Equivariant Lagrangian Floer cohomology via semi-global Kuranishi structures
aa r X i v : . [ m a t h . S G ] M a y EQUIVARIANT LAGRANGIAN FLOER COHOMOLOGY VIASEMI-GLOBAL KURANISHI STRUCTURES
ERKAO BAO AND KO HONDAA
BSTRACT . Using a simplified version of Kuranishi perturbation theory that wecall semi-global Kuranishi structures, we give a definition of the equivariant La-grangian Floer cohomology of a pair of Lagrangian submanifolds that are fixedunder a finite symplectic group action and satisfy certain simplifying assump-tions. C ONTENTS
1. Introduction 12. Equivariant semi-global Kuranishi structure 53. Orientations 214. Equivariance of curve counting 265. Equivariant Lagrangian Floer cohomology 32References 361. I
NTRODUCTION
Let G be a finite group. The equivariant Lagrangian Floer cohomology for apair of Lagrangians fixed under a symplectic G -action was first defined and stud-ied in [KS] and later in [SS, He1, He2, He3, HLS]. One of the main difficul-ties in defining such a theory is achieving transversality of the moduli spaces of J -holomorphic curves using an equivariant almost complex structure J . Indeed,there are obstructions to the existence of equivariant regular almost complex struc-tures; see [KS, SS]. The paper [HLS] uses an infinite family of non-equivariantregular almost complex structures and an algebraic approach to define equivariantcohomology.The goal of this paper is to give an alternate definition of equivariant LagrangianFloer cohomology using an equivariant almost complex structure J that is not nec-essarily regular. This involves constructing an equivariant version of a semi-globalKuranishi structure, which is a simplified version of the Kuranishi structures of[FOn, FO3] used in [BH2]; compare to [MW] for the Kuranishi atlas formulation. Date : This version: April 22, 2020.2000
Mathematics Subject Classification.
Primary 57M50; Secondary 53D10,53D40.
Key words and phrases. symplectic structure, Floer homology.
It is worth mentioning that there is a construction of equivariant Kuranishi chartsin [Fu] in a more general situation via a quite different approach.Let ( M, ω ) be a compact symplectic manifold of dimension n , and let L and L be oriented Lagrangian submanifolds of M that intersect transversely. Suppose G acts on ( M, ω ) symplectically and satisfies g ( L i ) = L i for all g ∈ G and i = 0 , ; and that G fixes the orientations of L i .We make the following simplifying assumption: (S) the maps π ( M ) R ω → R and π ( M, L i ) R ω → R for i = 0 , have image .More informally, (S) says that for all almost complex structures we consider wewant to avoid disk and sphere bubbles.We also assume that either M is closed or M has contact type boundary , i.e.,on a neighborhood of ∂M there exists a -form σ such that ω = dσ and the vectorfield X defined by ι X ω = σ is positively transverse to ∂M . Note that because ω is G -invariant, by averaging σ over G , we can take σ to be G -invariant.The time- r flow φ r of X gives a diffeomorphism Φ from ( − ǫ, × ∂M to aneighborhood of ∂M defined by ( r, m ) φ r ( m ) . Since L X σ = dι X σ + ι X dσ = dι X ι X ω + ι X ω = σ, we have ( φ r ) ∗ σ = e r σ . Setting α = σ | ∂M , we obtain Φ ∗ σ = e r α . Since α ∧ ( dα ) n − = ( ι X ω n ) | ∂M and X is transverse to ∂M , α ∧ ( dα ) n − is a volumeform on ∂M , and hence α is a contact form. We denote by ξ = ker α the contactstructure and R α the Reeb vector field of α .Let J be a G -invariant, ω -compatible almost complex structure on M , i.e., ω ( · , J · ) is a G -invariant Riemannian metric. Near ∂M we assume that J is convexat the boundary. More specifically: (J) on the collar neighborhood (( − ǫ, × ∂M, e r α ) , J is compatible with dα and maps ξ to ξ and ∂ r to the Reeb vector field R α of α . Exact case.
One special case for which (S) holds is: • ( M, ω = dσ ) is a Liouville domain , i.e., M is compact and the Liouvillevector field X defined by ι X ω = σ points out of ∂M ; • L and L are compact exact Lagrangians in M with Legendrian boundary,where exactness means that σ | L i is an exact -form on L i for i = 0 , .To orient the relevant moduli spaces of J -holomorphic strips, following [FO3,Section 8.1] we assume that: (O) the pair ( L , L ) is equipped with a relative spin structure which is pre-served by G .See Section 3 for more details on the auxiliary orientation data including relativespin structures, and Section 3.6 for the notion of a G -invariant relative spin struc-ture. In particular we assume that L and L are oriented. (See Seidel [Se] fororientations using Pin structures and Solomon [So] for orientations using relativePin structures.) QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 3
We denote by R the Novikov ring n ∞ X i =0 a i T λ i | a i ∈ Z , λ i ∈ R ≥ , λ = 0 and lim i →∞ λ i = ∞ o , where T is a formal parameter.The Lagrangian Floer cochain complex CF • ( L , L ) of the pair ( L , L ) is thefree module over the coefficient ring R generated by L ∩ L with differential d whose definition we give below.For p, q ∈ L ∩ L , let π ( p, q ) be the set of homotopy classes of continuousmaps u : [0 , × [0 , → M with boundary conditions u (0 , t ) = q , u (1 , t ) = p , u ( s, ∈ L , u ( s, ∈ L . Let f M J ( p, q ; A ) , where A ∈ π ( p, q ) , be the space ofsmooth maps u : R × [0 , → M that satisfy:(A1) ∂ J u := u s + J ( u ) u t = 0 ;(A2) u | R ×{ i } ⊆ L i for i ∈ { , } ;(A3) lim s →−∞ u ( s, t ) = q and lim s → + ∞ u ( s, t ) = p ; and(A4) [ u ] = A .Note that R acts on f M J ( p, q ; A ) by translation in the domain and we denote M J ( p, q ; A ) := f M J ( p, q ; A ) / R . We also denote the virtual (= expected) dimension of M J ( p, q ; A ) by vdim M J ( p, q ; A ) . Notation 1.0.1.
We use the notation M J ( p, q ; A ) to mean the space of possiblybroken strips from p to q in the class A and ∂ M J ( p, q ; A ) = M J ( p, q ; A ) − M J ( p, q ; A ) . Suppose for the moment that J is regular. Then we have the differential d : CF • ( L , L ) → CF • ( L , L ) d [ p ] = X q ∈ L ∩ L ,A ∈ π ( p,q ) M J ( p, q ; A ) · T R A ω [ q ] , where M J ( p, q ; A ) = 0 if vdim M J ( p, q ; A ) = 0 . Note that for each λ ≥ ,the number of ( q, A ) such that ω ( A ) ≤ λ and M J ( p, q ; A ) = ∅ is finite. Then asusual one shows that d = 0 and defines the usual Lagrangian Floer homology by HF • ( L , L ) := ker d/ Im d .We recall the definition of equivariant cohomology of a space Y with a G -action.Let BG be the classifying space of G and let EG be the universal bundle over BG . The diagonal action of G on EG × Y is free and the quotient is denoted by EG × G Y . The G -equivariant cohomology of Y with coefficient ring R is definedto be H • ( EG × G Y ; R ) . Let C • ( A ) be the singular chain complex of the space A over R and C • ( A ) = Hom R ( C • ( A ) , R ) be the singular cochain complex of A . Since the singular chain complexes and cochain complexes of EG and Y areinvariant under the G -action and their boundary maps are G -equivariant, they can ERKAO BAO AND KO HONDA be viewed as complexes over the group ring R [ G ] . Then we have H k ( EG × G Y ; R ) ∼ = H k (Hom R ( C • ( EG × G Y ) , R )) ∼ = H k (Hom R ( C • ( EG ) ⊗ R [ G ] C • ( Y ) , R )) ∼ = H k (Hom R [ G ] ( C • ( EG ) , C • ( Y ))) . In the second and third lines we are taking the k -th cohomology of the total com-plex of a double complex. We are also viewing C • ( EG ) as a complex of right R [ G ] -modules and C • ( Y ) as a complex of left R [ G ] -modules. We can also take asmaller model for C • ( EG ) : The projective resolution P • of R over R [ G ] is chainhomotopic to C • ( EG ) and(1.0.1) H k ( EG × G Y ; R ) ∼ = H k (Hom R [ G ] ( P • , C • ( Y ))) . Returning to the “usual” definition of equivariant Lagrangian Floer cohomologyassuming J is regular, we replace C • ( Y ) by CF • ( L , L ) in Equation (1.0.1).More precisely, since J is invariant under the G -action, we have M J ( p, q ; A ) = M J ( g ( p ) , g ( q ); g ( A )) for all g ∈ G . Hence d is a R [ G ] -linear map on CF • ( L , L ) . We can thendefine the G -equivariant Lagrangian Floer cohomology group HF • G ( L , L ) asthe cohomology of the total complex of Hom R [ G ] ( P • , CF • ( L , L )) . Example . [FO] Let f : L → R be a G -equivariant Morse function and let L be graph( ǫ · df ) ⊂ T ∗ L for some small ǫ > . Then HF • G ( L , L ) ∼ = H • G ( L ; R ) .In general, a G -invariant J is not regular and the moduli space M J ( p, q ; A ) is not transversely cut out. The main contribution of this paper is to obtain a G -equivariant cochain complex CF • ( L , L ) when J is not regular by construct-ing an equivariant version of a semi-global Kuranishi structure, initially developedin [BH2] for contact homology. The equivariant semi-global Kuranishi structurecomes with a section s , and while the Kuranishi structure itself is G -equivariant, thesection is not. This creates some difficulties, but interestingly enough there is a per-turbed count of M J ( p, q ; A ) that still remains G -invariant (cf. Theorem 4.1.2).Our main theorem is therefore the following: Theorem 1.0.3 (Equivariant Lagrangian Floer cohomology) . Suppose G acts on ( M, ω ) symplectically and for each i = 0 , , L i is oriented, g ( L i ) = L i , for each g ∈ G , and G fixes the orientation of L i . If (S) and (O) hold, then there existsan R -module HF • G ( L , L ) which is an invariant of ( L , L ) under G -equivariantHamiltonian isotopy. Moreover, when there exists a regular G -invariant ω -compatiblealmost complex structure on M satisfying (J) , the usual definition of equivariantLagrangian Floer cohomology can be made and agrees with HF • G ( L , L ) . If we want to equip the Lagrangian Floer homology groups with a Z -grading,we assume that ( L , L ) is a G -equivariant graded Lagrangian pair; see Section 5.1for details. QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 5
The definition of HF • G ( L , L ) is given in Section 5.2 and its invariance under G -equivariant Hamiltonian isotopy is given in Section 5.4. Most of the work isdevoted to the construction of the semi-global Kuranishi structure in Section 2 andthe equivariance of the curve count in Section 4. The agreement with the usualdefinition for regular J is automatic. Acknowledgements.
We thank Kristen Hendricks, Robert Lipshitz, and SucharitSarkar for explaining to us their approach to equivariant Lagrangian Floer coho-mology in [HLS]. The first author thanks Garrett Alston and Cecilia Karlsson fordiscussions on orientations and Vincent Colin for providing him a great visitingopportunity at the Lebesgue Center of Mathematics and the Universit´e de Nantes,where part of this work was carried out. The first author also thanks the SimonsCenter for Geometry and Physics, where he worked on this paper.2. E
QUIVARIANT SEMI - GLOBAL K URANISHI STRUCTURE
The construction of the equivariant semi-global Kuranishi structure follows thesame steps as that of [BH2]. The only differences are that (i) we consider sections,not multisections, and (ii) we pay attention to G -equivariance.2.1. G -invariant almost complex structure. The following lemma is well-known.
Lemma 2.1.1.
There exists an almost complex structure J which is ω -compatible, G -invariant, and satisfies (J) if ∂M = ∅ .Proof. If ∂M = ∅ , then on the collar neighborhood U = ( − ǫ, × ∂M , ∂ r , R α ,and ξ are preserved by G . Choose a Riemannian metric ˆ g on M such that ( ⋆ ) ∂ r , R α , and ξ are mutually orthogonal on U and ∂ r and R α have length e r/ .Let g be the average of ˆ g under the group action G . Then g is preserved by G and ( ⋆ ) holds.From ω and g , we obtain the canonical ω -compatible almost complex structure J on M by the usual polar decomposition argument; see for example [Si, Proposition12.3] and [MS1, Proposition 2.50]. More precisely, we define A : T M → T M by ω ( u, v ) = g ( Au, v ) and the almost complex structure J by J = ( √ A ∗ A ) − A ,where A ∗ is the g -adjoint of A . It is not hard to check that J is ω -compatibleand G -invariant and that J maps ∂ r R α and J ξ = ξ . Hence (J) is satisfied if ∂M = ∅ . (cid:3) Lemma 2.1.2.
