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The Z/pZ Gysin sequence in symplectic cohomology

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The Z/pZ Gysin sequence in symplectic cohomology

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Content THEZ/pZ GYSIN SEQUENCE FOR SYMPLECTIC COHOMOLOGY by Debtanu Sen A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) May 2023 Copyright 2023 Debtanu Sen Acknowledgements I would first and foremost like to thank my family for being extremely supportive in all of my pursuits and inspiring me to follow my dreams. Their outlook towards life has greatly shaped the person I am today and it serves as a beacon in all my creative endeavors. My sincerest gratitude goes to my advisor Professor Sheel Ganatra for his overwhelming support. Sheel has been an excellent mentor and has offered crucial support during all phases of my graduate years. His help has been formative not only in realizing this work but also in my overall development as a mathematician. I cannot thank him enough for his great patience, care and earnestness. Without him, this work simply would not exist. I would also like to extend my gratitude towards Professors Kyler Siegel, Joseph Helfer and Itzhak Bars for agreeing to read my thesis and serve on the dissertation committee. I would like to thank all the faculty members, my fellow graduate students and the staff in the department for creating a supportive and inclusive environment where I felt at home. ii Table of Contents Acknowledgements ii Abstract iv Chapter 1: Introduction 1 1.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Chapter 2: Modules on the algebra of singular chains on S 1 andZ/p as used in Floer theory and relationships between them 8 2.1 Algebraic preliminaries and C ∗ (S 1 )-modules . . . . . . . . . . . . . . . . . . 9 2.2 The algebraic Gysin sequence in the category of C ∗ (Z/p) modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 A small model for the restriction map from S 1 -complexes toZ/p complexes . 22 Chapter 3: S 1 andZ/p actions in Symplectic cohomology and their compatibility 31 3.1 Review of Symplectic cohomology . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 The circle action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 TheZ/p action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4 Compatibility of the S 1 action and theZ/p action . . . . . . . . . . . . . . . 50 Chapter 4: Proof of our main result and an application 58 4.1 Proof of Theorem 1.0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 An application of our main result . . . . . . . . . . . . . . . . . . . . . . . . 59 Bibliography 62 iii Abstract Symplectic cohomology is used as an invariant of a class of exact symplectic manifolds with boundary. There’s also an S 1 -equivariant refinement of the theory that has been defined in literature. Equivariant theories with respect to the subgroup Z/p of roots of unity in S 1 are also of interest in Floer Theory. In my thesis, I define Z/p-equivariant symplectic cohomology for any prime p extending work done in the literature so far in the case of p=2. The main result of this thesis is the existence of a Gysin type long exact sequence relating the S 1 andZ/p equivariant versions of symplectic cohomology. We use this result to answer affirmatively a conjecture made by Seidel regarding the structure of localized S 1 -equivariant symplectic cohomology. iv Chapter 1 Introduction Symplectic cohomology is a powerful invariant of a class exact symplectic manifolds with boundary. ItwasfirstdefinedbyCielibak-Floer-Hofer-Wysocki(in[12],[15],[16],[11])forcer- tainsubsetsofC n andthenextendedbyViterbo([38])toamoregeneralclassofnoncompact exact symplectic manifolds called Liouville domains (a class which includes cotangent bun- dles and Stein domains). It is with Liouville domains M that we shall concern ourselves in this paper. As with usual Hamiltonian Floer cohomology (see [32] for definition), sym- plectic cohomology (denoted as SH ∗ (M)) can be roughly defined as the Morse cohomology of an action functional A H on the free loop space with respect to a metric induced by an almost complex structure on the manifold. It is a well known result in Hamiltonian Floer cohomology that the above construction is independent of the choice of the