale-smooth – ∂Z κ ι ( α ) = ∂ M ( p − , M ) ev × ev Z λ GW A ev × ev ∂ M ( M, p + ) ⊔ ( − dim M ( p − ,M ) ∂ M ( p − , M ) ev × ev Z λ GW A ev × ev ∂ M ( M, p + )= (cid:0)F q ∈ Crit f M ( p − , q ) × M ( q, M ) (cid:1) ev × ev Z λ GW A ev × ev M ( M, p + )(31) ⊔ ( − | p − | + | p + | +1 M ( p − , M ) ev × ev Z λ GW A ev × ev (cid:0)F q ∈ Crit f M ( M, q ) × M ( q, p + ) (cid:1) = F q ∈ Crit f M ( p − , q ) × Z κ ι ( q, p + ; A ) ⊔ F q ∈ Crit f Z κ ι ( p − , q ; A ) × M ( q, p + ) . Here the signs in the first equality arise from the ambient Cartesian product ∂ ( M − × Z × M + ) ⊂ ( − dim( M − × Z ) M − × Z × M + ; in the second equality we used Remark 3.5; and in the final equalitywe use | p − | + | p + | + 1 ≡ I ι ( α ) = 1 ≡ ∂Z κ ι ( α ) for I ι ( α ) = 1 that proves (30) and thus yields a chain map. (cid:3) Admissible perturbations for isomorphism property.
In this section we prove Theo-rem 1.2 (ii), i.e. construct ι = ( − ∗ ι κ ι : C ∗ M → C ∗ M in Definition 5.7 and Lemma 6.4 as aΛ-module isomorphism on the chain complex CM = CM Λ over the Novikov field as in (5). Thisrequires a construction of the perturbations κ ι that preserves the properties of the zero sets inRemark 5.1 for nonpositive symplectic area ω ( A ) ≤ Lemma 6.5.
The coherent collection of sc + -multisections κ ι in Lemma 6.4 can be chosen suchthat Z κ ι ( p − , p + ; A ) = 0 for A ∈ H ( M ) r { } with ω ( A ) ≤ , or for A = 0 and p − = p + , and Z κ ι ( p, p ; 0) = 0 . As a consequence, ι = ( − ∗ ι κ ι : CM Λ → CM Λ is a Λ -module isomorphism.Proof. The sc + -multisections κ ι in Lemma 6.4 are obtained from choices of sc + -multisections ( κ A : W ιA → Q + ) A ∈ H ( M ) that are in general position to sc-Fredholm sections S A : X GW A → W A which cutout the Gromov-Witten moduli space M GW ( A ) = | S − A (0) | , and such that moreover the evaluationmaps restricted to the perturbed zero sets, ev − × ev + : Z ( κ A ) → CP × M × CP × M are transverseto the unstable and stable manifolds { z − } × W − p − × { z +0 } × W + p + ⊂ CP × M × CP × M for anypair of critical points p − , p + ∈ Crit( f ).We will first consider α = ( p − , p + ; A ) ∈ I for nontrivial homology classes A ∈ H ( M ) r { } withnonpositive symplectic area ω ( A ) ≤
0. Recall from Remark 5.1 that these moduli spaces are empty | S − A (0) | = ∅ , so as in Corollary 4.6 we can choose empty neighbourhoods ∅ = |U A | ⊂ |X GW A | tocontrol compactness. Then the perturbed zero set Z ( κ A ) = |{ x ∈ X A | κ A ( S A ( x )) > }| ⊂ |U A | is forced to be empty, i.e. κ A ◦ S A ≡
0. This is an allowed choice in Lemma 6.4 since evaluationmaps from an empty set are trivially transverse to any submanifold. This choice induces for any p ± ∈ Crit( f ) in α = ( p − , p + ; A ) an induced sc + -multisection κ ια = κ A ◦ pr ∗ α : W ια → Q + . Itsperturbed zero set is Z κ ια ( α ) = (cid:12)(cid:12)(cid:8) ( τ − , x, τ + ) ∈ X α (cid:12)(cid:12) κ ια (cid:0) S ια ( τ − , x, τ + ) (cid:1) > (cid:9)(cid:12)(cid:12) = ∅ since the coherence in Lemma 6.4 implies κ ια ◦ S ια = κ A ◦ pr ∗ α ◦ S ια = κ A ◦ S A ◦ pr α ≡
0, ormore concretely κ ι ( p − ,p + ; A ) (cid:0) S ιp − ,p + ; A ( τ − , x, τ + ) (cid:1) = κ A ( S A ( x )) = 0. Thus we have ensured vanishingcounts Z κ ι ( p − , p + ; A ) = 0 for A ∈ H ( M ) r { } with ω ( A ) ≤ I ι ( p − , p + ; A ) = 0.Next we consider A = 0 ∈ H ( M ) and recall from Remark 5.1 and Assumption 5.5 (ii) that theGromov-Witten moduli space M GW (0) = Z ( κ ) is already compact and transversely cut out. Thusthe trivial sc + -multisection κ : W → Q + , given by κ (0 x ) = 1 on zero vectors 0 x ∈ ( W ) x and κ | ( W ) x r { x } ≡
0, is an admissible sc + -multisection in general position to S : X GW → W . Recallmoreover that the evaluation maps on the unperturbed zero set areev − × ev + : Z ( κ ) ≃ CP × CP × M → CP × M × CP × M, ( z − , z + , x ) ( z − , x, z + , x ) . In the CP -factors this is submersive so transverse to the fixed points ( z − , z +0 ) ∈ CP × CP . Inthe M -factors this is the diagonal map, which is transverse to the unstable and stable manifolds W − p − × W + p + ⊂ M × M for any pair p − , p + ∈ Crit( f ) by the Morse-Smale condition on the metric POLYFOLD PROOF OF THE ARNOLD CONJECTURE 27 on M chosen in §
3. Thus the trivial multisection κ is in fact an allowed choice in Lemma 6.4. Nowwith this choice, the tuples ( p − , p + ; 0) ∈ I for which we need to compute Z κ ι ( p − , p + ; 0) = (cid:12)(cid:12)(cid:8) ( τ − , [ v ] , τ + ) ∈ M ( p − , M ) × Z ( κ ) × M ( M, p + ) (cid:12)(cid:12) (cid:0) z ± , ev( τ ± ) (cid:1) = ev ± ([ v ]) (cid:9)(cid:12)(cid:12) ∼ = (cid:12)(cid:12)(cid:8) ( τ − , τ + ) ∈ M ( p − , M ) × M ( M, p + ) (cid:12)(cid:12) ev( τ − ) = ev( τ + ) (cid:9)(cid:12)(cid:12) are those with 0 = I ι ( p − , p + ; 0) = 2 c (0) + | p − | − | p + | , i.e. | p − | = | p + | . These are the fiber productsidentified in Remark 3.5 (ii) as either empty or a one point set, M ( p − , M ) ev × ev M ( M, p + ) = ( ∅ ; p − = p + , ( τ − ≡ p − , τ + ≡ p + ) ; p − = p + . Thus we have counts Z κ ι ( p − , p + ; 0) = 0 for p − = p + and Z κ ι ( p, p ; 0) = 0 for each p ∈ Crit( f ).Finally, we will use these computations of Z κ ( p − , p + ; A ) for ω ( A ) ≤ ι := ( − ∗ ι κ ι : CM Λ → CM Λ is a Λ-module isomorphism. For that purpose we choose anarbitrary total order of the critical points Crit( f ) = { p , . . . , p ℓ } and for i, j ∈ { , . . . , ℓ } denotethe coefficients of ι ( h p j i ) = P ℓi =1 λ ij h p i i by λ ij ∈ Λ. We claim that the ( ℓ × ℓ )-matrix with entries λ ij = P r ∈ Γ λ ijr T r satisfies the conditions of Lemma 2.1. To check this recall that we have byconstruction in Definition 5.7 and change of signs in Lemma 6.4 λ ijr = P A ∈ H ( M ) ,ω ( A )= rI ι ( p j ,p i ; A )=0 ( − | p j | Z κ ι ( p j , p i ; A ) . For r < λ ijr = 0 since each coefficient Z κ ι ( p j , p i ; A ) = 0 vanishes for ω ( A ) = r < r = 0 and i = j we also have λ ij = 0 since Z κ ι ( p j , p i ; A ) = 0 also holds for ω ( A ) = 0 and p j = p i . Finally, for r = 0 and i = j we use Z κ ι ( p j , p i ; A ) = 0 for A = 0 with ω ( A ) = 0 tocompute λ ii = Z κ ι ( p j , p i ; 0) = 0. This confirms that Lemma 2.1 applies, and thus ι ∼ = ( λ ij ) ≤ i,j ≤ ℓ is invertible. This finishes the proof. (cid:3) Coherent perturbations for chain homotopy.
In this section we prove Theorem 1.2 (iii)by constructing h κ : CM → CM in Definition 5.7 as a chain homotopy between SSP κ + ◦ P SS κ − and ι κ ι from Definitions 4.8,5.7, with appropriate sign adjustments as in Lemma 6.4. This requiresa coherent construction of perturbations κ, κ ι , κ − , κ + over the indexing sets I = I ι := (cid:8) α = ( p − , p + , A ) (cid:12)(cid:12) p − , p + ∈ Crit( f ) , A ∈ H ( M ) (cid:9) , I + := (cid:8) α = ( p, γ, A ) (cid:12)(cid:12) p ∈ Crit( f ) , γ ∈ P ( H ) , A ∈ H ( M ) (cid:9) , I − := (cid:8) α = ( γ, p, A ) (cid:12)(cid:12) p ∈ Crit( f ) , γ ∈ P ( H ) , A ∈ H ( M ) (cid:9) . Here we will use notation from Lemma A.7 for Cartesian products of multisections.
Lemma 6.6.
There is a choice of κ + = ( κ + α ) α ∈I + , κ − = ( κ − α ) α ∈I − , κ ι = ( κ ια ) α ∈I , κ = ( κ α ) α ∈I inDefinitions 4.8,5.7 that is coherent in the following sense.(i) Each κ ··· α : W ··· α → Q + for α ∈ I + ⊔I − ⊔I ι ⊔I is an admissible sc + -multisection of a strong bundle P ··· α : W ··· α → X ··· α that is in general position to a sc-Fredholm section functor S ··· α : X ··· α → W ··· α which represents σ ··· α | V ··· α on an open neighbourhood V ··· α ⊂ B ··· ( α ) of the zero set σ ··· α − (0) . The tuple κ ι = ( κ ια ) α ∈I ι satisfies the conclusions of Lemma 6.4 and 6.5.(ii) The smooth level of the first boundary stratum of X p − ,p + ,A for every ( p − , p + , A ) ∈ I is naturallyidentified – on the level of object spaces, and compatible with morphisms – with ∂ X ∞ p − ,p + ,A ∼ = ∂ X ι, ∞ p − ,p + ,A ⊔ [ γ ∈P ( H ) ,A = A − + A + ∂ X + , ∞ p − ,γ,A + × ∂ X − , ∞ γ,p + ,A − ⊔ [ q ∈ Crit( f ) M ( p − , q ) × ∂ X ∞ q,p + ,A ⊔ [ q ∈ Crit( f ) ∂ X ∞ p − ,q,A × M ( q, p + ) , (32) and the oriented section functors S ··· α are compatible with these identifications in the sense that therestriction of S p − ,p + ,A to any of these faces F ∞ ⊂ ∂ X ∞ p − ,p + ,A is given by pullback S p − ,p + ,A | F ∞ = pr ∗F S F of another sc-Fredholm section of a strong bundle over an ep-groupoid S F : X F → W F givenby S q,p + ,A , S p − ,q,A , S ιp − ,p + ,A , resp. S F = S + p − ,γ,A + × S − γ,p + ,A − : X + p − ,γ,A + × X − γ,p + ,A − → W + p − ,γ,A + × W − γ,p + ,A − via the projection pr F : F → X F given by the natural maps M ( p − , q ) × ∂ X q,p + ,A → X q,p + ,A , ∂ X ιp − ,p + ,A → X ιp − ,p + ,A ,∂ X p − ,q,A × M ( q, p + ) → X p − ,q,A , ∂ X + p − ,γ,A + × ∂ X − γ,p + ,A − → X + p − ,γ,A + × X − γ,p + ,A − . (iii) Each restriction κ α | P − α ( F ∞ ) for α = ( p − , p + , A ) ∈ I to one of the faces F ∞ ⊂ ∂ X α is givenvia the identification P − α ( F ∞ ) ∼ = pr ∗F W F | ∂ X F and natural map pr ∗F : pr ∗F W F → W F by κ α | P − α ( F ∞ ) = κ q,p + ,A ◦ pr ∗F for F = M ( p − , q ) × ∂ X q,p + ,A ,κ p − ,q,A ◦ pr ∗F for F = ∂ X p − ,q,A × M ( q, p + ) ,κ ιp − ,p + ,A ◦ pr ∗F for F = ∂ X ιp − ,p + ,A , ( κ + p − ,γ,A + · κ − γ,p + ,A − ) ◦ pr ∗F for F = ∂ X + p − ,γ,A + × ∂ X − γ,p + ,A − . For any such choice of κ ι = ( κ ια ) α ∈I , the resulting maps P SS κ + , SSP κ − , ι κ ι , h κ in Definitions 4.8,5.7 satisfy ( − | p | ι κ ι h p i = ( − | p | SSP κ − (cid:0) P SS κ + h p i (cid:1) + h κ (d h p i ) + d( h κ h p i ) , where d is theMorse differential from §
3. By setting ι h p i := ( − | p | ι κ ι h p i as in Lemma 6.4, P SS h p i :=( − | p | P SS κ + h p i , SSP := SSP κ − , and h := h κ we then obtain a chain homotopy between ι and SSP ◦ P SS , that is ι − SSP ◦ P SS = d ◦ h + h ◦ d .Proof. This proof is similar to Lemma 6.4, with more complicated combinatorics of the boundaryfaces due to the boundary of B SFT described in Assumption 6.3, and presented in different order:We will first make the coherent constructions and then deduce the algebraic consequences.