Given almost complex structures J and J that are ω -compatible, G -invariant, and satisfy (J) if ∂M = ∅ , there exists a -parameter family of almostcomplex structures { J τ } τ ∈ [0 , connecting J and J such that for each τ ∈ [0 , , J τ is ω -compatible, G -invariant, and satisfies (J) if ∂M = ∅ .Proof. Define the metrics g i ( · , · ) := ω ( · , J i · ) for i ∈ { , } . We can connect g and g by a -parameter family of G -invariant metrics { g τ } τ ∈ [0 , . It is not hard tosee that we can take the g τ so that ( ⋆ ) holds for each τ ∈ [0 , . Then we can define { J τ } τ ∈ [0 , as in the proof of Lemma 2.1.1. (cid:3) ERKAO BAO AND KO HONDA
From now on we assume ω , J , and g are compatible and G -invariant. It is easyto check that we can further choose J such that for any p ∈ L ∩ L , J ( T p L ) = T p L . In later calculations, we implicitly use an identification of ( T p M, J ) with ( R n ⊕ i R n , i ) that maps T p L to the R n factor and T p L to the i R n factor.2.2. Fredholm setup.
Let S = R × [0 , with coordinates ( s, t ) and the standardcomplex structure j which maps ∂ s ∂ t . Let p, q ∈ L ∩ L . For k ≥ , let B k +1 ,p = B k +1 ,p ( p, q ; A ) be the space of maps u : S → M in W k +1 ,p ( S, M ) satisfying (A2)–(A4) and such that there exist ρ + , ρ − ∈ R , ξ + ∈ W k +1 ,p ( S, T p M ) , and ξ − ∈ W k +1 ,p ( S, T q M ) for which • u ( s, t ) = exp p ξ + ( s, t ) for s ≥ ρ + , • u ( s, t ) = exp q ξ − ( s, t ) for s ≤ ρ − .Here the exponential map exp is taken with respect to the G -invariant g . Let π : E k,p = E k,p ( p, q ; A ) → B k +1 ,p be the smooth Banach bundle with fiber E k,pu = π − ( u ) = W k,p ( S, ∧ , S ⊗ J u ∗ T M ) . Then ∂ J : B k +1 ,p → E k,p , u ( u s + J ( u ) u t ) ⊗ J ( ds − idt ) is a Fredholm section and ∂ − J (0) = f M J ( p, q ) .Let ∇ be the Levi-Civita connection on M with respect to g . Let D u be thedifferential ( ∂ J ) ∗ : T u B k +1 ,p → T ( u,∂ J u ) E k,p postcomposed with the projection to E k,p ( u,∂ J u ) . Let us write W k +1 ,p ( S, u ∗ T M ) for ξ ∈ W k +1 ,p ( S, u ∗ T M ) satisfying ξ ( s, ∈ T u ( s, L and ξ ( s, ∈ T u ( s, L .Then, by [MS2, Proposition 3.1.1], D u : W k +1 ,p ( S, u ∗ T M ) → W k,p ( S, ∧ , S ⊗ J u ∗ T M ) is given by D u ξ = ( ∇ ξ + J ∇ ξ ◦ j ) − J ( ∇ ξ J ) ∂ J u (2.2.1) = (cid:2) ( ∇ s ξ + J ∇ t ξ ) − J ( ∇ ξ J )( u s − J u t ) (cid:3) ⊗ J ( ds − idt ) . By abuse of notation, we are not distinguishing between sections of u ∗ T M andsections of
T M along u .In what follows we will usually write π : E → B . Note that, as s → ±∞ , ( ∇ s ξ + J ∇ t ξ ) → ∂ s ξ + J ( p ) ∂ t ξ and u s , u t → . This motivates the followingdefinition. We will be using p for both a point in L ∩ L and the L p -exponent. Hopefully this will notcreate any confusion. QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 7
The asymptotic operator.
Consider W p = { ξ ∈ C ∞ ([0 , , T p M ) | ξ ( i ) ∈ T p L i , i = 0 , } , with inner product h ξ , ξ i = Z g p ( ξ ( t ) , ξ ( t )) dt. The asymptotic operator A = A p : W p → W p is the self-adjoint operator Aξ ( t ) = − J ( p ) ∂∂t ξ ( t ) . We list the eigenvalues of A · · · ≤ λ − ≤ λ − < < λ ≤ λ ≤ · · · with corresponding eigenfunctions · · · , f p − , f p − , f p , f p · · · , chosen so that the f pi form an L -orthonormal basis of W p . Model calculation for the adjoint.
Consider the map u : R × [0 , → C withboundary conditions u ( s, ∈ L = R and u ( s, ∈ L = i R = J R anddecay conditions lim s →±∞ u ( s, t ) = 0 . Consider the Cauchy-Riemann operator Du = ∂u∂s + J ∂u∂t .We calculate the adjoint operator D ∗ v , for any compactly supported v : R × [0 , → C . It satisfies h Du, v i = h u, D ∗ v i , where h , i denotes the L -norm. Moreprecisely, we have Z R × [0 , ( ∂u∂s + J ∂u∂t ,v ) dsdt = Z R × [0 , ( ∂u∂s , v ) + ( J ∂u∂t , v ) dsdt = Z R × [0 , (cid:0) ∂∂s ( u, v ) − ( u, ∂v∂s ) + ∂∂t ( J u, v ) − ( J u, ∂v∂t ) (cid:1) dsdt = − Z R × [0 , ( u, ∂v∂s − J ∂v∂t ) dsdt + Z R × [0 , ∂∂t ( J u, v ) dsdt (2.3.1)Here ( · , · ) is the real part of the standard Hermitian inner product on C . Observethat: Z ∞−∞ ∂∂s ( u, v ) ds = ( u, v ) | s =+ ∞ s = −∞ = 0 by the decay conditions at s = ±∞ . We also have(2.3.2) Z ∂∂t ( J u, v ) dt = ( J u, v ) | t =1 t =0 = ω ( J u, J v ) | t =1 t =0 = ω ( u, v ) | t =1 t =0 . The following claim implies the adjoint is D ∗ v = − ( ∂v∂s − J ∂v∂t ) , subject to therestriction of the domain to v satisfying v ( s, ∈ L = R and v ( s, ∈ L = J R . Claim 2.3.1. If h Du, v i = 0 for all u , then v satisfies D ∗ v = 0 and boundaryconditions v ( s, ∈ L = R and v ( s, ∈ L = J R . ERKAO BAO AND KO HONDA
Proof.
By Equations (2.3.1) and (2.3.2), if h Du, v i = 0 for all u , then Z R × [0 , ( u, D ∗ v ) dsdt + Z R (( J u ( s, , v ( s, − ( J u ( s, , v ( s, ds = 0 for all u . We can decouple this equation into two pieces by considering u that aresupported in the interior of R × [0 , and on small neighborhoods of boundarypoints. Hence we obtain the conditions D ∗ v = 0 and v ( s, ∈ L = R and v ( s, ∈ L = J R . (cid:3) Interior semi-global Kuranishi charts.
Let us first consider a single modulispace M J = M J ( p, q ; A ) = f M J ( p, q ; A ) / R . We will often suppress the almost complex structure J from the notation when it isclear from the context. Let us also abbreviate f M = f M ( p, q ; A ) , B = B ( p, q ; A ) ,and E = E ( p, q ; A ) . Definition 2.4.1. An interior semi-global Kuranishi chart is a quadruple ( K , π : E → V , ∂, ψ ) , where:(i) K ⊂ M is a large compact subset; if M is compact, we take K = M ;(ii) π : E → V , called the obstruction bundle , is a finite rank vector bundleover a finite-dimensional manifold;(iii) ∂ : V → E is a section;(iv) ψ : ∂ − (0) → M is a homeomorphism onto an open subset of M and K ⊂ Im( ψ ) ;(v) dim V − rk E = vdim M .If the group G acts on ( K , π : E → V , ∂, ψ ) , then the Kuranishi chart is G -invariant .A section s of π : E → V that is transverse to ∂ is an obstruction section . Notation 2.4.2.
In (iii) we are abusing notation and writing ∂ for the section toindicate that it descends from ∂ : B → E ; for the charts we construct, the sections ∂ are consistent with one another. We will also often abuse notation and write K ⊂ V without referring to the map ψ .The goal of this subsection is to construct a G -equivariant interior semi-globalKuranishi chart over a large G -invariant compact subset K ⊂ M .Let B q ⊂ M be a sufficiently small disk neighborhood of q ∈ L ∩ L . Given m ∈ B q , let Γ qm : T q B q → T m B q be the parallel transport with respect to the Levi-Civita connection of g along theshortest geodesic from q to m . Next we define the t ∈ [0 , -dependent section F qj : [0 , × B q → T B q of T B q → B q by F qj ( t, m ) = Γ qm ( f qj ( t )) , where f qj are the eigenfunctions of A q . QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 9 s qu,ε β qu −∞ + ∞ u L L pq F IGURE Definition 2.4.3 (The map a q ) . Let P ( B q ) be the space of C -paths γ : [0 , → B q satisfying γ ( i ) ∈ L i for i ∈ { , } . We then define a map a q : P ( B q ) → R as follows: Let v γ : ( −∞ , × [0 , → M be a C -map such that v γ (0 , t ) = γ ( t ) , v γ ( s, i ) ∈ L i for i ∈ { , } , and lim s →−∞ v γ ( s, t ) = q . Then v γ is a path in P ( B q ) from the constant path at q to γ . Then let(2.4.1) a q ( γ ) = Z ( −∞ , × [0 , v ∗ γ ω. Note that a q ( γ ) does not depend on the choice of path v γ .By the monotonicity lemma, there exists ε > such that for each nonconstant v ∈ f M , there exists a unique value s qv,ε of s ∈ R which satisfies the following: (sq) the path γ v,s ( t ) = v ( s, t ) is contained in B q and a q ( γ v,s ) = ε .Note that if v ′ ( s, t ) = v ( s + s , t ) then s qv ′ ,ε = s qv,ε − s .We can also define s qu,ε for u ∈ B which is C -close to v .Let U = U v ⊂ B be a sufficiently small open neighborhood of v ∈ f M . Fix δ > small. We pick a smooth bump function β : R → [0 , such that(a) β ( s ) = 1 for s ∈ [ − , , and(b) β ( s ) = 0 for s [ − , .We construct a section ˜ f qj = ˜ f q,δj of E| U → U as follows. For each u ∈ U , wedefine(2.4.2) ˜ f qj ( u ) = β qu · u ∗ F qj ⊗ C ( ds − idt ) ∈ W k,p ( S, Λ , S ⊗ u ∗ T M ) , where β qu : R → [0 , is a smooth bump function of s defined by β qu ( s ) = β ( δ − ( s − s qu,ε )) . We denote by E ℓ = E q,ℓ → U the vector subbundle of E| U spanned by thesections ˜ f q − , . . . , ˜ f q − ℓ . The R -translation of S = R × [0 , induces an R -action on E → B with respect to which the sections ∂ and ˜ f qi are equivariant. We denote by E ℓ → U the quotient of the bundle E ℓ → U by the R -action. We also introducethe vector space(2.4.3) e ℓ = e q,ℓ = R h f q − , . . . , f q − ℓ i . Proposition 2.4.4.
There exist a sufficiently large ℓ and a sufficiently small openneighborhood N ( K ) ⊆ B / R of K such that the vector bundle E ℓ → N ( K ) , ob-tained by patching together charts of the form E ℓ → U with U = U [ v ] := U v / R and [ v ] ∈ K , is transverse to the section ∂ | N ( K ) and is trivial with fibers that arecanonically identified with e ℓ . The proof is similar to that of [BH2, Theorem 5.1.2] and will be omitted. In anutshell, this is because D ∗ u ( ζ ⊗ ( ds − idt )) is approximated by − ( ∂ s ζ + Aζ ) for s ≪ and each nonzero element of Ker D ∗ u has a negative end that is dominatedby e − λ j s f j ( t ) for some j < .We then define V := ∂ − ( E ℓ | N ( K ) ) ⊂ N ( K ) and restrict E ℓ → N ( K ) to V . By shrinking V if necessary, we may assume that V is G -equivariant. This completes the construction of a G -equivariant interiorsemi-global Kuranishi chart for K .In view of the identification of the fibers of E ℓ with e ℓ , we will usually takean obstruction section s on E ℓ → V to be a generic point in e ℓ = e q,ℓ which issufficiently close to the origin. A more specific choice of the generic point s q ∈ e q,ℓ will be made in Section 4.1.2.5. Boundary semi-global Kuranishi charts.