Hamiltonian H and the almost complex structure(see [32]) for compact symplectic manifolds. In case of a Liouville domains M, the construction only makes sense for special kinds of Hamil- tonians (for example, Hamiltonians which are linear near the boundary). More precisely, take a Hamiltonian function H τ which is linear of slope τ near the boundary ∂M. Denote by SH τ (M) = HF(M,H τ ,J), the Hamiltonian Floer cohomology of H τ . It can be shown that for different values of τ the resulting groups are not always the same. However for two 1 different choices τ 1 < τ 2 , we can construct ”continuation maps” SH τ 1 (M)→ SH τ 2 (M) and we define SH ∗ (M)= lim − → τ →∞ SH τ (M) (1.1) with respect to this directed system. The resulting group SH ∗ (M) is an invariant of the un- derlying manifold up to symplectomorphisms, and has been used to great effect in detecting exotic symplectic structures, that is manifolds which are diffeomorphic but not symplecto- morphic to one another (see McLean [31] , Seidel-Smith [36] and Abouzaid-Seidel [2]). We shall veer in a different direction and study equivariant versions of this theory that arise from the natural S 1 -action of rotation on the free loop space on any manifold. S 1 -equivariantFloercohomologyandcorrespondingly,symplecticcohomologycanbedefined in several ways; for details on these see [7]. One approach is to take the Morse cohomology of the H-perturbed action functional on LM × S 1 ES 1 , where ES 1 is modelled concretely as lim − → N S 2N+1 (this generalizes the standard Borel construction, see [7]). We will follow an approach which extends more simply to the symplectic co-chain complex (that is, the chain level homotopy direct limit of Floer complexes), which goes through geometrically defining a (homotopical) action of C ∗ (S 1 ) on the (Floer or symplectic) co-chain complex. Concretely, the rotation on the free loop space induces an A ∞ action of the algebra of singular chains C ∗ (S 1 ) on the co-chain complex underlying symplectic cohomology which in turn gives rise to an infinite sequence of maps δ k :SC ∗ (M)→SC ∗ (M)[1− 2k], k≥ 0 (1.2) arising from the action of chains of S 1 and satisfying some A ∞ -relations. The first map δ 0 is the differential on SC ∗ (M) and the second map is multiplication by [S 1 ] (the fundamental chain on S 1 ). The higher δ k maps which roughly constitute A ∞ -multiplication by a series of [S 1 ] classes are explicitly constructed for example in Ganatra ([17]) or Zhao ([41]). The A ∞ relations imply that if one introduces a formal variable u of degree 2, the associated 2 equivariant differential δ eq =δ 0 +uδ 1 +u 2 δ 2 +··· (1.3) thought of acting on the space SC ∗ (M)⊗ K[u] (suitably completed) squares to zero. From here, there are then two ways of defining S 1 -equivariant symplectic cohomology (the cohomologyof“homotopyfixedpoints”intheterminologyofGanatra[17]),oneistotakethe cohomologyoftheabovecomplexwithrespecttoδ eq (thisistheapproachtaken,forexample, in [7]), and the next is to define the equivariant differential for each individual CF ∗ (H τ ) and then take a direct limit of the resulting equivariant groups. This latter approach is adopted by Zhao in her paper [41] and this is the construction that we shall adopt too. That is, Zhao defines SH ∗ S 1(M)=lim − → τ H ∗ (CF ∗ (M,H τ )[[u]],δ eq ). (1.4) SinceZ/p embeds into S 1 via roots of unity, it can be reasonably expected thatZ/p equiv- ariant version of symplectic cohomology should exist too. In fact, for a closed manifold P Seidel([35]) has defined the Z/2 equivariant ordinary Floer cohomology of P with respect to a symplectomorphism ϕ of P and Shelukhin-Zhao ([37]) has defined the Z/p equivariant ordinary Floer homology groups HF ∗ Z/p (P,ϕ ) for arbitrary primes p. Wilkins [40] extends these constructions to the non compact case defining the Z/2 equivariant symplectic coho- mology groups SH ∗ Z/2 (M). We extend it further by constructing the equivariant symplectic cohomology groups for any prime p. This follows from geometrically defining an A ∞ -action of C ∗ (Z/p) on the co-chain complexes CF ∗ (M,H τ ) using the same principle as (1.4). That is we have, Proposition 1.0.1. There exists a sequence of maps defined geometrically µ k|1 : (K[Z/p]) ⊗ k ⊗ CF ∗ (M,H τ )→CF ∗ (M,H τ )[1− k], k≥ 0 (1.5) using the quasi-isomorphism