Coherent ep-groupoids and sections:
To construct coherent representatives S ··· α : X ··· α → W ··· α for α ∈ I + ⊔ I − ⊔ I ι ⊔ I as claimed in (ii) recall that the fiber product construction in Lemma 5.6defines each bundle W α = pr ∗ α W SFT A for α = ( p − , p + , A ) ∈ I as the pullback of a strong bundle P A : W SFT A → X SFT A under the natural projection of ep-groupoidspr p − ,p + ,A : X p − ,p + ,A = M ( p − , M ) ev +0 × ev + X SFT A ev − × ev − M ( M, p + ) −→ X SFT A . Here ev ± : M ( . . . ) → C ± × M, τ (0 , ev( τ )) arise from Morse evaluation (8). The ep-groupoid X SFT A ⊂ e X SFT A is a full subcategory – determined by the open subset B + , − SFT ( A ) = (ev + ) − ( C + × M ) ∩ (ev − ) − ( C − × M ) ⊂ B SFT ( A ) – of an ep-groupoid e X SFT A from Assumption 5.5 that represents B SFT ( A )and thus contains the compactified SFT neck stretching moduli space M SFT ( A ) = | ( S SFT A ) − (0) | as zero set of a sc-Fredholm section S SFT A : e X SFT A → f W SFT A . We will work with both groupoids:Multisection perturbations are constructed over e X SFT A since we need a compact zero set to specifythe admissibility that guarantees preservation of compactness under perturbations – both for S SFT A and its fiber product restrictions S α . On the other hand, | e X SFT A | = B SFT ( A ) has more complicatedboundary than B + , − SFT ( A ) – due to the distribution of marked points into building levels – and does notsupport a sc ∞ evaluation map. Thus we discuss coherence only over subgroupoids X SFT A ⊂ e X SFT A withthe boundary stratification of B + , − SFT ( A ), and which support sc ∞ functors ev ± : X SFT A → C ± × M representing the evaluation maps (29). Here we may even use subgroupoids X SFT A representing asmaller open subset (ev + ) − ( D + r × M ) ∩ (ev − ) − ( D − r × M ) ⊂ B SFT ( A ) of preimages of the disks D ± r := { z ∈ C ± | | z | < r } ⊂ C ± , which contain the standard marked points z ± ∼ = 0 ∈ C ± . Thepolyfold structure on the fiber products X α in Lemma 5.6 is independent of the choice of openneighbourhood in B SFT ( A ) of the subset satisfying the fiber product condition. After obtaining thesubgroupoid X SFT A ⊂ e X SFT A from such an open subset, we obtain the bundle W SFT A = f W SFT A | X SFT A andsection S SFT A | X SFT A : X SFT A → W SFT A by restriction. Finally, each section S α = S SFT A ◦ pr α is inducedby the above projection pr α : X α → X SFT A ⊂ e X SFT A . These disks should not be confused with the closed disks D ± in the construction of CP R , as e.g. D + ⊂ C + ∼ =( D + ⊔ [ − R, × S ) / ∼ R is a precompact subset of the first hemisphere in CP R ∼ = C + ∪ S ∪ C − for any R ≥ POLYFOLD PROOF OF THE ARNOLD CONJECTURE 29
Next, restriction to the boundary faces given in Assumption 6.3 (i) induces representatives S GW A : e X GW A → f W GW A resp. S ± γ,A ± : e X ± γ,A ± → f W ± γ,A ± of the sections σ GW : B GW ( A ) → E GW ( A ) resp. σ SFT : B ± SFT ( γ ; A ± ) → E ± SFT ( γ ; A ± ) from Assumption 4.3 resp. 5.5. Moreover, the boundary of theopen subset (ev + × ev − ) − ( D + r × M × D − r × M ) for 0 < r ≤ ∞ (with D ±∞ := C ± ) yields subgroupoids X GW A ⊂ e X GW A representing (ev + × ev − ) − ( D + r × M × D − r × M ) ⊂ B GW ( A ) resp. X ± γ,A ± ⊂ e X ± γ,A ± representing (ev ± ) − ( D ± r × M ) ⊂ B ± SFT ( A ), along with restricted sections S GW A : X GW A → W GW A = f W GW A | X GW A resp. S ± γ,A ± : X ± γ,A ± → W ± γ,A ± = f W ± γ,A ± | X ± γ,A ± . Then the evaluation maps restrict to sc ∞ functors ev ± : X GW A → D ± r × M resp. ev ± : X ± γ,A → D ± r × M , which yield – again independent of r > B ± ( α ) in Lemma 4.5, and of B ι ( α ) in Lemma 5.6.Now the identification of the top boundary strata ∂ X ∞ p − ,p + ,A will proceed similar to the proofof Lemma 6.1 with B GW ( A ) replaced by B + , − SFT ( A ), apart from the fact that the SFT polyfold hasboundary. This boundary is identified in Assumption 6.3 (ii) as(33) ∂ X SFT A ∼ = X GW A ⊔ F γ ∈P ( H ) A − + A += A ∂ X + γ,A + × ∂ X − γ,A − . By the fiber product construction [Fi1, Cor.7.3] of B ( p − , p + ; A ) in Lemma 5.6, the degeneracyindex satisfies d B ( p − ,p + ; A ) ( τ − , u, t + ) = d M ( p − ,M ) ( τ − ) + d B SFT ( A ( u ) + d M ( M,p + ) ( τ + ). Hence we have d B ( p − ,p + ; A ) ( τ − , u, τ + ) = 1 if and only if the degeneracy index of exactly one of the three arguments τ − , u, τ + is 1 and the other two are 0. This identifies | ∂ X p − ,p + ,A | = ∂ B ( p − , p + ; A ) as in thefirst line of the displayed equation below. Then the subsequent identifications result by comparingthe resulting expressions with the interiors in Lemma 4.5, 5.6. We obtain an identification thatthroughout is to be interpreted on the smooth level (as fiber product constructions drop some none-smooth points) ∂ X p − ,p + ,A ∼ = ∂ M ( p − , M ) ev +0 × ev + ∂ X SFT A ev − × ev − ∂ M ( M, p + ) ⊔ ∂ M ( p − , M ) ev +0 × ev + ∂ X SFT A ev − × ev − ∂ M ( M, p + ) ⊔ ∂ M ( p − , M ) ev +0 × ev + ∂ X SFT A ev − × ev − ∂ M ( M, p + )= M ( p − , M ) ev +0 × ev + X GW A ev − × ev − M ( M, p + ) ⊔ S γ ∈P ( H ) A − + A += A M ( p − , M ) ev +0 × ev + ∂ X + γ,A + × ∂ X − γ,A − ev − × ev − M ( M, p + ) ⊔ S q ∈ Crit( f ) M ( p − , q ) × M ( q, M ) ev +0 × ev + ∂ X SFT A ev − × ev − ∂ M ( M, p + ) ⊔ S q ∈ Crit( f ) M ( p − , M ) ev +0 × ev + ∂ X SFT A ev − × ev − M ( M, q ) × M ( q, p + )= ∂ X ιp − ,p + ,A ⊔ S γ ∈P ( H ) A − + A += A ∂ X + p − ,γ,A + × ∂ X − γ,p + ,A − ⊔ S q ∈ Crit( f ) M ( p − , q ) × ∂ X q,p + ,A ⊔ S q ∈ Crit( f ) ∂ X p − ,q,A × M ( q, p + ) . Here we also used the identification of evaluation maps in Assumption 6.3 (iii)(a). Then compatibilityin (ii) of the oriented section functors S ··· α with the identification of these (smooth levels of) faces F ∞ ⊂ ∂ X ∞ p − ,p + ,A follows from compatibility of pr p − ,p + ,A : X p − ,p + ,A → X SFT A with the projectionspr ± α : X ± α → X ± γ,A ± for α ∈ I ± used in Lemma 4.5 and pr ια : X ια → X GW A used in Lemma 5.6. Moreprecisely, S p − ,p + ,A | F ∞ = pr ∗F S F follows from compatibility of the sections in Assumption 6.3 (iii)and pr p − ,p + ,A | F ∞ = pr ιp − ,p + ,A ◦ pr F for F = ∂ X ιp − ,p + ,A , (pr + p − ,γ,A − × pr − γ,p + ,A + ) ◦ pr F for F = ∂ X + p − ,γ,A + × ∂ X − γ,p + ,A − , pr q,p + ,A ◦ pr F for F = M ( p − , q ) × ∂ X q,p + ,A , pr p − ,q,A ◦ pr F for F = ∂ X p − ,q,A × M ( q, p + ) . Construction of coherent perturbations:
Next, we construct admissible sc + -multisections κ ··· α : W ··· α → Q + for α ∈ I + ∪ I − ∪ I ι ∪ I as claimed in (i), i.e. in general position to the respectivesections S ··· α : X ··· α → : W ··· α , while also coherent as claimed in (iii). The existence of such coherenttransverse perturbations will ultimately be guaranteed by an abstract perturbation theorem for coherent systems of sc-Fredholm sections, as developed in the case of trivial isotropy in [Fi2]. Sincewe have to deal with nontrivial isotropy and the SFT perturbation package [FH1] has only beenoutlined in [FH], we give a detailed construction of the perturbations for our purposes. We proceedas in Lemma 6.4 and construct them all as pullbacks κ ··· α := λ ··· A ◦ (pr ··· α ) ∗ of a collection of sc + -multisections on the SFT resp. Gromov-Witten polyfold bundles – without Morse trajectories – λ = (cid:0) λ + γ,A : f W + γ,A → Q + (cid:1) γ ∈P ( H ) ,A ∈ H ( M ) (cid:0) λ GW A : f W GW A → Q + (cid:1) A ∈ H ( M ) (cid:0) λ − γ,A : f W − γ,A → Q + (cid:1) γ ∈P ( H ) ,A ∈ H ( M ) (cid:0) λ SFT A : f W SFT A → Q + (cid:1) A ∈ H ( M ) . For this to induce a coherent collection of sc + -multisections as required in (iii), (cid:0) κ + p,γ,A := λ + γ,A ◦ (pr + p,γ,A ) ∗ (cid:1) ( p,γ,A ) ∈I + , (cid:0) κ ιp − ,p + ,A := λ GW A ◦ (pr ιp − ,p + ,A ) ∗ (cid:1) ( p − ,p + ,A ) ∈I ι , (cid:0) κ − γ,p,A := λ − γ,A ◦ (pr − γ,p,A ) ∗ (cid:1) ( γ,p,A ) ∈I − , (cid:0) κ p − ,p + ,A := λ SFT A ◦ (pr p − ,p + ,A ) ∗ (cid:1) ( p − ,p + ,A ) ∈I , it suffices to pick λ compatible with respect to the faces of the SFT neck stretching polyfolds X SFT A in (33). More precisely, using the natural identifications of bundles from Assumption 6.3 (iii),we will construct λ coherent in the sense that – for some choice of r > |X ± γ,A | = (ev ± ) − ( D ± r × M ) ⊂ B ± SFT ( γ ; A ) and W ± γ,A = f W ± γ,A | X ± γ,A – we have λ SFT A ( w ) = λ GW A ( w ) ∀ w ∈ f W GW A , (34) λ SFT A (( l γ,A ± ) ∗ ( w + , w − )) = λ + γ,A + ( w + ) · λ − γ,A − ( w − ) ∀ ( w + , w − ) ∈ W + γ,A + × W − γ,A − . (35)So to finish this proof it remains to choose the sc + -multisections λ so that each induced sc + -multisection in the induced coherent collection for ( κ ··· α ) α ∈I + ∪I − ∪I ι ∪I is admissible and in generalposition, while also satisfying the coherence requirements (34), (35) and the requirements on κ ι inthe proofs of Lemma 6.4 and 6.5. The construction of coherent perturbations for the SFT polyfoldsoutlined in [FH] proceeds by first choosing coherent compactness controlling data, i.e. pairs ( N, U )of auxiliary norms on all the bundles and saturated neighbourhoods of the compact zero sets in allthe ep-groupoids e X ± γ,A , e X GW A , e X SFT A (c.f. Definition A.5), which are compatible with the immersionsto boundary faces in (33). Then it constructs the perturbations λ GW A as in Lemma 6.5 and also λ ± γ,A ± to be in general position, admissible w.r.t. the coherent data (2 N, U ), and coherent in the sense thatcontinuous extension of (34)–(35) induces a well defined multisection λ ∂A : f W SFT A | ∂ X SFT A → Q + . Herecoherence of the perturbations on the intersection of faces (see Remark 6.2) is required to guaranteeexistence of scale-smooth extensions of λ ∂A to multisections λ SFT A : f W SFT A → Q + . Coherence of thecompactness controlling pairs guarantees that the multisection λ ∂A over ∂ X SFT A ⊂ e X SFT A satisfiesthe auxiliary norm bounds N ( λ ∂A ) ≤ and support requirements that guarantee compactness forextensions λ SFT A of λ ∂A with N ( λ SFT A ) ≤ λ SFT A using Theorem A.9 to ensure – as in Lemma 6.4 – that theinduced multisections κ ··· α are in general position as well. The latter will automatically be admissiblewith respect to pullback of the pair controlling compactness. In more detail (but without specifyingthe auxiliary norm bounds) the inductive construction of perturbations in [FH1] – simplified to thesubset of SFT moduli spaces considered here – proceeds as follows: Construction of λ GW A and κ ι : Since the Gromov-Witten ep-groupoids e X GW A are boundaryless byAssumption 5.5 (iii), the sc + -multisections λ GW A can be chosen independently of all other multisec-tions. So we construct λ GW A as in the proofs of Lemma 6.4 and 6.5, to ensure that the conclusions inthese lemmas hold, as required by (i). This prescribes (34) on the boundary face e X GW A ⊂ ∂ e X SFT A .Moreover, recall that λ GW A is obtained by applying Theorem A.9 to the sc-Fredholm section func-tors S GW A , the sc ∞ submersion ev + × ev − : e X GW A → CP × M × CP × M , and the collection ofCartesian products of stable and unstable manifolds { z +0 } × W − p − × { z − } × W + p + . As in the proofof Lemma 6.4 this ensures that the pullbacks κ ι = ( κ ια = λ GW A ◦ (pr ια ) ∗ ) α ∈I ι are in general position.Moreover, these pullbacks are admissible w.r.t. the pairs controlling compactness on W ια → X ια that result by pullback from the coherent compactness controlling pair on f W GW A → e X GW A , which isconstructed in a preliminary step by [FH, Lecture 13]. POLYFOLD PROOF OF THE ARNOLD CONJECTURE 31