In this subsection, we explain howto construct Kuranishi charts for curves that are close to breaking.2.5.1.
Simplest case.
Let us consider the simplest situation where M = M ( p, r ; A ) , M = M ( r, q ; A ) , M = M ( p, q ; A + A ) ,∂ M = M × M , and M is G -invariant. Let K , K , K be compact subsetsof M , M , M , respectively, ( K , E → V , ∂ ) , ( K , E → V , ∂ ) , ( K , E → V , ∂ ) be the corresponding interior G -equivariant semi-global Kuranishi charts, and s ∈ e r,ℓ , s ∈ e q,ℓ , s ∈ e q,ℓ be the obstruction sections.We will construct a boundary semi-global Kuranishi chart E (12) → V (12) overthe curves of M that are close to breaking. Let σ > be small. Definition 2.5.1 (Close to breaking) . An element u ∈ B ( p, q ; A + A ) (resp. [ u ] ∈ B ( p, q ; A + A ) / R ) is σ -close to a broken strip ([ u ] , [ u ]) ∈ V × V ifthere exist representatives u , u of [ u ] , [ u ] (resp. u , u , u of [ u ] , [ u ] , [ u ] )such that QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 11 • u | [ σ − , ∞ ) × [0 , is σ -close in the C -norm to u | [ σ − , ∞ ) × [0 , , • u | ( −∞ , − σ − ] × [0 , is σ -close in the C -norm to u | ( −∞ , − σ − ] × [0 , , • u | [ − σ − ,σ − ] × [0 , , u | ( −∞ ,σ − ] × [0 , , and u | [ − σ − , ∞ ) × [0 , are σ -close inthe C -norm to the constant map to the point r .Let e G σ ( V , V ) ⊂ B ( p, q ; A + A ) be the subset of maps u that are σ -close tosome broken strip ([ u ] , [ u ]) ∈ V × V and let G σ ( V , V ) := e G σ ( V , V ) / R .For each u ∈ e G σ ( V , V ) , there exists a unique value s ru,ε of s ∈ R whichsatisfies the following: (sr) the path γ u,s ( t ) = u ( s, t ) is contained in B r and a r ( γ u,s ) = ε .Then for each u ∈ e G σ ( V , V ) we define ˜ f rj ( u ) = β ru · u ∗ F rj ⊗ C ( ds − idt ) ∈ W k,p ( S, Λ , S ⊗ u ∗ T M ) , where β ru : R → [0 , is the smooth bump function β ru ( s ) = β ( δ − ( s − s ru,ε )) ,and β is as before.Let E ℓ ( V , V ) → e G σ ( V , V ) be the vector subbundle of E| e G σ ( V , V ) spannedby the sections ˜ f r − , . . . , ˜ f r − ℓ and ˜ f q − , . . . , ˜ f q − ℓ . By linear gluing (a simpler versionof Theorem 2.6.1 described below) for σ > sufficiently small, E ℓ ( V , V ) → e G σ ( V , V ) is transverse to ∂ . We then define V (1 , := ∂ − ( E ℓ ( V , V )) ⊂ e G σ ( V , V ) . The quotient of E ℓ ( V , V ) | V (1 , → V (1 , by the R -translation is denoted by: π (1 , : E ℓ (1 , → V (1 , . Observe that E ℓ (1 , is a trivial vector bundle whose fibers are canonically identifiedwith e r,ℓ ⊕ e q,ℓ .Let us fix ε ′ satisfying < ε ′ ≪ ε . Suppose σ = σ ( ε ′ ) > is sufficiently small. Definition 2.5.2 (Neck length) . The neck length function is the function nl : e G σ ( V , V ) → R + ,u s ru, − ε ′ − s ru,ε ′ , (2.5.1)where s ru, − ε ′ and s ru,ε ′ are the unique values defined as in (sr) above.Observe that nl : e G σ ( V , V ) → R + descends to nl : G σ ( V , V ) → R + . Pick L = L ( ε ′ , σ ) > large and ε ′′ > small. After some modifications we mayassume that:(C) V and V (1 , cover M ;(C ) V ∩ M consists of [ u ] ∈ M − G σ ( V , V ) and [ u ] ∈ G σ ( V , V ) ∩ M satisfying nl ([ u ]) < L ;(C (1 , ) V (1 , ∩M consists of [ u ] ∈ G σ ( V , V ) ∩M satisfying nl ([ u ]) > L− ε ′′ ;(G) G acts equivariantly on E (1 , → V (1 , . The bundles E → V and E (1 , → V (1 , are related by the restriction-inclusion morphism: we first restrict E → V to V , (1 , := V ∩ {L − ε ′′ < nl ([ u ]) < L} and take the natural inclusion into E (1 , → V (1 , , recalling that the fibers of E are canonically identified with e q,ℓ and the fibers of E (1 , are canonically identifiedwith e r,ℓ ⊕ e q,ℓ . Definition 2.5.3 (The function ζ ) . Choose < ε ′′′ ≪ ε ′′ . Let ζ : [0 , ∞ ) → [0 , be a smooth function such that • ζ ([0 , L + ε ′′′ ]) = 0 , • ζ ([ L + ε ′′ − ε ′′′ , ∞ )) = 1 , and • its restriction to ( L + ε ′′′ , L + ε ′′ − ε ′′′ ) is a diffeomorphism onto (0 , .We then set s (1 , = ( ζ ( nl ) s r , s q ) ∈ e r,ℓ ⊕ e q,ℓ . In particular, s (1 , = ( s r , s q ) on nl ≥ L + ε ′′ and s (1 , = (0 , s q ) on nl ≤ L . Bythe restriction-inclusion, s (1 , is consistent with s .2.5.2. Order of choice of constants.
We outline the order in which the auxiliaryconstants are chosen.(1) Choose ε > small such that s ru,ε ′ is defined for any < ε ′ < ε and any u ∈ f M ( p, r ; A ) . Then choose ℓ and V .(2) Choose ε > small such that s qv,ε ′ and s qw,ε ′ are defined for any < ε ′ <ε and any v ∈ f M ( r, q ; A ) and w ∈ f M ( p, q ; A + A ) . Then choose ℓ and V .(3) Choose ε ′ > such that < ε ′ ≪ ε and then σ > small such that s rw,ε ′ and s rw, − ε ′ are defined for any w ∈ e G σ ( V , V ) . Here σ can be chosen to beindependent of ℓ and ℓ , but we may need to shrink V i satisfying K i ⊆ V i for i = 0 , . This is because if w is close to breaking into V × V and V × V is a sufficiently small neighborhood of K × K , then w is closeto breaking into K × K , which is ℓ , ℓ -independent. The neck length nl ( w ) is then given by s rw, − ε ′ − s rw,ε ′ .(4) Define V (1 , ⊂ G σ ( V , V ) .(5) Choose a compact K ⊆ M ( p, q ; A + A ) such that if [ w ] ∈ M ( p, q ; A + A ) − K , then [ w ] is σ -close to breaking into V × V , i.e., [ w ] ∈G σ ( V , V ) .(6) Pick L > large and ε ′′ > small, and enlarge K if necessary so that K ∩ V (1 , ⊇ { [ w ] ∈ M ( p, q ; A + A ) | L − ε ′′ < nl ([ w ]) < L} . (7) Choose ℓ ∈ N such that E ℓ | K is transverse to ∂ . QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 13 (8) With the choice of ℓ , we may need to increase ℓ so that ℓ = ℓ . By (3)this update does not affect σ , L , ε ′′ . To reduce the number of constants, wealso choose to update ℓ so that ℓ = ℓ = ℓ .(9) Trim V and V (1 , so that (C ) and (C (1 , ) are satisfied.(10) The constant < ε ′′′ ≪ ε ′′ is as defined in Definition 2.5.3. Then weinterpolate between the sections s and s (1 , .2.5.3. General case.
In general, we construct boundary semi-global Kuranishicharts by induction on the energy.
Ordering the moduli spaces.
We will explain how to order the moduli spaces M ( p, q ; A ) . We remark that even if M ( p, q ; A ) = ∅ , we still need to constructa Kuranishi chart for M ( p, q ; A ) (i.e., include M ( p, q ; A ) in our list), if A = A + A for some A ∈ π ( p, r ) and A ∈ π ( r, q ) , and M ( p, r ; A ) = ∅ and M ( r, q ; A ) = ∅ .We define M = { ( p, q, A ) | p, q ∈ L ∩ L and A ∈ π ( p, q ) } , M = { ( p, q, A ) ∈ M | M ( p, q ; A ) = ∅} ,λ = inf { ω ( A ) | ( p, q ; A ) ∈ M } , and M = { ( p, q ; A ) ∈ M | ω ( A ) = λ } . Suppose that M k has been inductively defined for all k < m . We then define M m + to be the set of ( p, q, A ) ∈ M \ (cid:0) ∪ m − i =1 M i (cid:1) such that either(1) M ( p, q ; A ) = ∅ ; or(2) There exist r ∈ L ∩ L , A ∈ π ( p, r ) and A ∈ π ( r, q ) satisfying(2a) A = A + A ,(2b) ( p, r ; A ) ∈ M k for some k < m , and(2c) ( r, q ; A ) ∈ M k for some k < m .Let λ m = inf { ω ( A ) | ( p, q, A ) ∈ M m + } , and M m = { ( p, q, A ) ∈ M m + | ω ( A ) = λ m } . By Gromov compactness, one can see that for each k , M k is finite, and { λ k | k ∈ N } ⊂ R ≥ is nowhere dense.We then order the elements of ∪ ∞ k =1 M k as(2.5.2) ( p , q , A ) , ( p , q , A ) , . . . so that it is consistent with the ordering M , M , . . . . We denote M i := M ( p i , q i ; A i ) . We choose an increasing sequence N , N , · · · → ∞ of integers and for each j we construct a semi-global Kuranishi structure K ( j ) using M , . . . , M N j and asection S ( j ) of K ( j ) . Later we will explain how to relate K ( j ) and K ( j +1) and theirsections. For the moment we choose N ≫ and such that the finite set {M , . . . , M N } of moduli spaces is G -invariant. Define the source , target , and homotopy class maps s ( M ( p, q ; A )) = p, t ( M ( p, q ; A )) = q, h ( M ( p, q ; A )) = A. Definition 2.5.4.
A tuple I = ( i , . . . , i k ) with i j ∈ { , , . . . , ρ } is called an index tuple if ω ( h ( M i j )) > for all j and t ( M i j ) = s ( M i j +1 ) for all j < k .If k = 1 , sometimes we write i instead of ( i ) . Definition 2.5.5.
Let I = ( i , i , . . . , i k ) be an index tuple.(1) An index tuple I ′ is a simple contraction of I if I ′ is obtained by replacinga consecutive pair i j , i j +1 by i ′ j such that h ( M i ′ j ) = h ( M i j )+ h ( M i j +1 ) . (2) An index tuple I ′ is a contraction of I if I ′ is obtained from I by a non-empty sequence of simple contractions. We write I ′ < I .(3) We write c ( I ) for the index tuple ( i ′ ) such that ( i ′ ) ≤ I (i.e., ( i ′ ) < I or ( i ′ ) = I ).(4) Given I ′ = ( i ′ , . . . , i ′ k ′ ) < I , the blocks of I relative to I ′ are groupings ( i , . . . , i l ) , ( i l +1 , . . . , i l ) , . . . , ( i l k ′− +1 , . . . , i l k ′ ) such that c applied to the j th block yields i ′ j . Note that the blocks are well-defined due to the requirement ω ( h ( M i ′ j ′ )) > for all j ′ ∈ { , , . . . , k ′ } (5) Given I ′ = ( i ′ , . . . , i ′ k ′ ) < I , let δ ( I, I ′ ) = { i , . . . , i l − , i l +1 , . . . , i l − , . . . , i l k ′− +1 , . . . , i l k ′ − } , where we are using block notation from (4).We can organize the set of index tuples as a category I , called the index tuplecategory , with objects which are index tuples and a unique morphism from I ′ to I if I ′ ≤ I .Let K i ⊆ M i be the large compact subsets over which we construct the equi-variant interior semi-global Kuranishi chart C i = ( K i , E i → V i , ∂ i , ψ i ) and theobstruction section s i .Let I = ( i , . . . , i k ) . The following construction of the boundary chart ( π I : E I → V I , ∂ I , ψ I ) is a straightforward generalization of Section 2.5.1: Let e G σ ( V i , . . . , V i k ) be theset of maps u that are σ -close to a broken strip in V i × · · · × V i k , defined in amanner analogous to Definition 2.5.1. For convenience we will also write e G σ ( V i ) for the set of maps u that are σ -close to a map in V i . Again we take L = L ( ε ′ , σ ) and ε ′′ > . QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 15
Definition 2.5.6 (Neck length) . Let u ∈ ∪ c ( i ,...,i k )=( i ) e G σ ( V i , . . . , V i k ) .(1) The neck length function satisfies nl ′ ( i ′ ,i ′ ) ( u ) = s ru, − ε ′ − s ru,ε ′ if u ∈ e G σ ( V i , . . . , V i k ) , ( i ) < ( i ′ , i ′ ) < ( i , . . . , i k ) , and r = t ( M i ′ ) = s ( M i ′ ) .(2) The modified neck length function satisfies nl ( i ′ ,i ′ ) ( u ) = (cid:26) λ ( nl ′ ( i ′ ,i ′ ) ( u )) if nl ′ ( i ′ ,i ′ ) ( u ) is defined; otherwise,where λ : R + → R ≥ is a smooth function such that λ ( x ) = x for x ≥L − ε ′′ , λ ( x ) = 0 for x ≤ L − ε ′′ , and λ ′ ( x ) > on ( L − ε ′′ , L − ε ′′ ) .We also write nl i j ( u ) = nl ( i ′ ,i ′ ) ( u ) if c ( i , . . . , i j ) = i ′ .We then define the boundary charts π I : E I → V I , I = ( i , . . . , i k ) , whosefibers are canonically identified with e r ,ℓ ⊕ · · · ⊕ e r k ,ℓ and such that:(C I ) V I ∩ M c ( I ) consists of [ u ] ∈ G σ ( V i , . . . , V i k ) ∩ M c ( I ) satisfying(a) nl i j ([ u ]) > L − ε ′′ for all j < k and(b) nl i ′′ j ([ u ]) < L for all i ′′ j ∈ δ ( I ′′ , I ) where I < I ′′ = ( i ′′ , . . . , i ′′ k ′′ ) .The section ∂ I is the ∂ -operator restricted to V I and ψ I : ∂ − I (0) → V I is theobvious inclusion.Observe that G acts on the set of index tuples. Let G I ⊂ G be the stabilizer of I . By trimming V I if necessary, we may assume that G I acts on E I → V I .Next we discuss the restriction-inclusion morphism φ I ′ ,I : ( π I ′ : E I ′ → V I ′ ) → ( π I : E I → V I ) , where I ′ = ( i ′ , . . . , i ′ k ′ ) < I = ( i , . . . , i k ) . We first restrict E I ′ → V I ′ to E I ′ ,I := E I ′ | V I ′ ,I → V I ′ ,I := V I ′ ∩ {L − ε ′′ < nl i j ([ u ]) < L , ∀ i j ∈ δ ( I, I ′ ) } . We then consider the inclusion of vector bundles given by the commutative diagram E I ′ ,I E I V I ′ ,I V Iφ ♯I ′ ,I φ ♭I ′ ,I where φ ♭I ′ ,I : V I ′ ,I → V I is the inclusion and the bundle map φ ♯I ′ ,I is defined bycanonically identifying the fibers of E I ′ and E I with e r ′ ,ℓ ⊕ · · · ⊕ e r ′ k ′ ,ℓ and e r ,ℓ ⊕ · · · ⊕ e r k ,ℓ , and including e r ′ ,ℓ ⊕ · · · ⊕ e r ′ k ′ ,ℓ ⊂ e r ,ℓ ⊕ · · · ⊕ e r k ,ℓ . Here r j = t ( M i j ) and r ′ j = t ( M i ′ j ) . We have (a) φ ♯I ′ ,I ◦ ∂ I ′ = ∂ I ◦ φ ♭I ′ ,I on V I ′ ,I ; and(b) ψ I ◦ φ ♭I ′ ,I = ψ I ′ on ∂ − I ′ (0) ∩ V I ′ ,I .For I = ( i , . . . , i k ) we set(2.5.3) s I = ( ζ ( nl i ) s r , . . . , ζ ( nl i k − ) s r k − , s r k ) ∈ e r ,ℓ ⊕ · · · ⊕ e r k ,ℓ , where the function ζ is as given in Definition 2.5.3. Denote s I ′ ,I := s I ′ | V I ′ ,I . It isimmediate that φ ♯I ′ ,I ◦ s I ′ ,I = s I ◦ φ ♭I ′ ,I and ∂ − I ( s I ) ∩ φ ♭I ′ ,I ( V I ′ ,I ) = φ ♭I ′ ,I ( ∂ − I ′ ( s I ′ ,I )) . Gluing.
The following gluing results can be proven in a manner similar toTheorems 6.4.1, 6.4.2 in [BH2] (in the contact case) and Theorem A.21 in [ES].
Theorem 2.6.1 (Gluing) . For sufficiently large
R > , there exists a gluing map (2.6.1) G ( i ,...,i m ) : V i × · · · × V i m × ( R, ∞ ) m − → V ( i ,...,i m ) which satisfies the following: Writing T , . . . , T m − for the coordinates on ( R, ∞ ) m − ,(1) G ( i ,...,i m ) is a C -diffeomorphism onto its image;(2) Im G ( i ,...,i m ) ⊃ V ( i ,...,i m ) ∩ { nl i j ≥ R + ε ′′ , ∀ j < m } ;(3) G ( i ,...,i m ) ([ u i ] , . . . , [ u i m ] , T , . . . , T m − ) is σ -close to the broken strip ([ u i ] , . . . , [ u i m ]) for some σ > (in the C -topology; see Definition 2.5.1);(4) for j = 1 , . . . , m − , the functions ( G ( i ,...,i m ) ) ∗ T j and nl i j are C -close;(5) ∂ ( G ( i ,...,i m ) ([ u i ] , . . . , [ u i m ] , T , . . . , T m − )) and ( ∂u i , . . . , ∂u i m ) , viewedas elements of e r ,ℓ ⊕ · · · ⊕ e r k ,ℓ , r j = t ( M i j ) , are C -close;(6) the errors in (3), (4), and (5) go to zero as all T j → ∞ . Theorem 2.6.2 (Iterated gluing) . For sufficiently large
R > , there is a gluingmap G ( i ,..., ( i a ,...,i b ) ,...i m ) : V i × · · · × V i a − × V ( i a ,...,i b ) × V i b +1 × · · · × V i m × ( R, ∞ ) m − ( b − a ) − → V ( i ,...,i m ) , satisfying properties analogous to those of Theorem 2.6.1 and such that G ( i ,...,i m ) and G ( i ,..., ( i a ,...,i b ) ,...i m ) ◦ (id , . . . , G ( i a ,...,i b ) , . . . , id) are C -close with error → as all the coordinates of ( R, ∞ ) m − ( b − a ) − go to ∞ . Equivariant semi-global Kuranishi structures.
The Kuranishi charts con-structed in Section 2.4 and 2.5 can be organized into a G -invariant semi-globalKuranishi structure. Again, for the moment we work with the G -invariant finite set {M , . . . , M N } of moduli spaces. Our definition is similar to McDuff-Wehrheim’s treatment of Kuranishi struc-tures (called atlases ) in [MW]. (1)–(3) are general properties of Kuranishi struc-tures/atlases and (4) and (5) are specific “semi-global” properties.
QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 17
Definition 2.7.1 (Semi-global Kuranishi structure) . A semi-global Kuranishi struc-ture K is a category consisting of the following data:(1) The objects are semi-global Kuranishi charts C I = ( π I : E I → V I , ∂ I , ψ I ) :(a) for each i , C i = ( π i : E i → V i , ∂ i , ψ i ) is an interior Kuranishi chartfor K i ⊂ M i ;(b) for each I = ( i , . . . , i m ) , π I : E I → V I is a finite rank vectorbundle over a finite-dimensional manifold, ∂ I : V I → E I is a section, ψ I : ∂ − I (0) → M c ( I ) is a homeomorphism onto an open subset of M c ( I ) , and dim V I − rk E I = vdim M c ( I ) ; and(c) for each i , ∪ c ( I )=( i ) Im( ψ I ) = M i .(2) For each I ′ ≤ I there is a specified morphism φ I ′ ,I : C I ′ → C I encoded bythe data ( V I ′ ,I , φ ♯I ′ ,I , φ ♭I ′ ,I ) and given by restriction-inclusion: first restrict E I ′ → V I ′ to an open subset V I ′ ,I ⊂ V I ′ and then take the inclusion ofvector bundles given by a commutative diagram E I ′ ,I := E I ′ | V I ′ ,I E I V I ′ ,I V I , φ ♯I ′ ,I φ ♭I ′ ,I subject to:(a) φ ♯I ′ ,I ◦ ∂ I ′ = ∂ I ◦ φ ♭I ′ ,I on V I ′ ,I ;(b) ψ I ◦ φ ♭I ′ ,I = ψ I ′ on ∂ − I ′ (0) ∩ V I ′ ,I ;(c) ( ∂ I ) ∗ : T V I → E I descends to an isomorphism T V I / ( φ ♭I ′ ,I ) ∗ ( T V I ′ ,I ) ∼ −→ E I / E I ′ . (3) The composition of morphisms is defined so that φ I ′′ ,I = φ I ′′ ,I ′ ◦ φ I ′ ,I .The following are strata compatibility conditions :(4) (Neck length functions) For each ( i ) < ( i ′ , i ′ ) , there exists a smooth(modified) neck length function nl ( i ′ ,i ′ ) : ∪ c ( I ′′ )=( i ) e G σ ( V i ′′ , . . . , V i ′′ k ) → R ≥ such that V I ′ ,I := { [ u ] ∈ V I ′ | L − ε ′′ ≤ nl ( c ( i ,...,i j ) ,c ( i j +1 ,...,i k )) ([ u ]) ≤ L , ∀ i j ∈ δ ( I, I ′ ) } . (5) For each I = ( I , . . . , I m ) there exists a C -bundle map ( e G I , G I ) : pr ∗ I E I ⊕ · · · ⊕ pr ∗ I m E I m E I V I × · · · × V I m × ( R, ∞ ) m − V I , e G I G I Here we abuse notation and refer both ( I , . . . , I m ) and ( i , . . . , i j , . . . , i m , . . . , i mj m ) by I , where I k = ( i k , . . . , i kj k ) . ( ∞ , ∞ ) nl nl L L − ε ′′ LL − ε ′′ V (1 , , V (4 , V (1 , V F IGURE
2. Corner structure. Suppose that c (1 ,
2) = 4 , c (2 ,
3) =5 , c (4 ,
3) = 6 = c (1 , .where R ≫ , pr I k : V I ×· · ·× V I m × ( R, ∞ ) m − → V I k is the projectionmap, T j is the coordinate for the j th ( R, ∞ ) factor, and(a) G I is a C -diffeomorphism onto its image;(b) Im G I ⊃ V I ∩ { nl ( c ( I ,...,I j ) ,c ( I j +1 ,...,I m )) ≥ R + ε ′′ , ∀ j < k } ;(c) G I ([ u I ] , . . . , [ u I m ] , T , . . . , T m − ) is close to the broken strip ([ u I ] , . . . , [ u I m ]) ;(d) for j = 1 , . . . , m − , the functions ( G I ) ∗ T j and nl ( c ( I ,...,I j ) ,c ( I j +1 ,...,I m )) are C -close;(e) e G I ◦ ( ∂ I , · · · , ∂ I m ) and ∂ I ◦ G I are C -close;(f) the errors of (d) and (e) go to zero as T j → ∞ for all j = 1 , . . . , m − ;(g) G ( I ,...,I m ) and G ( I ,..., ( I a ,...,I b ) ,...I m ) ◦ (id , . . . , G ( I a ,...,I b ) , . . . , id) are C -close with error → as T j → ∞ for all j = 1 , . . . , m − ( b − a ) − .We say that K is G -invariant if, for each g ∈ G , g induces an isomorphism ( V I → E I ) ∼ −→ ( V g ( I ) → E g ( I ) ) such that ∂ I , ψ I , nl ( i ′ ,i ′ ) , G I are taken to ∂ g ( I ) , ψ g ( I ) , nl g ( i ′ ,i ′ ) , G g ( I ) .A section of K is a collection { s I : V I → E I } I of obstruction sections suchthat:(1) φ ♯I ′ ,I ◦ s I ′ ,I = s I ◦ φ ♭I ′ ,I , where s I ′ ,I := s I | V I ′ ,I ;(2) ∂ − I ( s I ) ∩ φ ♭I ′ ,I ( V I ′ ,I ) = φ ♭I ′ ,I ( ∂ − I ′ ( s I ′ ,I )) ;(3) for each I = ( i , · · · , i m ) , e G I ◦ ( s i , · · · , s i m ) and s I ◦ G I are C -closeand the error goes to as T j → ∞ for all i = 1 , · · · , m − . Remark . There is no reason to expect the sections { s I } I to be G -invariant.This will be treated in Section 4.1.One can also view K as a functor from the index tuple category I to the “cate-gory of Kuranishi charts”. QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 19
Let K ( M i ) (also written as K ( p, q ; A ) if M i = M ( p, q ; A ) ) be the full sub-category of K with objects I such that c ( I ) = i .Given a section S = { s I } c ( I )=( i ) of K ( M i ) , we define Z ( K ( M i ) , S ) = a c ( I )=( i ) ∂ − I ( s I ) / ∼ K , where ∼ K is the identification given by the morphisms.We now come to an important point: There is no reason to expect Z ( K ( M i ) , S ) for an abstract semi-global Kuranishi structure to be a manifold, i.e., the Hausdorffproperty is not automatic. However, in our case the existence of the neck lengthfunctions implies the following analog of [BH2, Lemma 8.8.1]:
Lemma 2.7.3. Z ( K ( M i ) , S ) is a manifold.Proof. Same as that of [BH2, Lemma 8.8.1]. (cid:3)
Taking limits.