Coherence for λ ± γ,A : The next step is to construct sc + -multisections λ ± γ,A : f W ± γ,A → Q + over theSFT ep-groupoids e X ± γ,A of planes with limit orbit γ ∈ P ( H ) from Assumption 4.3, which then inducethe perturbations κ ± for the P SS/SSP moduli spaces. These constructions are independent of thechoice of λ GW A since the corresponding boundary faces of e X SFT A do not intersect by Assumption 6.3 (ii).However, to enable the subsequent construction of λ SFT A as extension of the boundary values pre-scribed in (34) and (35), we need to make sure that each sc + -multisection ( λ + γ,A + · λ − γ,A − ) ◦ ( l γ,A ± ) − ∗ is well defined on the (open subset of) face F γ,A ± := l γ,A ± ( X + γ,A + × X − γ,A − ) ⊂ ∂ e X SFT A and coincideswith the other sc + -multisections ( λ + γ ′ ,A ′ + · λ − γ ′ ,A ′− ) ◦ ( l γ ′ ,A ′± ) − ∗ on their intersection F γ,A ± ∩ F γ ′ ,A ′± .Then this yields a well defined sc + -multisection on S F γ,A ± = ∂ X SFT A ⊂ ∂ e X SFT A . To describe theseintersections we note that [FH] constructs the ep-groupoids e X ± γ,A ± with coherent boundaries – involv-ing ep-groupoids ( X Fl γ − ,γ + ,B ) γ ± ∈P ( H ) ,B ∈ H ( M ) which contain the moduli spaces of Floer trajectoriesbetween periodic orbits γ ± , as well as further ep-groupoids for Floer trajectories carrying a markedpoint. We will avoid dealing with the latter by specifying values r < ∞ when pulling back perturba-tions from the ep-groupoids X ± γ,A ⊂ e X ± γ,A given by |X ± γ,A | = (ev ± ) − ( D ± r × M ) ⊂ B ± SFT ( γ ; A ), as thiswill prevent the appearance of marked Floer trajectories even in the closure. For any fixed value0 < r ≤ ∞ , the j -th boundary stratum is given by j Floer trajectories breaking off, ∂ j X + γ,A = F γ ,...,γj = γ ∈P ( H ) A ++ B ... + Bj = A ∂ X + γ ,A + × ∂ X Fl γ ,γ ,B × . . . × ∂ X Fl γ j − ,γ j ,B j , (36) ∂ k − j X − γ,A = F γ = γj,...,γk ∈P ( H ) Bj +1+ ... + Bk + A − = A ∂ X Fl γ j ,γ j +1 ,B j +1 × . . . × ∂ X Fl γ k − ,γ k ,B k × ∂ X − γ k ,A − . Now, for example, ∂ X + γ ,A + × ∂ X Fl γ ,γ ,B × ∂ X − γ ,A − is both a subset of ∂ X + γ ,A + × ∂ X − γ ,A − + B ⊂ ∂ (cid:0) X + γ ,A + ×X − γ ,A − + B (cid:1) and of ∂ X + γ ,A + + B × ∂ X − γ ,A − ⊂ ∂ (cid:0) X + γ ,A + + B ×X − γ ,A − (cid:1) , and the embeddings l γ ,A ± and λ γ ,A ± for the two splittings A + +( A − + B ) = A + A − = A = A + A − = ( A + + B )+ A − coincide under this identification. Generally, the boundary of the Floer ep-groupoids is given bybroken trajectories, and this yields a disjoint cover of ∂ R = ∞ X SFT A ⊂ ∂ X SFT A , ∂ R = ∞ X SFT A = G γ ,...,γk ∈P ( H ) A ++ B ... + Bk + A − = A l γ,A ± ,B (cid:0) ∂ X + γ ,A + × ∂ X Fl γ ,γ ,B × . . . × ∂ X Fl γ k − ,γ k ,B k × ∂ X − γ k ,A − (cid:1) , in which the embeddings l γ,A ± ,B coincide with each of the embeddings l γ j ,A j ± for 0 ≤ j ≤ k and A j + = A + + P i ≤ j B i , A j − = A − + P i>j B i – when restricted to the subsets ∂ X + γ ,A + × ∂ X Fl γ ,γ ,B × . . . × ∂ X Fl γ k − ,γ k ,B k × ∂ X − γ k ,A − ⊂ ∂ j X + γ j ,A j + × ∂ k − j X − γ j ,A j − . Now on these subsets we require coherence λ + γ j ,A j + · λ − γ j ,A j − = λ + γ j ′ ,A j ′ + · λ − γ j ′ ,A j ′− for all 0 ≤ j = j ′ ≤ k ,as this is equivalent to (35) being well defined on im l γ,A ± ,B = T kj =0 F γ j ,A j ± . This will be achievedby constructing the sc + -multisections ( λ ± γ,A ± ) to have product structure on the boundary – wherethe bundles P γ,A : W ± γ,A → X ± γ,A are restricted to various faces of ∂ X ± γ,A – λ + γ j ,A j + (cid:12)(cid:12) P − γj,Aj + (cid:0) X + γ ,A + ×X Fl γ ,γ ,B × ... ×X Fl γj − ,γj,Bj (cid:1) = λ + γ ,A + · λ Fl γ ,γ ,B · . . . · λ Fl γ j − ,γ j ,B j , (37) λ − γ j ,A j − (cid:12)(cid:12) P − γj,Aj − (cid:0) X Fl γj,γj +1 ,Bj +1 × ... ×X Fl γk − ,γk,Bk ×X − γk,A − (cid:1) = λ Fl γ j ,γ j +1 ,B j +1 · . . . · λ Fl γ k − ,γ k ,B k · λ − γ k ,A − , for a collection of sc + -multisections λ Fl γ − ,γ + ,B : W Fl γ − ,γ + ,B → Q over the Floer ep-groupoids X Fl γ − ,γ + ,B .While this guarantees coherence on each overlap of embeddings im l γ,A ± ,B ⊂ F γ j ,A j ± ∩ F γ j ′ ,A j ′± , λ + γ j ,A j + · λ − γ j ,A j − = λ + γ ,A + · λ Fl γ ,γ ,B · . . . · λ Fl γ k − ,γ k ,B j · λ − γ k ,A − = λ + γ j ′ ,A j ′ + · λ − γ j ′ ,A j ′− , we are now faced with the challenge of satisfying the coherence conditions in (37). These condi-tions uniquely determine the boundary restrictions λ ± γ,A ± (cid:12)(cid:12) P − γ,A ± ( ∂ X ± γ,A ± ) via the identification of the boundaries with Cartesian products of interiors in (36). Thus (36) on Cartesian products involvingboundary strata poses coherence conditions on the choice of λ Fl β for β ∈ I Fl := P ( H ) ×P ( H ) × H ( M ). Construction of λ Fl γ − ,γ + ,B : To achieve the coherence in (37), [FH] first constructs the sc + -multisections ( λ Fl β ) β ∈I Fl by iteration over the maximal degeneracy k β := max { k ∈ N | ( S Fl β ) − (0) ∩ ∂ k X Fl β = ∅} of unperturbed solutions (which is finite by Gromov compactness): For k β = −∞ thesection S Fl β has no zeros so is already transverse, so that λ Fl β can be chosen as the trivial perturba-tion. (The trivial multivalued section functor λ : W → Q + is given by λ (0) = 1 and λ ( w = 0) = 0.)For k β = 0 the section S Fl β has all zeros in the interior, so that λ Fl β can be chosen admissible andtrivial on the boundary – by applying Corollary 4.6 (i) with a neighbourhood of the unperturbedzero set in the interior, | ( S Fl β ) − (0) | ⊂ V β ⊂ | ∂ X Fl β | . For β = ( γ − , γ + , B ) ∈ I Fl with k β ≥ λ Fl β | P − β ( ∂ X Fl β ) to the boundary ∂ X Fl β = S γ − = γ ,γ ,...,γ k − ,γ k = γ + ,B = B + ... + B k ∂ X Fl γ ,γ ,B × . . . × ∂ X Fl γ k − ,γ k ,B k is prescribed by the previous iteration steps λ Fl β (cid:12)(cid:12) P − β ( X Fl γ ,γ ,B ... ×X Fl γk − ,γk,Bk ) := λ Fl γ ,γ ,B . . . · λ Fl γ k − ,γ k ,B k on all boundary faces that contain unperturbed solutions in their closure. In-deed, existence of a solution in X Fl γ ,γ ,B × . . . ×X Fl γ k − ,γ k ,B k implies k γ i − ,γ i ,B i ≥ i = 1 , . . . , k , andthe Cartesian product of solutions of maximal degeneracy yields 1+ k γ ,γ ,B + . . . + k γ k − ,γ k ,B k ≤ k β .Thus these prescriptions are made for 0 ≤ k γ i − ,γ i ,B i ≤ k β −
1, and on boundary faces with no so-lutions in their closure we prescribe the trivial perturbation throughout. This yields a well definedsc + -multisection λ Fl β | P − β ( ∂ X Fl β ) by coherence in the prior iteration steps, so that λ Fl β can be con-structed by applying the extension result [HWZ, Thm.15.5] which provides general position andadmissibility with respect to a pair controlling compactness that extends the pair which was chosenon the boundary in prior iteration steps. Construction of λ ± γ,A and κ ± : With the Floer perturbations in place, [FH] next constructs thecollections of sc + -multisections ( λ ± γ,A ) γ ∈P ( H ) ,A ∈ H ( M ) to satisfy (37) by iteration over degeneracy k γ,A := max { k ∈ N | ( S ± γ,A ) − (0) ∩ ∂ k e X ± γ,A = ∅} . For k γ,A = −∞ one takes λ ± γ,A to be trivial. For k γ,A = 0 one applies Theorem A.9 to the sc-Fredholm section functor S ± γ,A : e X ± γ,A → f W ± γ,A , the mapev ± : e X ± γ,A → C ± × M , and the collection of stable resp. unstable manifolds { } × W ± p for all criticalpoints p ∈ Crit( f ). These satisfy the assumptions as the zero set | ( S ± γ,A ) − (0) | is compact and thepreimages (ev ± ) − ( { } × W ± p ) lie within the open subset X ± γ,A ⊂ e X ± γ,A on which ev ± restricts to asc ∞ submersion ev ± : X ± γ,A → C ± × M . We can moreover prescribe λ ± γ,A | P − γ,A ( ∂ e X ± γ,A ) to be trivial,since in the absence of solutions the trivial perturbation is in general position. Then Theorem A.9provides λ ± γ,A that is supported in the interior and transverse to each submanifold { } × W ± p in thesense that these submanifolds are transverse to the evaluation from the perturbed zero set(38) ev ± : (cid:12)(cid:12) { x ∈ X ± γ,A | λ ± γ,A ( S ± γ,A ( x )) > } (cid:12)(cid:12) → C ± × M. Now suppose that admissible λ ± γ ′ ,A ′ in general position have been constructed for k γ ′ ,A ′ ≤ k ∈ N ,and satisfy both the transversality in (38) and the coherence condition (37) over the ep-groupoids |X ± γ ′ ,A ′ | = (ev ± ) − ( D ± r k × M ) with r k := 2 + 2 − k . Then for k γ,A = k + 1 we will construct λ ± γ,A to sat-isfy (37) over (ev ± ) − ( D ± r k +1 × M ) by first noting that the previous iteration – and requiring trivialityon boundary faces without solutions – determines a well defined sc + -multisection λ ± γ,A | P − γ,A ( ∂ X ± γ,A ) over the r = r k boundary ∂ X ± γ,A ≃ S γ ′ ,A = A ± + B ∂ X ± γ ′ ,A ± × X Fl γ ′ ,γ,B . For faces (w.r.t. ∂ X ± γ,A ) withsolutions it is given by λ ± γ,A (cid:12)(cid:12) P − γ,A ( X ± γ ′ ,A ± ×X Fl γ ′ ,γ,B ) = λ ± γ ′ ,A ± × λ Fl γ ′ ,γ,B where k γ,A ≥ k γ ′ ,A ± + k γ ′ ,γ,B .This is well