So far we have constructed a semi-global Kuranishi structure K anda section S for the G -invariant finite set {M , . . . , M N } . If {M , M , . . . } isinfinite, we choose increasing sequences N , N , · · · → ∞ and ℓ , ℓ , · · · → ∞ ofintegers such that {M , . . . , M N j } is G -invariant for each N j and construct K ( j ) such that the fibers of the obstruction bundles E ( j ) i → V ( j ) i (i.e., E i → V i for j )are e q,ℓ j , where q = t ( M i ) . Since e q,ℓ j naturally includes into e q,ℓ j +1 , there arenatural inclusions E ( j ) I E ( j +1) I V ( j ) I V ( j +1) I that commute with the morphisms φ ( j ) I ′ ,I and φ ( j +1) I ′ ,I . Assuming we have alreadyconstructed the section S ( j ) , we construct S ( j +1) such that s ( j +1) I is the image of s ( j ) I under the appropriate inclusions ⊕ q e q,ℓ j → ⊕ q e q,ℓ j +1 whenever the entriesof I are ≤ N j . This is sufficient to ensure that, for i ≤ N j , there is a naturalidentification Z ( K ( j ) ( M i ) , S ( j ) ) ≃ Z ( K ( j +1) ( M i ) , S ( j +1) ) . We write Z ( K ( M i ) , S ) for any of the Z ( K ( j ) ( M i ) , S ( j ) ) such that i ≤ N j . From now on we will assume {M , M , . . . } is finite, making the appropriatemodifications as above, if it is not.Implicit charts. Our semi-global Kuranishi structure K ( M i ) , M i = M ( p i , q i ; A i ) ,can be converted into a single global implicit chart in the sense of Pardon [Pa]. Let S ( p i , q i ) be the set of all the q j that appear before ( p i , q i , A i ) in the list (2.5.2) andtake the global fiber to be e ( p i , q i ) := ⊕ r ∈S ( p i ,q i ) e r,ℓ . We consider solutions ( u, ξ ) , u ∈ ∪ c ( i ,...,i k )=( i ) e G σ ( V i , . . . , V i k ) , ξ = ( ξ r ) r ∈S ( p i ,q i ) ∈ e ( p i , q i ) , to the equation(2.7.1) ∂u = X r ∈S ( p i ,q i ) ( ζ ◦ nl r ( u )) · ξ r , where ζ and nl r are as given in Definitions 2.5.3 and 2.5.6. Roughly speaking, weturn off the perturbations for e r,ℓ when nl r ( u ) ≤ L but still remember the data for e r,ℓ .2.8. Equivariant semi-global Kuranishi structures for chain maps and chainhomotopies.
Chain maps.
Let H s : M → R , s ∈ [0 , , be a compactly supported,time-dependent, G -invariant Hamiltonian function and let φ s , s ∈ [0 , , be thecorresponding -parameter family of Hamiltonian symplectomorphisms of ( M, ω ) with φ = id ; we call such a φ s a G -equivariant Hamiltonian isotopy . Writing L ′ i = φ ( L i ) , i = 0 , , we assume that L ′ ⋔ L ′ . Let { J s } s ∈ [0 , be a -parameterfamily of almost complex structures that are ω -compatible, G -invariant, and satisfy (J) .Define a smooth function ϑ : R → [0 , such that ϑ ( s ) = 0 for s ≤ and ϑ ( s ) = 1 for s ≥ .Given p ∈ L ∩ L , q ∈ L ′ ∩ L ′ , and A ∈ π ( p, q ) , let M ◦ ( p, q ; A ) (we aresuppressing { J s } ) be the space of smooth maps u : R × [0 , → M that satisfy(A3) and (A4), in addition to:(A1 ′ ) u s ( s, t ) + J ϑ ( s ) ( u ( s, t )) u t ( s, t ) = 0 , and(A2 ′ ) u ( s, ∈ φ ϑ ( s ) ( L ) and u ( s, ∈ φ ϑ ( s ) ( L ) .When we are defining chain maps and chain homotopies, the moduli spaces for ( L , L ) will have superscripts − as in M − ( p, p ′ ; A ) and the moduli spaces for ( L ′ , L ′ ) will have superscripts + as in M + ( q, q ′ ; A ) .The construction of the Kuranishi charts and the Kuranishi structure from Sec-tions 2.4 to 2.7 carry over with very few modifications: Under our assumptionsthere are finitely many moduli spaces of type M ◦ ( p, q ; A ) , M − ( p, q ; A ) , and M + ( p, q ; A ) , which we list as M , . . . , M ρ as before so that ω ( A ) is in nondecreasing order. The type of M i is given by thesuperscript ◦ , − , or + . Definition 2.8.1.
A tuple I = ( i , . . . , i k ) is a c -index tuple (where c stands forchain map), if it satisfies the conditions of Definition 2.5.4 and • there exists i j such that M i j has type ◦ and all i l with l < j have type − and all i l with l > j have type + . QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 21
The charts ( π I : E I → V I , ∂ I , ψ I ) are constructed in exactly the same way asbefore, where I is now a c -index tuple. By construction the Kuranishi structure is G -invariant.2.8.2. Chain homotopies.
Fix T ≫ . We define a smooth function Θ : R × [0 , → [0 , with coordinates ( s, τ ) for R × [0 , such that: • Θ( s,
0) = 1 for s ∈ [ − T + 1 , T − ; • Θ( s,
1) = 0 for all s ; • Θ( s, τ ) = 0 for all s > T and s < − T and τ ∈ [0 , .For each τ ∈ [0 , , let M ◦ τ ( p, q ; A ) be the space of smooth maps u : R × [0 , → M that satisfy (A3), (A4),(A1 τ ) u s ( s, t ) + J Θ( s,τ ) ( u ( s, t )) u t ( s, t ) = 0 , and(A2 τ ) u ( s, ∈ φ Θ( s,τ ) ( L ) and u ( s, ∈ φ Θ( s,τ ) ( L ) .We also write M ◦{ τ } ( p, q ; A ) = a τ ∈ [0 , M ◦ τ ( p, q ; A ) . For each c -index tuple and each τ ∈ [0 , we construct a chart ( π I,τ : E I,τ → V I,τ , ∂
I,τ , ψ
I,τ ) , which can be combined into a family ( π I, [0 , : E I, [0 , → V I, [0 , , ∂ I, [0 , , ψ I, [0 , ) . By construction the family of Kuranishi structures is G -invariant.3. O RIENTATIONS
The goal of this section is to review the definition of a coherent (= compati-ble with gluing) system of orientations on the moduli space of (finite energy) J -holomorphic strips for a pair ( L , L ) of Lagrangians, following [FO3] and thenadapt it to the case with a G -action. We will see that in general, g ∈ G onlypreserves the orientation of M ( p, q ; A ) up to a sign σ ( g, p, q ) ∈ {− , } that isindependent of A . But this is enough to define a G -action on the CF • ( L , L ) . Cauchy-Riemann tuples. A Cauchy-Riemann tuple is a quadruple (Σ , ξ, η, D ) satisfying (CR1)–(CR4):(CR1) Σ = B \ X , where B is the closed unit disk in C and X is a finite subset of ∂B .For each x ∈ X , let I x ⊂ ∂B be a small interval neighborhood of x and let I x − and I x + be the two connected components of I x \ x .(CR2) ξ is a trivial C -vector bundle over Σ = B .(CR3) η is a real subbundle of ξ | ∂ Σ − X such that η | I x ± extends smoothly to a realsubspace η x ± ⊂ ξ x over x . Moreover, ξ x = η x + ⊕ η x − . Let
Γ(Σ , ξ ) be the space of compactly supported smooth sections of ξ | Σ that restrictto sections of η along ∂ Σ \ X . For each x ∈ X , choose a neighborhood N ( x ) ⊂ B and a holomorphic identification of Σ ∩ N ( x ) with a strip-like end [0 , ∞ ) × [0 , with coordinates ( s, t ) . Let W k +1 ,p (Σ , ξ ) be the closure of Γ(Σ , ξ ) in the W k +1 ,p -norm with respect to a metric on Σ consistent with the strip-like ends and a metricon ξ . The space W k,p (Σ , ∧ , Σ ⊗ C ξ ) is defined similarly.(CR4) The operator D : W k +1 ,p (Σ , ξ ) → W k,p (Σ , ∧ , Σ ⊗ C ξ ) is a real-linearCauchy-Riemann operator such that on each strip-like end Dw = ( ∇ s w + J ∇ t w ) ⊗ ( ds + idt ) , where J is the complex structure on ξ and ∇ is a connection of ξ .See [MS2, Appendix C] for the definition of a real-linear Cauchy-Riemann opera-tor over a compact Riemann surface.3.2. Auxiliary orientation data.
Recall the determinant line of (Σ , ξ, η, D ) is a -dimensional vector space defined by det D := ∧ top ker D ⊗ R ∧ top (coker D ) ∗ . Let π : E ( p, q ; A ) → V ( p, q ; A ) be an interior semi-global Kuranishi chart for M ( p, q ; A ) . Given u with [ u ] ∈ V , we define the Cauchy-Riemann tuple (Σ u , ξ u , η u , D u ) := ( S, u ∗ T M, ⊔ i ∈{ , } u ( · , i ) ∗ T L i , D u ) , where S = R × [0 , and D u is the linearized ∂ -operator at u .A coherent system of orientations o ( D u ) of det D u will depend on the following auxiliary orientation data ; see Theorem 3.4.1. Definition 3.2.1.
A choice of auxiliary orientation data consists of:(O1) a relative spin structure for the pair ( L , L ) ;and for each p ∈ L ∩ L ,(O2) a capping Lagrangian path;(O3) a capping orientation; and(O4) a stable capping trivialization.We will explain (O2) and (O3), leaving (O1) and (O4) for the next subsection.A capping Lagrangian path (O2) is a path {L p,t } ≤ t ≤ in the oriented La-grangian Grassmannian Lag( T p M, ω p ) such that L p,i = T p L i with orientations,for i = 0 , .For each p ∈ L ∩ L , we define a Cauchy-Riemann tuple (Σ p + , ξ p + , η p + , D p + ) as follows: Let Σ p + be the closed unit disk in C with one boundary puncture,identified with the upper half plane H = { z | Im z ≥ } , and let π p : Σ p + → M be the constant map to p . We then define: • ξ p + = π ∗ p ( T p M ) , • η p + z = L p, for z ∈ ( −∞ , , η p + z = L p,z for z ∈ [0 , , and η p + z = L p, for z ∈ (1 , + ∞ ) , and • D p + is a fixed real linear Cauchy-Riemann operator (the choice is uniqueup to homotopy). QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 23
We can similarly choose the Cauchy-Riemann tuple (Σ p − , ξ p − , η p − , D p − ) by swap-ping the roles of L and L .Finally, a capping orientation (O3) is a choice of orientation o ( D p + ) (but not o ( D p − ) ).3.3. Relative spin structures.