defined at ( x ± , x, x ′ ) ∈ ∂ X ± γ ′ ,A ± × X Fl γ ′ ,γ ′′ ,B ′ × X Fl γ ′′ ,γ,B − B ′ , which appears both as( x ± , ( x, x ′ )) ∈ ∂ X ± γ ′ ,A ± × ∂ X Fl γ ′ ,γ,B and (( x ± , x ) , x ′ ) ∈ ∂ X ± γ ′′ ,A ± + B ′ × X Fl γ ′′ ,γ,B − B ′ , by the coherenceof the Floer multisections and the prior iteration: For vectors in the respective fibers ( w ± , w, w ′ ) ∈ POLYFOLD PROOF OF THE ARNOLD CONJECTURE 33 P − γ ′ ,A ± ( x ± ) × P − γ ′ ,γ ′′ ,B ′ ( x ) × P − γ ′′ ,γ,B − B ′ ( x ′ ) we have λ ± γ ′ ,A ± ( w ± ) · λ Fl γ ′ ,γ,B ( w, w ′ ) = λ ± γ ′ ,A ± ( w ± ) · λ Fl γ ′ ,γ ′′ ,B ′ ( w ) · λ Fl γ ′′ ,γ,B − B ′ ( w ′ )= λ ± γ ′′ ,A ± + B ′ ( w ± , w ) · λ Fl γ ′′ ,γ,B − B ′ ( w ′ ) . Moreover, ev ± : |{ x ∈ ∂ X ± γ,A | λ ± γ,A ( S ± γ,A ( x )) > }| → C ± × M is transverse to the submanifolds { } × W ± p . However, this defines an admissible sc + -multisection in general position only over theopen subset of the boundary ∂ X ± γ,A = (ev ± ) − ( D ± r k × M ) ∩ ∂ e X ± γ,A . We multiply the given databy a scale-smooth cutoff function – guaranteed by the existence of partitions of unity for the opencover | e X ± γ,A | = (ev ± ) − ( D ± r k × M ) ∪ (ev ± ) − (( C ± rD ± r k + 12 ) × M ); see Remark A.6 – to obtain anadmissible sc + -multisection λ ∂γ,A : f W ± γ,A | ∂ e X ± γ,A → Q + which coincides with the prescribed data –thus in general position and with evaluation transverse to each { } × W ± p – over the closed subset(ev ± ) − ( D ± r k +1 × M ) ∩ ∂ e X ± γ,A . Then λ ± γ,A : f W ± γ,A → Q + is constructed with these given boundaryvalues using Theorem A.9 to achieve not just general position but also transversality as in (38).By admissibility of the prior iteration and coherence of the pairs controlling compactness, λ ± γ,A canmoreover be chosen admissible.As required in the coherence discussion, this determines right hand sides of (35) which agree onoverlaps of different immersions l γ,A ± ( X + γ,A + × X − γ,A − ) for r = 2. Thus it constructs a well definedsc + -multisection on ∂ R = ∞ X SFT A = S l γ,A ± ( X + γ,A + × X − γ,A − ) ⊂ ∂ e X SFT A that is admissible and hasevaluations transverse to the submanifolds { } × W − p − × { } × W + p + for all pairs p − , p + ∈ Crit( f ).Moreover, for α ∈ I ± we obtain a pair controlling compactness by pullback of the coherentpairs constructed in [FH] on the bundles W ± γ,A . Then the pullback multisections κ ± = ( κ ± α = λ ± γ,A ◦ (pr ια ) ∗ ) α ∈I ± are sc + , admissible w.r.t. the pullback pair, and in general position by thearguments in the proof of Lemma 6.4. Construction of λ SFT A and κ : The above constructions determine the right hand sides in the coher-ence requirements λ SFT A | P − A ( e X GW A ) = λ GW A over e X GW A ⊂ e X SFT A in (34), as well as λ SFT A | P − A ( F γ,A ± (2)) =( λ + γ,A + · λ − γ,A − ) ◦ ( l γ,A ± ) − ∗ on S γ ∈P ( H ) ,A − + A + = A F γ,A ± (2) ⊂ ∂ e X SFT A in (35), where we denote by F γ,A ± ( r ) := l γ,A ± ( X + γ,A + × X − γ,A − ) ⊂ ∂ e X SFT A the image of the immersion l γ,A ± on the ep-groupoidsrepresenting |X ± γ,A ± | = (ev ± ) − ( D ± r × M ) ⊂ B ± SFT ( γ ; A ). By admissibility in the prior steps andexistence of scale-smooth partitions of unity (see Remark A.6) these induce for every A ∈ H ( M )an admissible sc + -multisection λ ∂A : f W ± A | ∂ e X A → Q + which coincides with the prescribed data over e X GW A ⊔ S γ,A ± F γ,A ± (1) ⊂ ∂ e X SFT A . Thus on this closed subset we have general position and transver-sality of the evaluation map(39) ev + × ev − : (cid:12)(cid:12) { x ∈ ∂ X SFT A | λ ∂A ( S SFT A ( x )) > } (cid:12)(cid:12) → C + × M × C − × M to { } × W − p − × { } × W + p + for any pair of critical points p − , p + ∈ Crit( f ). Then the admissiblesc + -multisection λ SFT A : f W SFT A → Q + is constructed with these given boundary values – and auxiliarynorm and support prescribed by the coherent pairs controlling compactness – using Theorem A.9to achieve general position on all of e X SFT A and extend transversality of the evaluation ev + × ev − to { } × W − p − × { } × W + p + to the entire perturbed zero set |{ x ∈ X SFT A | λ SFT A ( S SFT A ( x )) > }| , where |X SFT A | = (ev + ) − ( D +1 × M ) ∩ (ev − ) − ( D − × M ) ⊂ B SFT ( A ).As in the proof of Lemma 6.4, the transversality of the evaluation maps implies that the pullbacks κ = ( κ α = λ A ◦ (pr ια ) ∗ ) α ∈I are in general position. They are also admissible with respect to thepullback of pairs controlling compactness. This finishes the construction of the sc + -multisectionsclaimed in (i) with the boundary restrictions required in (iii). Proof of identity:
By Λ-linearity of all maps involved, it suffices to fix two generators p − , p + ∈ Crit( f ) of CM and check that ι κ ι h p − i and ( SSP κ − ◦ P SS κ + ) h p − i + ( − | p − | (d ◦ h κ ) h p − i + ( − | p − | ( h κ ◦ d ) h p − i have the same coefficient in Λ on h p + i . That is, we claim X A ∈ H ( M ) Iι ( p − ,p +; A )=0 Z κ ι ( p − , p + ; A ) · T ω ( A ) = X γ ∈P ( H ) ,A − ,A + ∈ H ( M ) I ( p − ,γ ; A +)= I ( γ,p +; A − )=0 Z κ + ( p − , γ ; A + ) Z κ − ( γ, p + ; A − ) · T ω ( A − )+ ω ( A + ) + ( − | p − | X q ∈ Crit( f ) ,A ∈ H ( M ) I ( p − ,q ; A )= | q |−| p + |− Z κ ( p − , q ; A ) M ( q, p + ) · T ω ( A ) + ( − | p − | X q ∈ Crit( f ) ,A ∈ H ( M ) | p −|−| q |− I ( q,p +; A )=0 M ( p − , q ) Z κ ( q, p + ; A ) · T ω ( A ) . Here the sums on the right hand side are over counts of pairs of moduli spaces of index 0. From § M ( q, p + ) = ∅ for | q | − | p + | − < M ( p − , q ) = ∅ for | p − | − | q | − <
0, and generalposition of the sc + -multisections κ ··· as in Corollary 4.6 (iii) implies Z κ ··· ( . . . ) = ∅ for I ( . . . ) < I ( p − , γ ; A + ) + I ( γ, p + ; A − ) = I ι ( p − , p + ; A − + A + ) = I ( p − , p + ; A − + A + ) − , I ( p − , q ; A ) + | q | − | p + | − I ( p − , p + ; A ) − , | p − | − | q | − I ( q, p + ; A ) = I ( p − , p + ; A ) − . So all sums can be rewritten with the index condition I ( p − , p + ; A ) = 1 for A = A − + A + ∈ H ( M ),and since the symplectic area is additive ω ( A − ) + ω ( A + ) = ω ( A − + A + ), it suffices to show thefollowing identity for each α = ( p − , p + ; A ) ∈ I with I ( p − , p + ; A ) = 1,( − | p − | Z κ ι ( p − , p + ; A ) = ( − | p − | X γ ∈P ( H ) A − + A += A Z κ + ( p − , γ ; A + ) Z κ − ( γ, p + ; A − )+ X q ∈ Crit( f ) Z κ ( p − , q ; A ) M ( q, p + ) + X q ∈ Crit( f ) M ( p − , q ) Z κ ( q, p + ; A ) . (40)This identity will follow from Corollary 4.6 (v) applied to the weighted branched 1-dimensionalorbifold Z κ ( α ) that arises from an admissible sc + -multisection κ α : W α → Q + . The boundary ∂Z κ ( α ) is given by the intersection with the top boundary stratum ∂ B ( α ) ∩ V α = | ∂ X α | , and willbe determined here – with orientations computed in (41) below. ∂Z κ ( α ) = Z κ ( α ) ∩ | ∂ X α | = Z κ ( α ) ∩ | ∂ X ιp − ,p + ; A | ⊔ [ γ ∈P ( H ) A − + A += A Z κ ( α ) ∩ | ∂ X + p − ,γ ; A + × ∂ X − γ,p + ; A − |⊔ [ q ∈ Crit( f ) Z κ ( α ) ∩ (cid:0) M ( p − , q ) × | ∂ X q,p + ; A | (cid:1) ⊔ [ q ∈ Crit( f ) Z κ ( α ) ∩ (cid:0) | ∂ X p − ,q ; A | × M ( q, p + ) (cid:1) = Z κ ι ( p − , p + ; A ) ⊔ [ γ ∈P ( H ) ,A = A − + A + Z κ + ( p − , γ ; A + ) × Z κ − ( γ, p + ; A − ) ⊔ [ q ∈ Crit( f ) M ( p − , q ) × Z κ ( q, p + ; A ) ⊔ [ q ∈ Crit( f ) Z κ ( p − , q ; A ) × M ( q, p + ) . Here the second identity uses coherence of the ep-groupoid as in (32). The third identity follows fromthe coherence of sections S ··· α and sc + multisections κ ··· α as stated in (ii), (iii), together with the factfrom Corollary 4.6 (iv) that perturbed zero sets Z κ ··· ( α ) ⊂ | ∂ X ··· α | are contained in the interior of thepolyfolds when the Fredholm index is 0. For the second summand we moreover use Lemma A.7 whichensures that each restriction κ α | P − α ( F ) to a face F = ∂ X + p − ,γ ; A + × ∂ X − γ,p + ; A − ⊂ ∂ X p − ,p + ; A , givenby κ + p − ,γ ; A + · κ − γ,p + ; A − , is in general position to the section S + p − ,γ ; A + × S − γ,p + ; A − . Then its perturbedzero set Z κ ι ( p − , p + ; A ) ∩ |F| is contained in the interior ∂ |X + p − ,γ ; A + × X − γ,p + ; A − | = | ∂ X + p − ,γ ; A + × POLYFOLD PROOF OF THE ARNOLD CONJECTURE 35 ∂ X − γ,p + ; A − | as the complement of the pairs of points ( x + , x − ) with0 = κ p − ,p + ; A ( S p − ,p + ; A ( x + , x − )) = ( κ + p − ,γ ; A + · κ − γ,p + ; A − ) (cid:0) ( S + p − ,γ ; A + × S − γ,p + ; A − )( x + , x − ) (cid:1) = κ + p − ,γ ; A + ( S + p − ,γ ; A + ( x + )) · κ − γ,p + ; A − ( S − γ,p + ; A − ( x − )) . Since a product in Q + = Q ∩ [0 , ∞ ) is nonzero exactly when both factors are nonzero, this identifiesthe objects of the perturbed zero set of κ p − ,p + ; A with the product of perturbed zero objects for κ ± , (cid:8) ( x + , x − ) ∈ F (cid:12)(cid:12) κ p − ,p + ; A ( S p − ,p + ; A ( x + , x − )) > (cid:9) = (cid:8) x + ∈ X + p − ,γ ; A + (cid:12)(cid:12) κ + p − ,γ ; A + ( S + p − ,γ ; A + ( x + )) > (cid:9) × (cid:8) x − ∈ X − γ,p + ; A − (cid:12)(cid:12) κ − γ,p + ; A − ( S − γ,p + ; A − ( x − )) > (cid:9) . And the realization of this set is precisely Z κ + ( p − , γ ; A + ) × Z κ − ( γ, p + ; A − ), as claimed above. Computation of orientations:
To prove the identity (40) it remains to compute the effect ofthe orientations in Remark 5.8 on the algebraic identity in Corollary 4.6 (v) that arises from theboundary ∂Z κ ( α ) of the 1-dimensional weighted branched orbifolds arising from regularization ofthe moduli spaces with index I ( α ) = I ( p − , p + ; A ) = 1. Here Z λ SFT A is of odd dimension with orientedboundary determined by the orientation relations in Assumption 6.3 (iii)(b) and (c) as ∂ Z λ SFT A = Z λ SFT A ∩ ∂ B SFT ( A ) = ( − Z λ GW A ⊔ F γ ∈P ( H ) A − + A += A Z λ + γ,A + × Z λ − γ,A − . Moreover, the index of σ SFT is I ( α ) = | p − | − | p + | + 2 c ( A ) + 1 = 1, so we compute orientations inclose analogy to (31) – while also giving an alternative identification of the boundary components – ∂Z κ ( α ) = ∂ M ( p − , M ) ev × ev Z λ SFT A ev × ev ∂ M ( M, p + ) ⊔ ( − dim M ( p − ,M ) ∂ M ( p − , M ) ev × ev ∂ Z λ SFT A ev × ev ∂ M ( M, p + ) ⊔ ( − dim M ( p − ,M )+1 ∂ M ( p − , M ) ev × ev ∂ Z λ SFT A ev × ev ∂ M ( M, p + )= (cid:0) F q ∈ Crit f M ( p − , q ) × M ( q, M ) (cid:1) ev × ev Z λ SFT A ev × ev M ( M, p + )(41) ⊔ ( − | p − | + | p + | M ( p − , M ) ev × ev Z λ SFT A ev × ev (cid:0) F q ∈ Crit f M ( M, q ) × M ( q, p + ) (cid:1) ⊔ ( − | p − | M ( p − , M ) ev × ev (cid:0) F γ ∈P ( H ) ,A = A − + A + Z λ + γ,A + × Z λ − γ,A − (cid:1) ev × ev M ( M, p + ) ⊔ ( − | p − | +1 M ( p − , M ) ev × ev Z λ GW A ev × ev M ( M, p + )= F q ∈ Crit f M ( p − , q ) × Z κ ( q, p + ; A ) ⊔ F q ∈ Crit f Z κ ( p − , q ; A ) × M ( q, p + ) ⊔ ( − | p − | F γ ∈P ( H ) ,A = A − + A + Z κ + ( p − , γ ; A + ) × Z κ − ( γ, p + ; A − ) ⊔ ( − | p − | +1 Z κ ι ( p − , p + ; A ) . This computation should be understood in a neighbourhood of a solution, so in particular withscale-smooth evaluation maps to C ± × M . Based on this, Corollary 4.6 (v) implies – as claimed –0 = h κ (d h p − i ) + d( h κ h p − i ) + ( − | p − | SSP κ − (cid:0) P SS κ + h p − i (cid:1) − ( − | p − | ι κ ι h p − i = (cid:0) h ◦ d + d ◦ h + SSP ◦ P SS − ι (cid:1) h p − i . (cid:3) Appendix A. Summary of Polyfold Theory
This section gives an overview of the main notions of polyfold theory that are used in this paper.The following language is used to describe settings with trivial isotropy. Remark A.1. (i) An
M-polyfold without boundary is analogous to the notion of a Banachmanifold: While the latter are locally homeomorphic to open subsets of a Banach space, an M-polyfold is locally homeomorphic to the image O = im ρ of a retract ρ : U → U of an open subset U ⊂ E of a Banach space E . While ρ is generally not classically differentiable, it is required tobe scale-smooth (sc ∞ ) with respect to a scale structure on E , which is indicated by E . Trivial isotropy would be guaranteed in our settings by an almost complex structure J for which there are nononconstant J -holomorphic spheres. (i’) An M-polyfold , as defined in [HWZ, Def.2.8], is a paracompact Hausdorff space X togetherwith an atlas of charts φ ι : U ι → O ι ⊂ [0 , ∞ ) s ι × E ι (i.e. homeomorphisms between open sets U ι ⊂ X and sc-retracts O ι such that ∪ ι U ι = X ), whose transition maps are sc-smooth.For k ∈ N the k -th boundary stratum ∂ k X is the set of all x ∈ X of degeneracy index d ( x ) = k given by the number of components equal to 0 for the point in a chart φ ι ( x ) ∈ [0 , ∞ ) s ι × E ι . Inparticular, ∂ X is the interior of X .(ii) A strong bundle over an M-polyfold X , as defined in [HWZ, Def.2.26], is a sc-smooth sur-jection P : W → X with linear structures on each fiber W x = P − ( x ) for x ∈ X , and anequivalence class of compatible strong bundle charts, which in particular encode a sc-smoothsubbundle W ⊃ W → X whose fiber inclusions W x ֒ → W x are compact and dense.(iii) The notion of sc-Fredholm for a scale smooth section S : X → W of a strong bundle in [HWZ,Def.3.8] encodes elliptic regularity and a nonlinear contraction property [HWZ, Def.3.6,3.7]. Thelatter is a stronger condition than the classical notion of linearizations being Fredholm operators,and is crucial to ensure an implicit function theorem; see [FWZ].A more detailed survey of these trivial isotropy notions can be found in [FFGW]. Then thegeneralization to nontrivial isotropy is directly analogous to the notion of smooth sections of orbi-bundles, in which orbifolds are realizations of ´etale proper groupoids [Mo]. Remark A.2.
A sc-Fredholm section σ : B → E of a strong polyfold bundle as introduced in[HWZ, Def.16.16,16.40] is a map between topological spaces together with an equivalence class ofsc-Fredholm section functors s : X → W of strong bundles W over ep-groupoids X , whose realization | s | : |X | → |W| together with homeomorphisms |X | := Obj X / Mor X ∼ = B and |W| ∼ = E induces σ .To summarize these notions we use conventions of [HWZ] in denoting object and morphism spacesas Obj X = X and Mor X = X . These will be equipped with M-polyfold structures, so that the k -thboundary stratum of a polyfold B ∼ = |X | is given as ∂ k B ∼ = ∂ k X/ X ⊂ |X | for all k ∈ N .(i) An ep-groupoid as in [HWZ, Def.7.3] is a groupoid X = ( X, X ) equipped with M-polyfoldstructures on the object and morphism sets such that all structure maps are local sc-diffeomorphismsand every x ∈ X has a neighbourhood V ( x ) such that t : s − (cid:0) cl X ( V ( x )) (cid:1) → X is proper. As in[HWZ, § |X | is paracompact and thus metrizable.(ii) A strong bundle as in [HWZ, Def.8.4] over the ep-groupoid X is a pair ( P, µ ) of a strongbundle P : W → X and a strong bundle map µ : X s × P W → W so that P lifts to a functor P : W → X from an ep-groupoid W = ( W, W ) induced by ( P, µ ). Then P restricts to a functor W → X on the full subcategory whose object space is the sc-smooth subbundle W ⊂ W .(iii) A sc-Fredholm section functor of the strong bundle P : W → X as in [HWZ, Def.8.7] is afunctor S : X → W that is sc-smooth on object and morphism spaces, satisfies
P ◦ S = id X , andsuch that S : X → W is sc-Fredholm on the M-polyfold X .Now a polyfold description of a compact moduli space M is a sc-Fredholm section σ : B → E of astrong polyfold bundle with zero set σ − (0) ∼ = M . The polyfold descriptions used in this paper areobtained as fiber products of existing polyfolds and sc-Fredholm sections over them. This requiresa technical shift in levels described in the following remark, and a notion of submersion below. Remark A.3.
Polyfolds carry a level structure B ∞ ⊂ . . . ⊂ B ⊂ B = B as follows: For anyM-polyfold X , in particular the object space of the ep-groupoid representing B = |X | , a sequence ofdense subsets X ∞ ⊂ . . . ⊂ X ⊂ X = X is induced by the scale structures E ι = ( E ιm ) m ∈ N of thecharts, that is X m = S ι φ − ι ( O ι ∩ R s ι × E ιm ). Then B m := X m / Mor X is well defined since morphismsof X – locally represented by scale-diffeomorphisms – preserve the levels on Obj X = X .The restriction σ | B m of a sc-Fredholm section σ : B → E is again sc-Fredholm with values in E m ,and the choice of such a shift in levels is irrelevant for applications since the zero set σ − (0) ⊂ B ∞ – as well as the perturbed zero set for any admissible perturbation – is always contained in theso-called “smooth part” that is densely contained in each level B ∞ ⊂ B m . The degeneracy index d ( x ) ∈ N in [HWZ, Def.2.13,Thm.2.3] is a priori independent of the choice of chart φ ι onlyfor points in a dense subset X ∞ ⊂ X specified in Remark A.3. With that d ( x ) := max { lim sup d ( x i ) | X ∞ ∋ x i → x } is well defined for all x ∈ X and can also be computed in any fixed chart. POLYFOLD PROOF OF THE ARNOLD CONJECTURE 37
For a finite dimensional manifold or orbifold M – such as the Morse trajectory spaces in § M ∞ = . . . = M = M = M . Definition A.4. [Fi1, Def.5.8]
A sc ∞ functor f : X → M from an ep-groupoid X = ( X, X ) to afinite dimensional manifold M is a submersion if for all x ∈ X ∞ the tangent map D x f : T Rx X → T f ( x ) M is surjective, where T Rx X is the reduced tangent space [HWZ, Def.2.15] .Consider in addition a sc-Fredholm section functor S : X → W . Then the sc ∞ functor f : X → M is S -compatibly submersive if for all x ∈ X ∞ there exists a sc-complement L ⊂ T Rx X of ker(D x f ) ∩ T Rx X and a tame sc-Fredholm chart for S at x [Fi1, Def.5.4] in which the change ofcoordinates ψ : O → [0 , ∞ ) s × R k − s × W that puts S in basic germ form – which by tameness hasthe form ψ ( v, e ) = ( v, ψ ( e )) for ( v, e ) ∈ O ⊂ [0 , ∞ ) s × E and a linear sc-isomorphism ψ – moreoversatisfies ψ ( L ) ⊂ { } k − s × W , where the chart identifies L ⊂ T Rx X ∼ = T R O = { } × E .More generally, given a smooth submanifold N ⊂ M , the sc ∞ functor f is transverse to N if forall x ∈ f − ( N ) ∩ X ∞ we have D x f (T Rx X )+ T f ( x ) N = T f ( x ) M , and f is S -compatibly transverse to N if there exists a sc-complement L of (D x f ) − (T f ( x ) ( N )) ∩ T Rx X satisfying the above condition. The purpose of giving a moduli space a polyfold description is to utilize the perturbation theory forsc-Fredholm sections over polyfolds, which allows to “regularize” the moduli space by associatingto it a well defined cobordism class of weighted branched orbifolds. (For a technical statementsee Corollary 4.6 and the references therein.) Since the ambient space |X | is almost never locallycompact, this requires “admissible perturbations” of the section to preserve compactness of the zeroset. This admissibility is determined by the following data introduced in [HWZ, Def.12.2,15.4].
Definition A.5. A saturated open subset U ⊂ X of an ep-groupoid X = ( X, X ) is an opensubset U ⊂ X with π − ( π ( U )) = U , where π : X → |X | = X / X is the projection to the realization.A pair controlling compactness for a sc-Fredholm section S : X → W of a strong bundle P : W → X consists of an auxiliary norm N : W [1] → [0 , ∞ ) (see [HWZ, Def.12.2] ) and a saturatedopen subset U ⊂ X that contains the zero set S − (0) ⊂ U , such that (cid:12)(cid:12) { x ∈ U | N ( S ( x )) ≤ } (cid:12)(cid:12) ⊂ |X | has compact closure.Given such a pair, a section s : X → W is ( N, U )-admissible if N ( s ( x )) ≤ and supp s ⊂ U . The construction of perturbations moreover requires scale-smooth partitions of unity, which willbe guaranteed by the following standing assumptions.
Remark A.6.
Throughout this paper we assume that the realizations |X | of ep-groupoids areparacompact, and the Banach spaces E in all M-polyfold charts are Hilbert spaces. This guaranteesthe existence of scale-smooth partitions of unity by [HWZ, § § B m as discussed in Remark A.3, we moreover assume that each scale structure E = ( E m ) m ∈ N consists of Hilbert spaces E m . These assumptions hold in applications, such as theones cited [HWZ1, FH]. Then paracompactness and thus existence of scale-smooth partitions ofunity on every level is guaranteed by [HWZ, Prop.7.12].When discussing coherence of perturbations of a system of sc-Fredholm sections, the boundariesare described in terms of Cartesian products of polyfolds, bundles, and sections. So we will makeuse of Cartesian products of multivalued perturbations as follows, to obtain multisections over theboundary as summarized in the subsequent remark. Lemma A.7.