A pair ( L , L ) is relatively spin if there exists st ∈ H ( M ; Z / such that w ( T L i ) = ι ∗ i st for i = 0 , : Fix a triangulation τ of M such that L , L , and L ∩ L are subcomplexes. Choose an oriented realvector bundle V of rank ≥ on the -skeleton M (3) of M such that w ( V ) = st .(Here we are using the notation X ( i ) for the i -skeleton of a triangulation of X .)Then the bundle T L i | L (2) i ⊕ V | L (2) i is spin and hence is a trivial bundle. Choosinga spin structure is equivalent to choosing a homotopy class of trivializations t i of T L i | L (1) i ⊕ V | L (1) i that extends to L (2) i . Since π ( SO ( m )) = 0 for m ≥ , theextension to L (2) i is unique and t i also extends to L (3) i . We will refer to choices of τ , V , and homotopy classes of t i , i = 0 , , as a relative spin structure ; see [FO3,Section 8.1] for an explanation of when two relative spin structures are equivalent.A more algebraic (and cleaner) definition of a relative spin structure can be foundin [WW, Section 3.1] and [Sc].Let e L p → [0 , be a vector bundle whose fiber over t ∈ [0 , is L p,t ⊕ V p . Thena stable capping trivialization (O4) is a trivialization e t p of e L p that agrees with thetrivializations t i of ( T L i | L (1) i ⊕ V | L (1) i ) | p that we have already chosen for i = 0 , .3.4. Coherent orientation system.
We review the following theorem from [FO3,Section 8.1]:
Theorem 3.4.1.
The moduli space of J -holomorphic strips admits a coherent ori-entation system if the pair of Lagrangians ( L , L ) is relative spin. Moreover, thechoice of auxiliary orientation data (O1)–(O4) determines the orientation. We give a sketch of the proof, partly to establish notation. The fundamental factthat we use is the following (cf. [FO3, Proposition 34.3]), stated without proof.
Fact 3.4.2.
Given a Cauchy-Riemann tuple (Σ , C n , η, D ) , if Σ has no puncturesand η is trivial, then any trivialization of η canonically determines an orientationof det D .Sketch of proof of Theorem 3.4.1.Step 1. Let (Σ , ξ , η , D ) and (Σ , ξ , η , D ) be two Cauchy-Riemann tuples.Given punctures x ∈ ∂ Σ and x ∈ ∂ Σ , suppose there is a C -linear isomor-phism Φ : ξ x ∼ −→ ξ x that maps η x ± to η x ∓ . Then there is an associatedCauchy-Riemann tuple (Σ , , ξ , , η , , D , ) defined by a straightforward preglu-ing which identifies x and x and the orientations of det D and det D inducean orientation of det D , .In particular, if we preglue (Σ q + , ξ q + , η q + , D q + ) and (Σ q − , ξ q − , η q − , D q − ) , weobtain the Cauchy-Riemann tuple (Σ q + ,q − , ξ q + ,q − , η q + ,q − , D q + ,q − ) and it has acanonical orientation by Fact 3.4.2. (Here we are taking the trivializations of η q + and η q − to come from the same trivialization of L p ; then the trivialization of η q + ,q − is independent of the choice of trivialization of L p .) Hence the capping orientation o ( D q + ) determines o ( D q − ) .For any u with [ u ] ∈ V ( p, q, A ) , we preglue (Σ p + , ξ p + , η p + , D p + ) , (Σ u , ξ u , η u , D u ) , (Σ q − , ξ q − , η q − , D q − ) along p and q to obtain (Σ p + ,u,q − , ξ p + ,u,q − , η p + ,u,q − , D p + ,u,q − ) . If we can orient det( D p + ,u,q − ) , then o ( D u ) is determined by o ( D p + ,u,q − ) and thecapping orientations o ( D p + ) and o ( D q − ) . Step 2.
By the simplicial approximation theorem, after a homotopy we can assumethat u (Σ) ⊆ M (2) and u ( ∂ Σ) ⊆ L (1)0 ∪ L (1)1 . Let V → M (3) be the bundleappearing in the definitions of (O1) and (O4).Define Cauchy-Riemann tuples (Σ p + , ξ p + V , η p + V , D p + V ) , (Σ u , ξ uV , η uV , D uV ) , (Σ p − , ξ p − V , η p − V , D p − V ) in the same way as the versions without V , except that we replace T M by V ⊕ iV , T L i by V for i = 0 , , and L p,t by V p . By pregluing as in Step 1, we obtain (Σ p + ,u,q − , ξ p + ,u,q − V , η p + ,u,q − V , D p + ,u,q − V ) . A key point to observe now is that, since V is oriented and defined over u (Σ) , thereis a canonical equivalence class of trivializations of η p + ,u,q − V and hence a canonicalorientation of det D p + ,u,q − V by Fact 3.4.2.We take the direct sum of ( ξ p + ,u,q − , η p + ,u,q − , D p + ,u,q − ) and ( ξ p + ,u,q − V , η p + ,u,q − V , D p + ,u,q − V ) , over Σ p + ,u,q − to obtain (Σ p + ,u,q − , ξ p + ,u,q − ⊕ ξ p + ,u,q − V , η p + ,u,q − ⊕ η p + ,u,q − V , D p + ,u,q − ⊕ D p + ,u,q − V ) . Now t i , e t p , and e t q give a trivialization of η p + ,u,q − ⊕ η p,u,q − V , so det( D p + ,u,q − ⊕ D p + ,u,q − V ) is canonically oriented by Fact 3.4.2. Since det( D p + ,u,q − ⊕ D p + ,u,q − V ) iscanonically isomorphic to det D p + ,u,q − ⊗ det D p + ,u,q − V and det D p + ,u,q − V is canon-ically orientated, we obtain a canonical orientation of det D p + ,u,q − . Step 3.
It remains to show that o ( D u ) is independent of the choices. We refer thereader to [FO3, Section 8.1] for a proof. (cid:3) Since det D u is canonically isomorphic to det D u , where D u is the linearizedoperator of ∂ J : V ( p, q ; A ) → E ( p, q ; A ) , a choice of auxiliary orientation datainduces a system of orientations on (Λ top E I ) ∗ ⊗ Λ top T V I . Next we study orientations under the group action. To do that, we first need toallow G to act on the obstruction bundle. QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 25
Orientations on e r,ℓ .Lemma 3.5.1. If ℓ is an even multiple of n , then e r,ℓ admits a canonical G -invariant orientation.Proof. Without loss of generality, we assume:(i) T r M ≃ R n ⊕ i R n = C n , where T r L is the R n factor and T r L is the i R n factor;(ii) J ( r ) = J is the standard complex structure that takes v ∈ R n to iv ∈ i R n and g r is the standard Euclidean structure on T r M ; and(iii) G leaves T r L invariant. Since G is compatible with J and g , it can bedescribed by a representation ρ : G → O ( R n ) .The asymptotic operator A is given by − J ∂∂t with boundary conditions R n at t = 0 and i R n at t = 1 . For each k = 0 , , . . . , there are n eigenfunctions ˜ e kj : [0 , → C n , t e i ( πk + π/ t e j , j = 1 , . . . , n, where e , . . . , e n is a basis for R n . Writing ℓ = 2 k n , we choose the orientation(3.5.1) ˜ e ∧ · · · ∧ ˜ e n ∧ · · · ∧ ˜ e k − ∧ · · · ∧ ˜ e k − n for e r,ℓ . Since G acts on each R h ˜ e k , . . . , ˜ e kn i in the same way as on R n using theidentification ˜ e kj e j , for any g ∈ G , g (˜ e ) ∧· · ·∧ g (˜ e n ) ∧· · ·∧ g (˜ e k − ) ∧· · ·∧ g (˜ e k − n ) = ˜ e ∧· · ·∧ ˜ e n ∧· · ·∧ ˜ e k − ∧· · ·∧ ˜ e k − n and G preserves the orientation. Note that the definition in Equation (3.5.1) doesnot depend on the orientation of R n . (cid:3) From now on let us assume that ℓ is an even multiple of n and hence all the e r,ℓ are canonically oriented, so G acts on E I → V I .3.6. Orientations under group action.
Now we study the action of G on theorientation of (Λ top E I ) ∗ ⊗ Λ top T V I . We assume Condition (O) from Section 1, i.e., that the relative spin structure ispreserved under G , whose definition we give presently.Let ( τ, V, t , t ) be a relative spin structure for ( L , L ) . Let τ be a G -equivarianttriangulation of M ; such a triangulation exists by the equivariant triangulation the-orem. Then ( τ, V, t , t ) is preserved by G , if for any g ∈ G , there exists anorientation-preserving bundle isomorphism θ g : V ∼ −→ V such that • π V ◦ θ g = g ◦ π V , where π V : V → M (3) is the projection to the base, and • for each i = 0 , , the trivialization t i : T L i | L (2) i ⊕ V | L (2) i → L (2) i × R d is homotopic to g ♯ t i := t i ◦ ( g ∗ ⊕ θ g | L (2) i ) − , where d = n + rank V and g ∗ ⊕ θ g | L (2) i : T L i | L (2) i ⊕ V | L (2) i → T L i | L (2) i ⊕ V | L (2) i . For p ∈ L ∩ L and g ∈ G , let s = gp . At s , we have the canonical isomorphism(3.6.1) det D s + ⊗ det D s − ⊗ det D s + ,s − V ≃ det( D s + ,s − ⊕ D s + ,s − V ) coming from gluing. Let o ( D p + ) and o ( D s − ) be the capping orientations of det D p + and det D s − and let o ( D s + ,s − V ) be the canonical orientation of D s + ,s − V .Then g o ( D p + ) ⊗ o ( D s − ) ⊗ o ( D s + ,s − V ) determines an orientation of the left-handside of Equation (3.6.1). On the right-hand side of Equation (3.6.1), we have acanonical orientation of det( D s + ,s − ⊕ D s + ,s − V ) coming from the concatenation ofthe stable capping trivializations g e t p and e t s . (The trivializations g e t p and e t s apriori do not agree at the endpoints. We assume that g t i has been homotoped to t i and by abuse of notation we refer to g e t p as the result of applying the homotopy to g e t p .) We compare these two orientations via the isomorphism of Equation (3.6.1),and define σ ( p, g ) ∈ {± } to be the difference. For u with [ u ] ∈ V ( p, q, A ) , let o ( D u ) be the orientation of u determined by the auxiliary orientation data (O1)–(O4). Then one can check that g o ( D u ) = σ ( p, g ) σ ( q, g ) o ( D gu ) .In general, g ∈ G may not preserve the orientation, but we can define the actionof g ∈ G on CF • ( L , L ) by sending [ p ] to σ ( p, g )[ gp ] . In the case when themoduli spaces that we count to define the differential d of CF • ( L , L ) are G -invariant, it is obvious that the G -action on CF • ( L , L ) commutes with d . InSection 4, we see this is true even when the moduli space is not G -invariant. From now on, we fix a choice of auxiliary orientation data (O1)-(O4) such thatthe relative spin structure (O1) is preserved under the G -action. This gives anorientation of (Λ top E I ) ∗ ⊗ Λ top T V I . Since the fiber of E I → V I is canonicallyoriented by Section 3.5, we also get an orientation of V I .4. E QUIVARIANCE OF CURVE COUNTING
Equivariance of curve counting.
Choice of S . We first describe how to choose S = { s I } I to be as G -equivariant aspossible . First decompose L ∩ L into a disjoint union of G -orbits O p , p ∈ L ∩ L .Given O p , pick a generic s p ∈ e p,ℓ which is sufficiently close to the origin and foreach q ∈ O p choose a single g ∈ G such that g ( p ) = q and set s q = g ( s p ) ∈ e q,ℓ .We then choose s i = s p ∈ e p,ℓ , where p = t ( M i ) , and construct s I as describedin Sections 2.4 and 2.5. We additionally assume that:(*) | s i | ≪ | s j | if i > j . Remark . Note that S is not expected to equal g ( S ) for all g ∈ G . If wereplace S by a G -equivariant collection of multisections , the Floer chain groupswill be defined over Q as in Cho-Hong [CH]. Since this leads to some loss ofinformation, we choose to work with collections of sections.The following key theorem makes the equivariant count work. Theorem 4.1.2. If vdim M i = 0 and g ( M i ) = M i , then Z ( K ( M i ) , S ) and Z ( K ( M i ) , g ( S )) are cobordant. QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 27
Proof. If M i = M i , i.e., there is no boundary, then Z ( K ( M i ) , S ) is given bythe preimage of s r , r = t ( M i ) , under the map ∂ i : V i → e r,ℓ . Similarly, Z ( K ( M i ) , g ( S )) is given by the preimage of g ( s r ) . Since ∂ i ( ∂ V i ) does not contain , and S and g ( S ) are sufficiently close to , the two preimagesare cobordant.The main point of the proof is to homotop S to g ( S ) near ∂ M i (i.e., for curvesin V i that are close to breaking) when it is nonempty. In order to simplify thecumbersome notation, let us assume without loss of generality that:(**) M i j = g ( M i j ) for all i j that appears in I = ( i , . . . , i m ) , m ≥ , suchthat c ( I ) = ( i ) and for all g ∈ G ; in particular M i is G -invariant. Step 1.
Given I = ( i , . . . , i m ) such that c ( I ) = ( i ) , consider the composition V i × · · · × V i m × ( R, ∞ ) m − S ( i ,...,im ) −−−−−−−→ V I ∂ I −→ e r ,ℓ ⊕ · · · ⊕ e r m ,ℓ , where r j = t ( M i j ) and in particular r m = r . As R → ∞ , its image approachesthe image of the product map ( ∂ i , . . . , ∂ i m ) : V i × · · · × V i m → e r ,ℓ ⊕ · · · ⊕ e r m ,ℓ . This implies that
Im( ∂ I ◦ S ( i ,...,i m ) ) is effectively Im( ∂ i , . . . , ∂ i m ) . We assumethat the generic point ( s r , . . . , s r m ) ∈ e r ,ℓ ⊕ · · · ⊕ e r m ,ℓ has been chosen to avoid Im( ∂ i , . . . , ∂ i m ) . Note that under our assumption vdim M i = 0 , we have(4.1.1) ( m −
1) + m X j =1 dim V i j = dim V I = m X j =1 dim e r j ,ℓ . Remark . We will see that Z ( K ( M i ) , S ) and Z ( K ( M i ) , g ( S )) are emptysets “near the boundary” unless m = 2 and (vdim M i , vdim M i ) = ( − , or (0 , − .We now continue the proof in steps based on the value of m . Step 2.
Suppose that m = 2 . Step 2A.
Suppose that (vdim M i , vdim M i ) = (0 , − or ( − , . We treat theformer; the latter is analogous. Consider the G -equivariant, codimension one map ∂ i : V i → e r ,ℓ . Let S r ,ℓ − ρ ⊂ e r ,ℓ (resp. B r ,ℓρ ⊂ e r ,ℓ ) be a sphere (resp. an open ball) of radius < ρ ≪ dist( { } , ∂ i ( ∂ V i )) . The action G → GL( e r ,ℓ ) factors through theorthogonal group and hence G acts on S r ,ℓ − ρ . Lemma 4.1.4. If s r ∈ e r ,ℓ is a point such that < | s r | < ρ and s r Im( ∂ i ) ,then for any path γ r : [0 , → B r ,ℓρ from s r to g ( s r ) , the signed intersectionnumber h γ r , ∂ i i between γ r and ∂ i is zero. Os r gs r S r ,ℓ − ρ Im ∂ i γ γ γ F IGURE Proof of Lemma 4.1.4.
We may slightly perturb ρ such that S r ,ℓ − ρ ⋔ ∂ i . Then N := ∂ − i ( S r ,ℓ − ρ ) is a submanifold of V i of dimension ( ℓ − . We homotop γ r to a concatenation γ γ γ , where(1) γ is a slightly perturbed radial ray from s r to a point x ∈ C := S r ,ℓ − ρ − ∂ i N ;(2) γ connects x to g ( x ) on S r ,ℓ − ρ ; and(3) γ = ( g ( γ )) − from g ( x ) to g ( s r ) .See Figure 3. The contributions to γ r ∩ ∂ i from γ and γ cancel, and it remainsto calculate the contribution from γ .There exists a locally constant weight function w : C → Z , such that the val-ues on adjacent connected components differ by ; more precisely, given any twopoints x, x ′ ∈ C , if δ is a path from x to x ′ in S r ,ℓ − ρ and δ intersects ∂ i | N pos-itively and only once, then w ( x ) − w ( x ′ ) = 1 . The existence of such a functionfollows from the existence of the winding number of the map ∂ i | N : N → S r ,ℓ − ρ − { z } ≃ R ℓ − , for any z ∈ C . More precisely, for any x ∈ R ℓ − \ ∂ i ( N ) , w ( x ) is given by thedegree of the mapping from N to R ℓ − \{ x } ∼ = S ℓ − . Any two weight functionsdiffer by an integer-valued constant function (depending on the choice of z ).Next we claim that w = w ◦ g for any g ∈ G . First observe that w ◦ g isalso a weight function. Arguing by contradiction, suppose there is a component C of C such that w ( g ( C )) = w ( C ) + k , k = 0 . Then w ◦ g = w + k ,and w ( g ( C )) = ( w ◦ g )( C ) + k = w ( C ) + 2 k . Applying this procedure tothe order m of the group G , w ( C ) = w ( g m ( C )) = w ( C ) + mk , which is acontradiction.Since h γ , ∂ i i = h γ , ∂ i | N i ◦ , where h· , ·i ◦ is the intersection number on S r ,ℓ − ρ ,and h γ , ∂ i | N i ◦ = w ( x ) − w ( g ( x )) = 0 , the lemma follows. (cid:3) QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 29
L L ′ L ′ + ε τ L ′ + 2 ε F IGURE S “near the boundary of” V i toanother section S ′ such that:(M1) S ′ and g ( S ) agree “near the boundary”; and(M2) S and S ′ have the same signed count of intersections with ∂ .In the m = 2 case, this means that we homotop s ( i ,i ) to another section s ′ ( i ,i ) such that:(1) s ′ ( i ,i ) and g ( s ( i ,i ) ) agree “near the boundary”; and(2) s ( i ,i ) and s ′ ( i ,i ) have the same signed count of intersections with ∂ ( i ,i ) .Pick L ′ ≫ L and let τ : [ L ′ , ∞ ) → [0 , be a smooth function such that • τ ([ L ′ , L ′ + ε ′′ ]) = 0 , • τ ([ L ′ + 2 ε ′′ , ∞ )) = 1 , and • its restriction to ( L ′ + ε ′′ , L ′ + 2 ε ′′ ) is a diffeomorphism onto (0 , .Let γ ∗ r j = γ r j ◦ τ , where we take γ r to be an arbitrary path in e r ,ℓ connecting s r to g ( s r ) and γ r to be as in Lemma 4.1.4. We then define s ′ ( i ,i ) = (cid:26) ( γ ∗ r ( nl i ) , γ ∗ r ( nl i )) , for nl i ≥ L ′ , s ( i ,i ) , for nl i ≤ L ′ . By Lemma 4.1.4, S and S ′ have the same signed count of intersections with ∂ near V ( i ,i ) . Step 2B.
Suppose that (vdim M i , vdim M i ) = ( a, − a − or ( − a − , a ) with a > ; we treat the former. By Equation (4.1.1), a generic path γ r from s r to g ( s r ) does not intersect ∂ i and the same construction of S ′ applies. This coversthe homotopy of S near V I for m = 2 . Step 3.
Suppose m = 3 . Let r j = t ( M i j ) , j = 1 , , . Step 3A.
Suppose that vdim M c ( i ,i ) ≤ − .Now we have the following variant of Lemma 4.1.4: Lemma 4.1.5.
There exists ρ > small such that if s r ∈ e r ,ℓ is a point such that < | s r | < ρ and s r Im( ∂ c ( i ,i ) ) , then there exists a path γ r : [0 , → B r ,ℓρ from s r to g ( s r ) such that:(1) the signed intersection number h γ r , ∂ c ( i ,i ) i is zero and(2) γ r is disjoint from ∂ c ( i ,i ) ( ∂ V c ( i ,i ) ) .Proof of Lemma 4.1.5. Case vdim M c ( i ,i ) = − . In this case the proof follows the same outline as thatof Lemma 4.1.4, but N := ∂ − c ( i ,i ) ( S r ,ℓ − ρ ) is now a manifold with boundary. Let us write N = N ′ ∪ N ′′ , where N ′ is closedand each component of N ′′ has nonempty boundary. Writing γ r as γ γ γ asbefore, h γ , ∂ c ( i ,i ) i = h γ , ∂ c ( i ,i ) | N i ◦ = h γ , ∂ c ( i ,i ) | N ′ + ∂ c ( i ,i ) | N ′′ i ◦ , where h· , ·i ◦ is the intersection number on S r ,ℓ − ρ . As before, h γ , ∂ c ( i ,i ) | N ′ i ◦ =0 . We can modify γ if h γ , ∂ c ( i ,i ) | N ′′ i ◦ = k by concatenating it with a loopin S r ,ℓ − ρ that winds − k times around ∂ c ( i ,i ) | ∂N ′′ . The resulting γ will havezero signed intersection with ∂ c ( i ,i ) | N ′′ , implying (1). (2) is immediate since ∂ c ( i ,i ) | ∂ V c ( i ,i is a codimension two map. Case vdim M c ( i ,i ) < − . In this case γ r can just be a generic arc from s r to g ( s r ) and it will have no intersections with ∂ c ( i ,i ) . (cid:3) We now explain how to modify S to S ′ near the codimension one and two“boundaries” of V i so that (M1) and (M2) hold. In other words, we modify thesections ( s ( i ,i ,i ) , s ( i ,c ( i ,i )) , s ( c ( i ,i ) ,i ) , s c ( i ,i ,i ) ) to ( s ′ ( i ,i ,i ) , s ′ ( i ,c ( i ,i )) , s ′ ( c ( i ,i ) ,i ) , s c ( i ,i ,i ) ) . The modifications will take place on the set X = { nl i ≥ L ′ } ∪ { nl i ≥ L ′ } , where L ′ ≫ L ; in other words, s ∗ = s ′∗ on the complement of X . In the rest of thisstep we encourage the reader to refer to Figure 2 for the picture of a corner, where i , i , i , c ( i , i ) , c ( i , i ) , c ( i , i , i ) are labeled – .First we define s ′ ( i ,c ( i ,i )) = (cid:26) ( γ ∗ r ( nl i ) , γ ∗ r ( nl i )) , for nl i ≥ L ′ , s ( i ,c ( i ,i )) , for nl i ≤ L ′ . By Lemma 4.1.5(1), the signed intersection number between ∂ ( i ,c ( i ,i )) and s ′ ( i ,c ( i ,i )) on nl i ≥ L ′ is zero.Next consider the pushforwards of s ( i ,c ( i ,i )) and s ′ ( i ,c ( i ,i )) under the mor-phism φ ( i ,c ( i ,i )) , ( i ,i ,i ) . On the overlap X , := { nl i ≥ L ′ , L − ε ′′ ≤ nl i ≤ L} , the section s ( i ,c ( i ,i )) = ( s r , s r ) is sent to s ( i ,i ,i ) = ( s r , , s r ) and the sec-tion s ′ ( i ,c ( i ,i )) = ( γ ∗ r ( nl i ) , γ ∗ r ( nl i )) is sent to s ′ ( i ,i ,i ) = ( γ ∗ r ( nl i ) , , γ ∗ r ( nl i )) .By applying Lemma 4.1.5(2) to the term γ ∗ r ( nl i ) , we see that ∂ ( i ,i ,i ) has no in-tersections with s ′ ( i ,i ,i ) on X , if we take ε ′′ > to be sufficiently small. QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 31 On X , := { nl i ≥ L ′ , L ≤ nl i ≤ L + ε ′′ } , (4.1.2) s ( i ,i ,i ) = ( ζ ( nl i ) s r , ζ ( nl i ) s r , s r ) = ( s r , ζ ( nl i ) s r , s r ) . We then set(4.1.3) s ′ ( i ,i ,i ) = ( γ ∗ r ( nl i ) , ζ ( nl i ) γ ∗ r ( nl i ) , γ ∗ r ( nl i )) , where γ r : [0 , → B r ,ℓρ is some path from s r to g ( s r ) with ρ = 2 | s r | .Now we come to the key point: s ( i ,i ,i ) and s ′ ( i ,i ,i ) do not intersect ∂ ( i ,i ,i ) on X , for ε ′′ > sufficiently small. This is due to | s r | ≪ | s r | by Condition (*).Since γ r does not intersect ∂ c ( i ,i ) ( ∂ V c ( i ,i ) ) by Lemma 4.1.5(2), ∂ ( i ,i ,i ) | X , does not intersect a small neighborhood of ( γ ∗ r ( nl i ) , , γ ∗ r ( nl i )) . In particular, if | s r | is sufficiently small, then ∂ ( i ,i ,i ) | X , does not intersect s ′ ( i ,i ,i ) ; s ( i ,i ,i ) is similar.The situation for s ′ ( c ( i ,i ) ,i ) and s ′ ( i ,i ,i ) on X , ∪ X , := { nl i ≥ L ′ , L − ε ′′ ≤ nl i ≤ L + ε ′′ } is analogous (for one of Steps 3A, 3B, or 3C).It remains to modify s ( i ,i ,i ) to s ′ ( i ,i ,i ) on X , ∪ X , ∪ X , , where: X , := { nl i ≥ L ′ , L + ε ′′ ≤ nl i ≤ L ′ } ,X , := {L + ε ′′ ≤ nl i ≤ L ′ , nl i ≥ L ′ } ,X , := { nl i , nl i ≥ L ′ } . On { nl i , nl i ≥ L + ε ′′ } we have s ( i ,i ,i ) = ( s r , s r , s r ) . We then define s ′ ( i ,i ,i ) as:(1) ( γ ∗ r ( nl i ) , γ ∗ r ( nl i ) , γ ∗ r ( nl i )) on X , ;(2) ( γ ∗ r ( nl i ) , γ ∗ r ( nl i ) , γ ∗ r ( nl i )) on X , ;(3) ( γ ∗ r ( β ( nl i , nl i )) , γ ∗ r ( β ( nl i , nl i )) , γ ∗ r ( β ( nl i , nl i ))) on X , , where β ( a, b ) = p ( a − L ′ ) + ( b − L ′ ) − L ′ . The images of the maps (1)–(3) are -dimensional, since each is a postcompo-sition by ( γ ∗ r , γ ∗ r , γ ∗ r ) , which has -dimensional image. On the other hand, byEquation (4.1.1), two of the three maps ∂ i j : V i j → e r j ,ℓ , j = 1 , , , have codi-mension at least one or one of the maps has codimension at least two. Hence if γ r j , j = 1 , , , are sufficiently generic, then s ( i ,i ,i ) and s ′ ( i ,i ,i ) have no intersec-tions with ∂ ( i ,i ,i ) on X , ∪ X , ∪ X , . Step 3B.