Let S : X → W and S : X → W be sc-Fredholm section of strong bundles P i : W i → X i over ep-groupoids. Then the Cartesian product X × X is naturally an ep-groupoidand ( S × S ) : X × X → W × W is a sc-Fredholm section of the strong bundle P × P .Moreover, if λ i : W i → Q + are sc + -multisections for i = 1 , , then there is a well defined sc + -multisection λ · λ : W × W → Q + given by ( λ · λ )( w , w ) = λ ( w ) · λ ( w ) . If, for i = 1 , , thesections λ i are ( N i , U i ) -admissible for some fixed pair controlling compactness as in Definition A.5,then λ · λ is (max( N , N ) , U ×U ) -admissible. Finally, if λ i is in general position to S i for i = 1 , then λ · λ is in general position to S × S .Proof. A detailed treatment of sc-Fredholmness of the product section S × S can be found in[Fi1, Lemma 7.2]. The remaining statements follow easily from the definitions in [HWZ] (as do thestatements in the first paragraph). Recall in particular from [HWZ, Def.13.4] that a sc + -multisection on a strong bundle P : W → X is a functor λ : W → Q + that is locally of the form λ ( w ) = P { j | w = p j ( P ( w )) } q j , represented bysc + -sections p , . . . , p k : V → P − ( V ) (i.e. sc ∞ sections of W ; see [HWZ, Def.2.27]) and weights q , . . . , q k ∈ Q ∩ [0 , ∞ ) with P j q j = 1. Then for local sections p ij and weights q ij representing λ i for i = 1 ,
2, the multisection λ · λ is locally represented by the sections ( p j , p j ′ ) with weights q j q j ′ ,and all admissibility and general position arguments are made at the level of these local sections.In particular, the ( N i , U i )-admissibility can be phrased as the existence of local representationsby sections with N i ( p ij ( x )) ≤ Z ( S i , p ij ) := { x ∈ V i | ∃ t ∈ [ − ,
1] : S i ( x ) = tp ij ( x ) } ⊂ U i . Then(max( N , N ) , U × U )-admissibility uses the observation { ( x , x ) | ∃ t ∈ [ − ,
1] : ( S , S )( x , x ) = t ( p j ( x ) , p j ′ ( x ) } ⊂ Z ( S , p j ) × Z ( S , p j ′ ) ⊂ U × U . (cid:3) Remark A.8.
Let P : W → X be a strong bundle over a tame ep-groupoid X = ( X, X ). Then forevery x ∈ X ∞ there is a chart φ : U x → O from a locally uniformizing neighbourhood U x ⊂ X of x to a sc-retract O ⊂ [0 , ∞ ) n × E , with φ ( x ) = 0 lying in the intersection of the n local faces F k := φ − ( { ( v, e ) ∈ [0 , ∞ ) n × E | v k = 0 } ) which cover the boundary ∂X ∩ U x = S nk =1 F k .Now a sc + -multisection over the boundary is a functor λ ∂ : P − ( ∂ X ) → Q + whose restriction λ ∂ | P − ( F k ) to each local face is a sc + -multisection of the strong bundle P − ( F k ) → F k . In thepresence of a sc-Fredholm section S : X → W , such a sc + -multisection is in general positionover the boundary if for each intersection of faces F K := T k ∈ K F k ⊂ ∂X the restriction of theperturbed multi-section λ ∂ ◦ S | F K : P − ( F K ) → Q + has surjective linearizations at all solutions. If,moreover, ( N, U ) is a pair controlling compactness, then λ ∂ is ( N, U )-admissible if each restriction λ ∂ | P − ( F k ) is admissible w.r.t. the pair ( N | P − ( F k ) , U ∩ F k ).In our applications, as described in Assumption 6.3, the local faces F k are images of open subsetsof global face immersions l F : F → ∂ X , where each F is a Cartesian product of two polyfolds, and therestriction to the interior l F | ∂ F is an embedding into the top boundary stratum ∂ X . The bundlesover each face are naturally identified with the pullbacks l ∗F W , and then the pushforwards of sc + -multisections λ F : l ∗F W → Q + form a sc + -multisection over the boundary λ ∂ : P − ( S im λ F ) → Q + if they agree on overlaps and self-intersections of the immersions l F , at the boundary ∂ F of thefaces. In this setting, general position of λ ∂ is equivalent to general position of the multisections λ F .The following perturbation theorem allows us to refine the construction of coherent perturbationsin [FH] for the SFT moduli spaces such that moreover the evaluation maps from the perturbedsolution sets are transverse to the unstable and stable manifolds in the symplectic manifold. Thisis a generalization of the polyfold perturbation theorem over ep-groupoids and the extension oftransverse perturbations from the boundary [HWZ, Theorems 15.4,15.5] (with norm bound givenby h ≡ Theorem A.9.
Suppose S : X → W is a sc-Fredholm section functor of a strong bundle P : W → X over a tame ep-groupoid X with compact solution set | S − (0) | ⊂ |X | , and let ( N, U ) be a paircontrolling compactness. Moreover, let e : X → M be a sc -map to a finite dimensional manifold M which has a sc ∞ submersive restriction e | V : V → M on a saturated open set V ⊂ X .Then, for any countable collection of smooth submanifolds ( C i ⊂ M ) i ∈ I with e − (cid:0) ∪ i ∈ I ( C i ) (cid:1) ⊂ V ,there exists an ( N, U ) -admissible sc + -multisection λ : W → Q + so that ( S, λ ) is in general position(see [HWZ, Definition 15.6] ) and the restriction e | Z λ : Z λ → M to the perturbed zero set Z λ = |{ x ∈ X | λ ( S ( x )) > }| is in general position to the submanifolds C i for all i ∈ I . A neighbourhood U x ⊂ X forms a local uniformizer as in [HWZ, Def.7.9] if the morphisms between points in U x are given by a local action of the isotropy group G x . General position to C i requires transversality to C i of each restriction e | Z λ ∩F K to the perturbed solution setwithin an intersection of local faces F K = T k ∈ K F k as defined in Remark A.8, including for F ∅ := Z λ . POLYFOLD PROOF OF THE ARNOLD CONJECTURE 39
Moreover, suppose I is finite and λ ∂ : P − ( ∂ X ) → Q + for some < α < is an ( α N, U ) -admissible structurable sc + -multisection in general position over the boundary such that the restric-tion e | Z ∂ : Z ∂ → M to the perturbed zero set in the boundary Z ∂ := |{ x ∈ ∂ X | λ ∂ ( S ( x )) > }| is in general position to the submanifolds C i for all i ∈ I . Then λ above can be chosen with λ | P − ( ∂ X ) = λ ∂ .Proof. Our proof follows the perturbation procedure of [HWZ, Theorem 15.4], which proves thespecial case when there is no condition on a map e : X → M , i.e. when M = { pt } and C i = { pt } .To obtain the desired transversality of e to the submanifolds C i ⊂ M we will go through theproof and indicate adjustments in three steps: A local stabilization construction, which adds afinite dimensional parameter space to cover the cokernels near a point x ∈ S − (0); a local-to-global argument which combines the local constructions into a global stabilization which covers thecokernels near S − (0); and a global Sard argument which shows that regular values yield transverseperturbations. Within these arguments we need to consider restrictions to any intersection of facesto ensure general position to the boundary, use submersivity of e to achieve transversality to the C i ,and work with multisections due to isotropy. The statement with prescribed boundary values λ ∂ generalizes the extension result [HWZ, Theorem 15.5], which hinges on the fact that general positionover the boundary persists in an open neighbourhood – something that is generally guaranteed onlyfor finitely many transversality conditions; see the end of this proof. The first step in any constructionof perturbations is the existence of local stabilizations which cover the cokernels, as follows. Local stabilization constructions:
For every zero x ∈ S − (0) of the unperturbed sc-Fredholmsection we construct a finite dimensional parameter space R l for l = l x ∈ N and sc + -multisection˜Λ x : R l × W → Q + , ( t, w ) Λ xt ( w )such that Λ x is the trivial multisection, i.e. Λ x (0) = 1, Λ x ( w ) = 0 for w ∈ W x r { } . This multisection˜Λ x is viewed as local perturbation near (0 , x ) of a sc-Fredholm section functor ˜ S x of a bundle ˜ P x ,˜ S x : R l × X → R l × W ˜ P x : R l × W → R l × X ( t, y ) ( t, S ( y )) ( t, w ) ( t, P ( w )) . It is constructed in [HWZ] to be structurable in the sense of [HWZ, Def.13.17], in general positionin the sense that the linearization T ( ˜ S x , ˜Λ x ) (0 , x ) : T R l × T Rx X → W x is surjective and admissiblein the sense that the domain support of ˜Λ x is contained in U and the auxiliary norm is boundedlinearly, N (Λ)( t, y ) ≤ c x | t | for some constant c x . In case x ∈ V ∩ S − (0) we refine this constructionto require surjectivity of the restrictions(42) T ( ˜ S x , ˜Λ x ) (0 , x ) | T R l × K x : T R l × K x → W x , where K x := ker(D x e | T Rx X ) ⊂ T Rx X is the kernel of the linearization D x e : T Rx X → T e ( x ) M restrictedto the reduced tangent space. For that purpose note that e is sc ∞ near x by assumption, so has a welldefined linearization, and since its codomain is finite dimensional, its kernel has finite codimension.Moreover im D x S ⊂ W x has finite codimension by the sc-Fredholm property of S , and the reducedtangent space T Rx X ⊂ T x X has finite codimension by the definition of M-polyfolds with corners.Thus we can find finitely many vectors w , . . . , w l ∈ W x which together with D x S ( K x ) span W x .These vectors are extended to sc + -sections of the form p j ( t, y ) = P t j w j ( y ), multiplied with sc ∞ cutoff functions of sufficiently small support, and pulled back by local isotropy actions to constructthe functor ˜Λ x as in [HWZ, Thm.15.4]. We claim that this yields the following local properties withrespect to the sc ∞ functor ˜ e x : R l × V → M, ( t, y ) e ( y ) . Local stabilization properties:
There exists ǫ x > and a locally uniformizing neighborhood Q ( x ) ⊂ X of x whose closure is contained in U , such that (43) Θ x : { t ∈ R l | | t | < ǫ x } × Q ( x ) → Q + , ( t, y ) Λ xt (cid:0) S ( y ) (cid:1) = ˜Λ x ( ˜ S x ( t, y )) This requires general position of each restriction e | Z λ ∩F k to a local face F k ⊂ ∂ X as defined in Remark A.8. This is shorthand for ˜ S x + p j having surjective linearization for every section p j in a local representation of ˜Λ x with ˜ S x (0 , x ) = 0 = p j (0 , x ), and restricted to the reduced tangent space T Rx X . is a tame ep + -subgroupoid, and for ( t, y ) ∈ supp Θ x = { ( t, y ) | Θ x ( t, y ) > } ⊂ R l × X the reducedlinearizations T R ( ˜ S x , ˜Λ x ) ( t, y ) := T ( ˜ S x , ˜Λ x ) ( t, y ) | T t R l × T Ry X are surjective. Moreover, if x ∈ V then wemay choose Q ( x ) ⊂ V such that for all ( t, y ) ∈ supp Θ x we have surjections D ( t,y ) ˜ e x | N xt,y : N xt,y := ker T R ( ˜ S x , ˜Λ x ) ( t, y ) → T e ( y ) M. In particular, the realization | supp Θ x | is a weighted branched orbifold and ˜ e x induces a submer-sion | supp Θ x | → M in the sense of Definition A.4. Moreover, for all y ∈ S − (0) ∩ U x we have (0 , y ) ∈ supp Θ x so that the reduced linearizations T R ( ˜ S x , ˜Λ x ) (0 , y ) and the restriction to their kernel D (0 ,y ) ˜ e x | N x ,y are surjective. These properties persist for y ∈ S − (0) with | y | ∈ | Q ( x ) | . The structure of supp Θ x and surjectivity of linearizations T R ( ˜ S x , ˜Λ x ) follows from the local implicitfunction theorem [HWZ, Theorems 15.2,15.3]. Then the kernels N xt,y = ker T R ( ˜ S x , ˜Λ x ) ( t, y ) representthe reduced tangent spaces at | ( t, y ) | to the weighted branched orbifold | supp Θ x | . Surjectivityof D (0 ,x ) ˜ e x | N x ,x holds since D (0 ,x ) ˜ e x is surjective by assumption, and the preimage of any givenvector in T e ( x ) M can be adjusted by vectors in ker D (0 ,x ) ˜ e x to lie in N x ,x = ker T R ( ˜ S x , ˜Λ x ) (0 , x ),because T ( ˜ S x , ˜Λ x ) ( t, y ) | ker D (0 ,x ) ˜ e x is surjective by (42). Then ˜ e x restricts to a map | supp Θ x | → M that is classically smooth on each (finite dimensional) branch of supp Θ x , and thus surjectivity ofD ( t,y ) ˜ e x | N xt,y is an open condition along each branch. Since supp Θ x is locally compact – in particularwith finitely many branches near x – we can then choose ǫ x and Q ( x ) sufficiently small to guaranteethat each D ( t,y ) ˜ e x | N xt,y is surjective. This proves submersivitiy in the sense of Definition A.4. From local to global stabilization:
In this portion of the proof, we proceed almost verba-tim to the corresponding portion of [HWZ, Thm.15.4], with extra considerations to deduce sub-mersivity of (46). By assumption, | S − (0) | is compact and | e | : |X | → M is continuous. Then | S − (0) | ∩ | e − ( C ) | is compact since C := ∪ i ∈ I ( C i ) ⊂ M is closed. We moreover have the identity | S − (0) ∩ e − ( C ) | = | S − (0) | ∩ | e − ( C ) | since both sets are saturated. Thus we have an open cov-ering (cid:0) | Q ( x ) | (cid:1) x ∈ S − (0) ∩ e − ( C ) by the open subsets chosen above, and can pick finitely many points x , . . . , x r ∈ S − (0) ∩ e − ( C ) to obtain a finite open cover | S − (0) ∩ e − ( C ) | ⊂ S ri =1 | Q ( x i ) | . Then | S − (0) | r S ri =1 | Q ( x i ) | is compact, with open cover by (cid:0) | Q ( x ) | (cid:1) x ∈ S − (0) , so we may pick further x r +1 , . . . , x k ∈ S − (0) to obtain the covers | S − (0) | ⊂ S ki =1 | Q ( x i ) | , | S − (0) ∩ e − ( C ) | ⊂ S ri =1 | Q ( x i ) | , (44) S − (0) ⊂ ˜ Q := π − (cid:0)S ki =1 | Q ( x i ) | (cid:1) ⊂ U . For each x = x i we constructed above a family of sc + -multisections (cid:0) Λ x i t : W → Q + (cid:1) t ∈ R lxi . Theseare summed up, using [HWZ, Def.13.11], to a sc + -multisection˜Λ : R ˜ l × W → Q + , (cid:0) t = ( t , . . . , t k ) , w (cid:1) Λ t ( w ) := (cid:0) Λ x t ⊕ · · · ⊕ Λ x k t k (cid:1) ( w )for ˜ l := l x + · · · + l x k . Here each Λ t : W → Q + for t ∈ R ˜ l is a structurable sc + -multisectionby [HWZ, Prop.13.3]. We view the multisection ˜Λ as global perturbation of a sc-Fredholm sectionfunctor ˜ S of a bundle ˜ P ,˜ S : R ˜ l × X → R ˜ l × W =: ˜ W ˜ P : R ˜ l × W → R ˜ l × X ( t, y ) ( t, S ( y )) ( t, w ) ( t, P ( w )) , and claim that e : X → M induces a submersion on its perturbed solution set in the following sense.Global stabilization properties: There exists ǫ > such that for every < ǫ < ǫ (45) ˜Θ : { t ∈ R ˜ l | | t | < ǫ } × X → Q + , ( t, y ) Λ t (cid:0) S ( y ) (cid:1) = ˜Λ( ˜ S ( t, y )) As before, this is shorthand for surjectivity on each reduced tangent space ker D ( t,y ) ( ˜ S x + p j ) | T t R l × T Ry X . POLYFOLD PROOF OF THE ARNOLD CONJECTURE 41 is a tame ep + -subgroupoid with surjective reduced linearizations T R ( ˜ S, ˜Λ) ( t, y ) for all ( t, y ) ∈ supp ˜Θ .In particular, the realization | supp ˜Θ | is a weighted branched orbifold. Moreover, there is a neigh-bourhood V ′ ⊂ X of S − (0) ∩ e − ( C ) such that (46) ˜ e | supp ˜Θ : supp ˜Θ → M, ( t, y ) e ( y ) satisfies (˜ e | supp ˜Θ ) − ( C ) ⊂ R ˜ l × V ′ , and its restriction to supp ˜Θ ∩ ( R ˜ l × V ′ ) is classically smooth andsubmersive as in Definition A.4. Note that the auxiliary norm N on W pulls back to an auxiliary norm ˜ N on ˜ W , and compactnessof ˜ S is controlled in the sense that for any compact subset K ⊂ R ˜ l we have compactness of(47) (cid:12)(cid:12) { ( t, x ) ∈ K × U | ˜ N ( ˜ S ( t, x )) ≤ } (cid:12)(cid:12) = K × (cid:12)(cid:12) { ( x ∈ U | N ( S ( x )) ≤ } (cid:12)(cid:12) ⊂ R ˜ l × |X | . Next, the restriction of ˜Λ to each R l xi × X ֒ → R ˜ l × X is the local perturbation ˜Λ x i of ˜ S x i , sincewe identify R l xi ∼ = { ( t , . . . , t k ) ∈ R ˜ l | t j = 0 ∀ j = i } and each Λ x j is trivial. In particular, Λ isthe trivial multisection, with N (Λ ) = 0. Moreover, we have an estimate N (Λ t ) ≤ c | t | that resultsfrom the linear estimates on each Λ x i t . Now for ǫ ≤ c we can deduce compactness of the stabilizedsolution set as closed subset of (47),(48) ˜ Z := (cid:12)(cid:12)(cid:8) ( t, x ) ∈ R ˜ l × X (cid:12)(cid:12) | t | ≤ ǫ , Λ t ( S ( x )) > (cid:9)(cid:12)(cid:12) . The next step is to argue that (48) is smooth in a neighbourhood of ˜ Z ∩ ( { } × |X | ) = { } × | S − (0) | .Recall here that ˜ Q = π − ( S ki =1 | Q ( x i ) | ) ⊂ X is an open neighbourhood of S − (0). So for any x ∈ ˜ Q we can use the local properties of some ˜Λ x i with | x | ∈ | Q ( x i ) | to deduce surjectivity ofT R ( ˜ S, ˜Λ) (0 , x ). Then the local implicit function theorems [HWZ, Thms 15.2,15.3, Rmk.15.2] yieldan open neighbourhood U (0 , x ) = {| t | < ǫ ′ x } × U ( x ) ⊂ R ˜ l × X of (0 , x ) for some 0 < ǫ ′ x < ǫ ,and hence a saturated neighbourhood ˜ U (0 , x ) := {| t | < ǫ ′ x } × π − ( | U ( x ) | ) ⊂ R ˜ l × X such that˜Θ | ˜ U (0 ,x ) = ˜Λ ◦ ˜ S | ˜ U (0 ,x ) is a tame branched ep + -subgroupoid of ˜ U (0 , x ). As a consequence, the orbitspace of the support (cid:12)(cid:12) supp ˜Θ | ˜ U (0 ,x ) (cid:12)(cid:12) is a weighted branched orbifold with boundary and corners.For x ∈ S − (0) r e − ( C ) we can moreover choose U ( x ) ∩ e − ( C ) = ∅ , since | e − ( C ) | ⊂ |X | isclosed. For x ∈ S − (0) ∩ e − ( C ) ⊂ V the covering (44) guarantees | x | ∈ | Q ( x i ) | for some 1 ≤ i ≤ r with Q ( x i ) ⊂ V and we choose U ( x ) ⊂ Q ( x i ). This guarantees that the restriction of ˜ e : R ˜ l × X → M, ( t, y ) e ( y ) to ˜ U (0 , x ) is sc ∞ , and surjectivity of D (0 ,x ) ˜ e x i | ker T R ( ˜ Sxi , ˜Λ xi ) (0 ,x ) implies surjectivityof D (0 ,x ) ˜ e | N ,x : N ,x → T e ( x ) M on N ,x := ker T R ( ˜ S, ˜Λ) (0 , x ). Here N ,x represents the reducedtangent space at | (0 , x ) | to the weighted branched orbifold | supp ˜Θ | ˜ U (0 ,x ) | . Now ˜ e | supp ˜Θ ∩ ˜ U (0 ,x ) :supp ˜Θ | ˜ U (0 ,x ) → M is classically smooth since it is a restriction of an sc ∞ map to finite dimensions,and we have shown it to be submersive at (0 , x ). Hence, by openness of submersivity along eachcorner stratum, and local compactness of supp ˜Θ | ˜ U (0 ,x ) ⊂ ˜ Z it follows that ˜ U (0 , x ) ⊂ R ˜ l × V can bechosen sufficiently small to ensure that ˜ e | supp ˜Θ ∩ ˜ U (0 ,x ) is submersive as in Definition A.4.Now compactness of | S − (0) ∩ e − ( C ) | and | S − (0) | again allows us to find finite covers | S − (0) | ⊂ S k ′ i =1 | U ( x ′ i ) | , | S − (0) ∩ e − ( C ) | ⊂ S r ′ i =1 | U ( x ′ i ) | with x ′ i ∈ S − (0) ∩ e − ( C ) for i = 1 , . . . , r ′ and U ( x ′ i ) ∩ e − ( C ) = ∅ for r ′ < i ≤ k ′ . Then wehave ǫ := min { ǫ ′ x ′ , . . . ǫ ′ x ′ k ′ } >
0, an open cover S − (0) ⊂ A := π − (cid:0)S k ′ i =1 | U ( x ′ i ) | (cid:1) , and the functor { t ∈ R ˜ l | | t | < ǫ } × A → Q + , ( t, y ) Λ t ( S ( y )) is a tame branched ep + -subgroupoid, since it is therestriction of ˜Θ = ˜Λ ◦ ˜ S to an open subset of S k ′ i =1 ˜ U (0 , x ′ i ). Moreover, we claim that for a possiblysmaller 0 < ǫ < ǫ we have(49) ( t, y ) ∈ {| t | < ǫ } × X, ˜Θ( t, y ) > ⇒ y ∈ A . By contradiction, consider a sequence R ˜ l ∋ t n → y n ∈ X with ˜Θ( t n , y n ) > y n ∈ X r A .Then compactness of (48) guarantees a convergent subsequence | ( t n , y n ) | → | (0 , y ∞ ) | ∈ ˜ Z , and since˜ Z ∩ { } × |X | = { } × | supp Λ ◦ S | = { } × | S − (0) | this contradicts the fact that | y n | ∈ |X | r |A| , where |A| = S k ′ i =1 | U ( x ′ i ) | ⊂ |X | is an open neighbourhood of | S − (0) | . Thus we have shown (49)and can deduce that ˜Θ = ˜Λ ◦ ˜ S : { t ∈ R ˜ l | | t | < ǫ } × X → Q + is a tame branched ep + -subgroupoidwith supp ˜Θ ⊂ R ˜ l × A , and thus (cid:12)(cid:12) supp ˜Θ (cid:12)(cid:12) ⊂ R ˜ l × S k ′ i =1 | U ( x ′ i ) | is a weighted branched orbifold withboundary and corners, as claimed.Moreover, from the properties of ˜ e | supp ˜Θ ∩ ˜ U (0 ,x ′ i ) for i = 1 , . . . , r ′ we know that the restriction of˜ e to supp ˜Θ ∩ ( R ˜ l × V ′ ) for V ′ := π − (cid:0)S r ′ i =1 U ( x ′ i ) (cid:1) ⊂ V is classically smooth and submersive. Herewe have e − ( C ) ∩ A ⊂ V ′ since U ( x i ) for i > r ′ was chosen disjoint from e − ( C ), and hence we have (cid:0) ˜ e | supp ˜Θ (cid:1) − ( C ) = supp ˜Θ ∩ (cid:0) R ˜ l × e − ( C ) (cid:1) ⊂ R ˜ l × (cid:0) e − ( C ) ∩ A (cid:1) ⊂ R ˜ l × V ′ , and thus ˜ e | supp ˜Θ : supp ˜Θ → M is classically smooth and submersive (in the sense of Definition A.4)in the open neighborhood supp ˜Θ ∩ ( R ˜ l × V ′ ) of (cid:0) ˜ e | supp ˜Θ (cid:1) − ( C i ) ⊂ supp ˜Θ for all i ∈ I . Global transversality from regular values:
As we continue to follow the proof of [HWZ,Thm.15.4], we replace each application of the Sard theorem by countably many Sard argumentsto obtain general position to the countably many submanifolds C i ⊂ M for i ∈ I . For that purposewe will consider various restrictions of the projectionsupp ˜Θ = (cid:8) ( t, y ) ∈ R ˜ l × X (cid:12)(cid:12) | t | < ǫ, Λ t ( S ( y )) > (cid:9) → R ˜ l , ( t, y ) t. The global properties of ˜Θ imply that every ( t , y ) ∈ supp ˜Θ has a saturated open neighborhood˜ U ( t , y ) = { t ∈ R ˜ l | | t − t | < δ } × π − ( | U ( y ) | ) ⊂ R ˜ l × X satisfying the following: • U ( y ) ⊂ X admits the natural action of the isotropy group G y ; see [HWZ, Thm.7.1], satisfies theproperness property [HWZ, Def.7.17], and has d X ( y ) local faces F y , . . . , F y d X ( y ) which contain y ; see [HWZ, Def.2.21, Prop.2.14]. • The branched ep + -subgroupoid supp ˜Θ ∩ ˜ U ( t , y ) has a local branching structure˜Θ( t, y ) = Λ t ( S ( y )) = | J | · (cid:12)(cid:12)(cid:8) j ∈ J | ( t, y ) ∈ M t ,y j (cid:9)(cid:12)(cid:12) , given by finitely many properly embedded submanifolds with boundary and corners M t ,y j ⊂ ˜ U ( t , y ), which intersect any intersection of local faces in a manifold with boundary and corners. • On each branch M t ,y j , the reduced linearizations T R ( ˜ S, ˜Λ) ( t, y ) are surjective for all ( t, y ) ∈ M t ,y j ,and the restriction of ˜ e | supp ˜Θ is a submersion M t ,y j ∩ ( R ˜ l × V ′ ) → M in general position to theboundary in the sense of Definition A.4. That is, D ( t,y ) ˜ e | N t,y : N t,y → T e ( y ) M is surjective on N t,y := ker T R ( ˜ S, ˜Λ) ( t, y ) for all ( t, y ) ∈ M t ,y j ∩ ( R ˜ l × V ′ ).There is a countable cover supp ˜Θ ⊂ S β ∈ Z ˜ U ( t β , y β ) indexed by ( t β , y β ) β ∈ Z ⊂ supp ˜Θ, since R ˜ l × X – and hence its subspace supp ˜Θ – is second-countable, and every open cover of a second-countablespace