Suppose that vdim M c ( i ,i ) = 0 . Then vdim M i = − . The onlydifferences with Step 3A are that, assuming genericity of γ r and γ r : • γ r : [0 , → e r ,ℓ satisfies the conditions of Lemma 4.1.4 (where wereplace i by i ) and intersects ∂ i at isolated points; • γ r : [0 , → e r ,ℓ intersects ∂ c ( i ,i ) | ∂ V c ( i ,i at isolated points since it isa codimension one map; and • the intersection points do not occur at the same time in [0 , .It implies that if | s r | ≪ | s r | , then s ( i ,i ,i ) and s ′ ( i ,i ,i ) , given by Equations (4.1.2)and (4.1.3), have no intersections with ∂ ( i ,i ,i ) on X , ∪ X , . Step 3C.
Suppose that vdim M c ( i ,i ) ≥ . Then vdim M i ≤ − and • γ r : [0 , → e r ,ℓ does not intersect Im ∂ i .If | s r | ≪ | s r | , then s ( i ,i ,i ) and s ′ ( i ,i ,i ) , given by Equations (4.1.2) and (4.1.3),have no intersections with ∂ ( i ,i ,i ) on X , ∪ X , .This implies the theorem for m = 3 . The general case is completely analogousand is only more complicated in notation. (cid:3)
5. E
QUIVARIANT L AGRANGIAN F LOER COHOMOLOGY
Grading.
In order to Z -grade our equivariant Lagrangian Floer cohomologygroups, we require L and L to be G -equivariantly graded , i.e., (G1)–(G3) tohold.(G1) c ( M, J ) = 0 .Then there exists a nowhere-vanishing section Ω of ∧ n C ( T ∗ M, J ) . Let det ,i : L i → S be the map given by det ,i ( p i ) = Ω( X i, ∧ · · · ∧ X i,n ) | Ω( X i, ∧ · · · ∧ X i,n ) | , where p i ∈ L i and X i, , . . . , X i,n is a basis of T p i L i .(G2) There exists a function θ i : L i → R that lifts det ,i , i.e., e π √− θ i ( p i ) = det ,i ( p i ) . Then for each p ∈ L ∩ L , we define an integer index µ ( p ) by the formula µ ( p ) = n + θ − θ − ∠ ( T p L , T p L ) . Here ∠ ( T p L , T p L ) = α + · · · + α n , where the α i ∈ (0 , ) are defined bychoosing a unitary basis { e , . . . , e n } of T p L with respect to ω and J and writing T p L = R h e π √− α e , . . . , e π √− α n e n i . (G3) µ ( gp ) = µ ( p ) for all p ∈ L ∩ L and g ∈ G .For more details on grading, we refer the reader to [Se2] or [AB, Section 2.3].5.2. Chain complex.
Recall the Novikov ring R = n ∞ X i =0 a i T λ i | a i ∈ Z , λ i ∈ R ≥ , λ = 0 and lim i →∞ λ i = ∞ o , where T is the formal parameter. We define the Z -graded R -module CF • ( L , L ) = L j CF j ( L , L ) ,CF j ( L , L ) = R h p ∈ L ∩ L | µ ( p ) = j i . QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 33
The differential d : CF j ( L , L ) → CF j +1 ( L , L ) is defined on the generators by d [ p ] = X Z ( K ( p, q ; A ) , S ) · T R A ω [ q ] , where the sum is over all q ∈ L ∩ L and A ∈ π ( p, q ) subject to the condition vdim( M ( p, q ; A )) = 0 . Lemma 5.2.1. d = 0 .Sketch of proof. We consider the ends of the -manifold Z ( K ( p, q ; A ) , S ) where vdim( M ( p, q ; A )) = 1 . First observe that, by codimension reasons, for any u ∈Z ( K ( p, q ; A ) , S ) , we have nl r ( u ) ≤ L + ε ′′ for all but possibly one r ∈ L ∩ L (cf. Equation (2.5.3) for the definition of the section s I and Definition 2.5.3for the definition of ζ and note that ζ ([ L + ε ′′ , ∞ )) = 1 ). Hence the ends of Z ( K ( p, q ; A ) , S ) are in bijection with a r Z ( K ( p, r ; A ) , S ) × Z ( K ( r, q ; A ) , S ) , where vdim( M ( p, r ; A )) = vdim( M ( r, q ; A ) = 0 and S and S are compat-ible with S . (cid:3) We define the usual Lagrangian Floer cohomology from L to L by HF • ( L , L ) = ker d/ Im d. By Theorem 4.1.2, if vdim( M ( p, q ; A )) = 0 and g ∈ G takes K ( p, q ; A ) toitself,then Z ( K ( p, q ; A ) , S ) = Z ( K ( p, q ; A ) , g ( S )) . Hence, dg [ p ] = σ ( g, p ) d [ gp ]= σ ( g, p ) X r,B Z ( K ( gp, r ; B ) , S )[ r ]= σ ( g, p ) X q,A Z ( K ( gp, gq ; gA ) , S )[ gq ]= σ ( g, p ) X q,A Z ( K ( gp, gq ; gA ) , g S )[ gq ]= σ ( g, p ) X q,A g ( Z ( K ( p, q ; A ) , S ))[ gq ]= σ ( g, p ) X q,A Z ( K ( p, q ; A ) , S ) σ ( g, p ) σ ( g, q )[ gq ]= gd [ p ] , i.e., d is R [ G ] -linear. Let ( P • , d P ) be a projective resolution of R over R [ G ] . We denote E i,j := Hom R [ G ] ( P i , CF j ( L , L )) , where P i and CF j ( L , L ) are regarded as R [ G ] -modules. Let d i,j> : E i,j → E i +1 ,j be the map induced by d P : P i +1 → P i , and d i,j ∧ : E i,j → E i,j +1 bethe map induced by d : CF j ( L , L ) → CF j +1 ( L , L ) multiplied by the factor ( − i . Then d i,j> and d i,j ∧ commute with the multiplication by elements in R [ G ] and form a double complex.The G -equivariant Lagrangian Floer cochain complex is the total complex ( CF • G ( L , L ) , d G ) , where CF kG ( L , L ) = L i + j = k E i,j , d G | E i,j = d i,j> + d i,j ∧ . The corresponding G -equivariant Lagrangian Floer cohomology group is: HF • G ( L , L ) = ker d G / Im d G . The cohomology H • (Hom R [ G ] ( P • , R )) ∼ = H • ( BG ) (taking Y = { pt } as inSection 1) is a ring whose product is the standard cup product. Similarly we candefine the following R [ G ] -bilinear map: H • ( BG ) × HF • G ( L , L ) → HF • G ( L , L ) , which makes HF • G ( L , L ) an H • ( BG ) -module. Indeed, it is easier to see themodule structure via the definition of HF • G ( L , L ) using the singular chain com-plex C • ( EG ) in place of P • . More precisely, the product on the chain level isinduced by the composition of the K ¨unneth map followed by the diagnal map: Hom R [ G ] ( C i ( EG ) , CF j ( L , L )) × Hom R [ G ] ( C k ( EG ) , R ) K¨unneth map −→ Hom R [ G ] ( C i + k ( EG × EG ) , CF j ( L , L )) ∆ ∗ −→ Hom R [ G ] ( C i + k ( EG ) , CF j ( L , L )) . From standard results on spectral sequences of double complexes, we obtain:
Lemma 5.2.2.
There exists a spectral sequence { E i,jr } r with second page E i,j = Ext iR [ G ] ( R, HF j ( L , L )) converging to HF • G ( L , L ) . Chain map.
Using the notation from Section 2.8, for p ∈ L ∩ L , q ∈ L ′ ∩ L ′ , and A ∈ π ( p, q ) , there exists a Kuranishi structure K ( p, q ; A ) and acollection of sections S such that chain map Φ : CF • ( L , L ) → CF • ( L ′ , L ′ ) is defined on the generators by Φ( p ) = X Z ( K ( p, q ; A ) , S ) · T R A ω q, QUIVARIANT LAGRANGIAN FLOER COHOMOLOGY 35 where the sum is over all q ∈ L ′ ∩ L ′ and A ∈ π ( p, q ) subject to the conditions vdim( M ◦ ( p, q ; A )) = 0 and ( p, q, A ) belongs to Sequence (2.5.2). We also havethe following, whose proof is similar to that of Lemma 5.2.1: Lemma 5.3.1. d ◦ Φ = Φ ◦ d . The proof of Theorem 4.1.2 carries over for chain maps. In other words, if vdim( M ◦ ( p, q ; A )) = 0 and g ∈ G preserves K ( p, q ; A ) , then Z ( K ( p, q ; A ) , S ) = Z ( K ( p, q ; A ) , g ( S )) . This implies that:
Lemma 5.3.2.
Φ : CF • ( L , L ) → CF • ( L ′ , L ′ ) is a chain map of R [ G ] -modules. It is clear from the definition that the chain map Φ induces the chain map Φ G : CF • G ( L , L ) → CF • G ( L ′ , L ′ ) . Chain homotopy.
Let
Φ : CF • ( L , L ) → CF • ( L ′ , L ′ ) and Ψ : CF • ( L ′ , L ′ ) → CF • ( L , L ) be chain maps of R [ G ] -modules, defined using φ s and φ − s .Fix p ∈ L ∩ L , q ∈ L ′ ∩ L ′ , and A ∈ π ( p, q ) . Using the function Θ from Section 2.8.2 we construct the bundles π I, [0 , : E I, [0 , → V I, [0 , and the -parameter family K [0 , ( p, q ; A ) := a τ ∈ [0 , K τ ( p, q ; A ) , of Kuranishi structures. Here each term K τ ( p, q ; A ) corresponds to K ( p, q ; A ) for τ ∈ [0 , . We can take the sections s I, [0 , of S [0 , , viewed as a map to a fixedvector space, to be “independent of τ ” or, more precisely, only dependent on necklengths.Now consider the ends of Z ( K [0 , ( p, q ; A ) , S [0 , ) for vdim( M ( p, q ; A )) = 0 .A similar argument as Lemma 5.3.2 gives Lemma 5.4.1. Φ ◦ Ψ − id = K ◦ d + d ◦ K, where K : CF • ( L , L ) → CF • ( L , L ) is a map of R [ G ] -modules. Let K G : CF • G ( L , L ) → CF • G ( L , L ) be the map induced by K . We obtain Φ G ◦ Ψ G − id = K G ◦ d G + d G ◦ K G . Summarizing, we have:
Corollary 5.4.2.
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TATE U NIVERSITY OF N EW Y ORK , S
TONY B ROOK , NY 11790
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