has a countable subcover. Moreover, for any given β ∈ Z there are finitely many choices˜ F K := {| t − t | < δ } × T k ∈ K F y β k ⊂ ˜ U ( t β , y β ) of intersections of finitely many local faces K ⊂{ , . . . , d X ( y β ) } , with ˜ F ∅ := ˜ U ( t β , y β ). Finally, for each β ∈ Z and intersection of faces ˜ F K , thereare finitely many smooth manifolds ˜ F K ∩ M t β ,y β j indexed by j ∈ J β . For each of these countablymany choices, Sard’s theorem asserts that ˜ F K ∩ M t β ,y β j → R ˜ l , ( t, y ) t has an open and densesubset R βK,j ⊂ R ˜ l of regular values. Then, since R ˜ l is a Baire space, the set of common regular values R := T β ∈ Z T K,j R βK,j ⊂ R ˜ l is still dense. For any t ∈ R , the sc + -multisection Λ t : W → Q + is in general position by the usual linear algebra for each restriction of the linearized operators tointersections of faces: Consider ( t , x ) ∈ ˜ F K ∩ M t β ,y β j ⊂ supp ˜Θ and a local section S + p j inthe representation of ˜Θ = ˜Λ ◦ ˜ S with M t β ,y β j ⊂ ( S + p j ) − (0). The surjective differential alongthis intersection of faces can be written as D t ,x ( S + p j ) | ˜ F K = D ⊕ L , where L is a boundedoperator (arising from differentiating p j in the direction of R ˜ l ) and D is the reduced linearization– on the intersection of faces F K := T k ∈ K F y β k ⊂ U ( y β ) ⊂ X – of the section S + p j ( t , · ) that is POLYFOLD PROOF OF THE ARNOLD CONJECTURE 43 a part of the representation of Λ t ◦ S . Then regularity of t implies surjectivity of the projectionΠ : ker( D ⊕ L ) → R ˜ l , which in turn is equivalent to surjectivity of D ; see e.g. [MS, Lemma A.3.6].Moreover, each Λ t for | t | < ǫ is ( N, U )-admissible, thus any sufficiently small regular t ∈ R yields an admissible sc + -multisection λ := Λ t in general position as in [HWZ, Thm.15.4]. To proveour theorem, we have to moreover choose t ∈ R so that the restriction e | Z λ : Z λ → M to thesolution set Z λ = | supp λ ◦ S | is in general position to C i ⊂ M for all i ∈ I . For that purpose weconsider the countably many projections(50) ˜ e − ( C i ) ∩ ˜ F K ∩ M t β ,y β j → R ˜ l , ( t, x ) t for any i ∈ I , index β ∈ Z of the countable cover, intersection of local faces ˜ F K , and smooth branch M t β ,y β j ⊂ supp ˜Θ ∩ ˜ U ( t β , y β ). Here we have ˜ e − ( C i ) ∩ M t β ,y β j ⊂ (˜ e | supp ˜Θ ) − ( C ), so that the restriction˜ e | ˜ F K ∩ M tβ,yβj : ˜ F K ∩ M t β ,y β j → M is smooth and submersive in a neighborhood of ˜ e − ( C i ). Inparticular, it is transverse to C i so that there is a natural smooth structure on ˜ e − ( C i ) ∩ ˜ F K ∩ M t β ,y β j .Thus we can apply the Sard theorem to each (50) to find open and dense subsets T i,βK,j ⊂ R ˜ l of regularvalues, and a dense set of common regular values T := T β ∈ Z T K,j R βK,j ∩ T i T i,βK,j ⊂ R ˜ l . Note that T ⊂ R , so sufficiently small t ∈ T yield admissible sc + -multisections λ := Λ t in general position.Moreover, general position of e | Z λ : Z λ → M to C i at x ∈ Z λ ∩ e − ( C i ) means that the linearizationsof e | F K ∩ Z λ map onto T e ( x ) M / T e ( x ) C i for each intersection of local faces F K ⊂ U ( y β ) ⊂ X thatcontains x . Here the tangent spaces of F K ∩ Z λ at x are given by those of ˜ F K ∩ M t β ,y β j ∩ ( { t } × X )for each branch with ( t , x ) ∈ M t β ,y β j ⊂ supp ˜Θ, so we need to ensure surjectivity of D ( t ,x ) ˜ e :ker Π → T e ( x ) M / T e ( x ) C i on the kernel of the projection Π : T ( t ,x ) (cid:0) ˜ F K ∩ M t β ,y β j (cid:1) → R ˜ l . HereD ( t ,x ) ˜ e : T ( t ,x ) (cid:0) ˜ F K ∩ M t β ,y β j (cid:1) → T e ( x ) M is surjective (since ˜ e | supp ˜Θ is submersive), and regularity t ∈ T i,βK,j means that we have Π (D ( t ,x ) ˜ e ) − (T e ( x ) C i ) = R ˜ l , so for any Y ∈ T e ( x ) M we find( T, X ) ∈ T ( t ,x ) (cid:0) ˜ F K ∩ M t β ,y β j (cid:1) with D ( t ,x ) ˜ e ( T, X ) = Y and ( T, X ′ ) ∈ (D ( t ,x ) ˜ e ) − (T e ( x ) C i ), so that(0 , X − X ′ ) ∈ ker Π proves the required surjectivity D ( t ,x ) ˜ e (0 , X − X ′ ) = Y − D ( t ,x ) ˜ e ( T, X ′ ) =[ Y ] ∈ T e ( x ) M / T e ( x ) C i . Thus this choice of sufficiently small t ∈ T also guarantees general position of e | Z λ to each of the countably many submanifolds C i , which finishes the proof of the theorem whenno boundary values are prescribed. Regular extension:
To prove the last paragraph of the theorem we consider a given ( α N, U )-admissible structurable sc + -multisection λ ∂ : P − ( ∂ X ) → Q + that is in general position over theboundary, and with e | Z ∂ : Z ∂ = supp λ ∂ ◦ S | ∂ X → M in general position to finitely many submani-folds C i . Then we will adjust the above construction of λ : W → Q + to also satisfy λ | P − ( ∂ X ) = λ ∂ ,by following the proof of the transversal extension theorem over ep-groupoids [HWZ, Thm.15.5].Since λ ∂ is supported in U ∩ ∂ X with N ( λ ∂ )( x ) < α for all x ∈ ∂ X we can find a continuousfunctor h : X → [0 ,
1) supported in U with N ( λ ∂ )( x ) < h ( x ) < N ( λ ∂ )( x ) + for all x ∈ ∂ X . Then[HWZ, Thm.14.2] yields a sc + -multisection Λ ′ : W → Q + with Λ ′ | P − ( ∂ X ) = λ ∂ , domain support in U , and N (Λ ′ )( x ) ≤ h ( x ) ≤ α +12 for all x ∈ X . This guarantees compactness of | supp Λ ′ ◦ S | ⊂ |X | and regularity of | supp Λ ′ ◦ S | ∩ | ∂ X | = | supp λ ∂ ◦ S | ∂ X | . To obtain regularity in the interior weconstruct λ = Λ ′ ⊕ Λ t by the above arguments with S − (0) replaced by S ′ := supp Λ ′ ◦ S ⊂ X , notingthat |S ′ | ⊂ |X | is also compact. To achieve general position to the C i we need further adjustments. Local constructions relative to boundary values:
For interior points x ∈ S ′ ∩ ∂ X we construct˜Λ x : R l × W → Q + with domain support in the interior R l × ( ∂ X ∩ U ) to cover the cokernels ofT R ( ˜ S x , ˜Λ ′ ) for the stabilized multisection ˜Λ ′ : R l × W → Q + , ( t, w ) Λ ′ ( w ). For x ∈ S ′ ∩ ∂ X we needno stabilization by a R l factor (i.e. take l = 0) due to the general position of λ ∂ at x . However, weonly obtain general position to the C i , rather than submersivity in the following claim.Local properties relative to boundary: For each x ∈ S ′ there exists l x ∈ N – with l x = 0 for x ∈ S ′ ∩ ∂ X – and a locally uniformizing neighborhood Q ( x ) ⊂ X of x whose closure is contained in U , such that for some ǫ x > we have a tame ep + -subgroupoid Θ x : { t ∈ R l | | t | < ǫ x } × Q ( x ) → Q + , ( t, y ) (cid:0) Λ ′ ⊕ Λ xt (cid:1) ( S ( y )) with surjective reduced linearizations, and thus a weighted branched orbifold | supp Θ x | . Moreover, if x ∈ S ′ ∩V then ˜ e x induces a smooth map | supp Θ x | → M , which is in generalposition to C i for each i ∈ I . The structure of Θ x is established in [HWZ, Thm.15.5.], and the general position to each C i for x ∈ ∂ X follows from submersivity. To achieve general position to the C i for x ∈ ∂ X , recall that C = ∪ i ∈ I ( C i ) ⊂ M is closed, so for x / ∈ e − ( C ) we can choose Q ( x ) disjoint from e − ( C ) so thatgeneral position to the C i ⊂ C is automatic. For x ∈ e − ( C ) ⊂ V we have e : supp Θ x ∩ ∂ X =supp λ ∂ ◦ S | ∂ X → M in general position to each C i by assumption on λ ∂ . Moreover, we choose Q ( x ) ⊂ V so that e : Q ( x ) ∩ supp Θ x → M is smooth, and thus general position to each C i extendsto a neighbourhood Q i ⊂ X of x . Then Q ′ := T i ∈ I Q i is a neighbourhood of x since I is finite, andwe can replace Q ( x ) by a uniformizing neighbourhood in Q ′ to achieve general position to all C i . From local to global relative to boundary:
This portion of the proof is started by picking afinite cover |S ′ | ∩ | ∂ X | = | supp λ ∂ ◦ S | ∂ X | = S i = − k ∂ | Q ( x i ) | ⊂ |X | by the above neighbourhoodsfor x i ∈ S ′ ∩ ∂ X . Next we cover |S ′ | \ S i = − k ∂ | Q ( x i ) | ⊂ S ki =1 | Q ( x i ) | with neighbourhoods ofinterior points x i ∈ S ′ ∩ ∂ X whose associated multisections Λ x i are supported in the interior,dom-supp Λ x i ⊂ R l x ∩ ∂ X . Then we define ˜Λ : R ˜ l × W → Q + by ˜Λ( t, w ) := Λ t ( w ) := (cid:0) Λ ′ ⊕ Λ x t ⊕ · · · ⊕ Λ x k t k (cid:1) ( w ). This multisection is constructed so that Λ = Λ ′ and Λ t | P − ( ∂ X ) = λ ∂ forany t ∈ R ˜ l . Moreover, the estimate N (Λ t ) ≤ N (Λ ′ ) + c | t | ≤ α + c | t | allows us to guaranteeadmissibility N (Λ t ) ≤ | t | ≤ − α c . Then compactness of ˜ Z in (48) follows as above,and its smoothness is established using a covering | S − (0) | ⊂ S k ′ i = − k ∂ | U ( x ′ i ) | where | U ( x ′ i ) | for i ≤ x ′ i ∈ S ′ ∩ ∂ X and cover a neighbourhood of | ∂ X | . Moreover, U ( x ′ i ) ⊂ R ˜ l × Q ( x ′ i ) canbe chosen as in the prior proof of the local properties such that ˜ e | supp Θ : U ( x ′ i ) → M is in generalposition to C i for each i ∈ I . This establishes the following.Global stabilization properties with fixed boundary values: There exists ǫ > such that ˜Θ := ˜Λ ◦ ˜ S : {| t | < ǫ } × X → Q + is a tame ep + -subgroupoid with surjective reduced linearizations for every <ǫ < ǫ . In particular, | supp ˜Θ | is a weighted branched orbifold. Moreover, there is a neighbourhood V ′ ⊂ X of S − (0) ∩ e − ( C ) such that ˜ e | supp ˜Θ : supp ˜Θ → M satisfies (˜ e | supp ˜Θ ) − ( C ) ⊂ R ˜ l × V ′ , andits restriction to supp ˜Θ ∩ ( R ˜ l × V ′ ) is classically smooth and in general position to each C i . Global transversality relative to boundary:
In this final step we use the fact that Λ t is( N, U )-admissible for | t | ≤ − α c and choose a common regular value of countably many projectionsas before. The only difference to the proof above is that the restriction of ˜ e | supp ˜Θ to a branch M t ,y j ∩ ( R ˜ l × V ′ ) → M is not necessarily submersive but still in general position to each of the C i , that is D ( t,y ) ˜ e | N t,y : N t,y → T e ( y ) M / T e ( y ) C i is surjective for each i ∈ I . When considering theprojections (50), this suffices to obtain smooth structures on ˜ e − ( C i ) ∩ ˜ F K ∩ M t β ,y β j for each branchand intersection of faces ˜ F K . Then general position of e | Z λ : Z λ → M to C i at x ∈ Z λ ∩ e − ( C i ) for λ = Λ t with a regular value t ∈ R l again requires surjectivity of D ( t ,x ) ˜ e : ker Π → T e ( x ) M / T e ( x ) C i on the kernel of the projection Π : T ( t ,x ) (cid:0) ˜ F K ∩ M t β ,y β j (cid:1) → R ˜ l . To see that [ Y ] ∈ T e ( x ) M / T e ( x ) C i isin the image we use the above surjectivity of D ( t ,x ) ˜ e | N t ,x to find ( T, X ) ∈ T ( t ,x ) (cid:0) ˜ F K ∩ M t β ,y β j (cid:1) with D ( t ,x ) ˜ e ( T, X ) ∈ [ Y ]. Then regularity of t yields ( T, X ′ ) ∈ (D ( t ,x ) ˜ e ) − (T e ( x ) C i ), so that(0 , X − X ′ ) ∈ ker Π solves [D ( t ,x ) ˜ e (0 , X − X ′ )] = [ Y − D ( t ,x ) ˜ e ( T, X ′ )] = [ Y ] ∈ T e ( x ) M / T e ( x ) C i . Thisfinishes the proof with prescribed boundary values. (cid:3) References [A] V.I. Arnold, Sur une propriet´e topologique des applications globalement canoniques et `a m´ecanicque classique,
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