C ∗ (Z/p) ∼ =K[Z/p] (explained in more detail in§2.2) satisfying 3 the A ∞ relations. Then one can define equivariant symplectic cohomology in the case p = 2 as in (1.4) by setting ϕ k =µ k|1 (1+τ, ··· ,1+τ, | {z } k copies − ) withZ/2={1,τ } as follows SH ∗ Z/2 (M)=lim − → τ H ∗ (CF ∗ (M,H τ )[[h]],δ eq ). (1.6) where h is a formal variable of degree 1 and δ eq =ϕ 0 + hϕ 1 + h 2 ϕ 2 + ··· . For a definition for general p, look at [37]. To make the definition in (1.6) rigorous we will also need to define equivariant continuation maps for τ 1 < τ 2 . These too shall be constructed in §3.3, Proposition 3.3.4 along with the maps in (1.5). Inthetopologicalcase, foraspaceX withaS 1 action, wehavealongexactsequence(called the Gysin sequence) relating H ∗ S 1 (X) and H ∗ (X) (see [8], equation (1.2)): ···→ H ∗ (X)→H ∗ S 1(X) [+2] − − → H ∗ +2 S 1 (X) [− 1] − − →··· (1.7) This is a special case of an exact sequence relating the base and fiber of a S 1 -bundle, which in this case is X× ES 1 →X× S 1 ES 1 . There is also a variant sequence relating theZ/p-equivariant cohomology of X (with respect to the subgroupZ/p in S 1 ) with S 1 -equivariant cohomology, given by ···→ H ∗ Z/p (X)→H ∗ S 1(X) [+2] − − → H ∗ +2 S 1 (X) [− 1] − − →··· (1.8) and coming from analyzing the S 1 -bundle X× Z/p ES 1 →X× S 1 (ES 1 ). In particular it can be shown that the map H ∗ S 1 (X) [+2] − − → H ∗ +2 S 1 (X) from (1.8) is a multiple of p, so withF p coefficients (1.8) splits yielding H ∗ Z/p (X;F p )=H ∗ S 1(X;F p )⊕ H ∗− 1 S 1 (X;F p ) (1.9) 4 BourgeoisandOancea[8],establishedaversionofthefirstGysinsequence(1.7)inthesetting of symplectic cohomology. Our main result a generalization of the second Gysin sequence (1.8) which was first proposed by Seidel [35]. Theorem 1.0.2. There exists a Gysin type long exact sequence relating SH ∗ Z/p with SH ∗ S 1 ···→ SH ∗ Z/p (M)−→ SH ∗ S 1(M) [+2] − −→ p.u SH ∗ S 1(M) [− 1] − −→··· (1.10) such that the map SH ∗ S 1 (M) [+2] − −→ SH ∗ S 1 (M) is p-divisible. Hence, withF/p coefficients the sequence splits yielding SH ∗ Z/p (M;F p )=SH ∗ S 1(M;F p )⊕ SH ∗− 1 S 1 (M;F p ) (1.11) The proof of Theorem 1.0.2 follows from some general algebraic facts about complexes that admit C ∗ (S 1 ) actions and their restricted C ∗ (Z/p) actions along geometric constructions ensuring the compatibility of these actions which are geometrically defined. These actions will be constructed geometrically in §3.2 and §3.3. The next chapter is devoted to laying down the algebraic framework. As an application of our main result, we can investigate certain properties of localised S 1 - equivariant symplectic cohomology. It is well known that ordinary equivariant cohomology exhibits nice fixed point localization properties (for example, see Atiyah and Bott [5], which relates equivariant homology of a manifold X equipped with a S 1 action to the homology of the fixed point set F of the S 1 action). However, the standard definition of S 1 -equivariant symplectic cohomology cannot do the same. In particular, Goodwillie [19], showed that the rotation on free loop spaces do not obey fixed point localization and hence neither does symplecticcohomologyofcotangentbundles(duetoaresultbyViterbo,in[39],whichstates thatwehaveanisomorphismSH ∗ (T ∗ Q) ∼ = H ∗ (LQ),whereLQdenotesthefreeloopspaceof Q,ifQisspin). Thisfailureisnotonlyrestrictedtocotangentbundles: e.g.,there’saspectral 5 sequence from SH ∗ (M)⊗ H ∗ (BS 1 ) to equivariant symplectic cohomology but the latter vanishesfore.g.,D 2 sotheequivarianttheorydoestoo,i.e.,itcannotrecovercohomology(for details see [34], section 8). To address the problem, Jones and Petrack [25] constructed, for freeloopspaces,aless-completedvariantofS 1 -equivariantcohomologywhichdoessatisfythe localization property. Zhao [41] and Albers, Cielibak and Frauenfelder [4] generalized Jones- Petrack’sconstructiontothesymplecticco-chaincomplex,constructinga“corrected”version ofS 1 -equivariantsymplecticcohomology(denotedSH ∗ S 1 (M))andprovedalocalizationresult as follows Theorem 1.0.3. (Zhao [41], Albers-Cieliebak-Frauenfelder [4])The natural inclusion ι : M ,→ LM via constant loops induces a map H ∗ (M)[u] → SH ∗ S 1 (M;Z) which becomes an isomorphism ofQ((u)) modules after inverting u and tensoring both sides withQ. Denote the u-inverted localized theory as u − 1 SH ∗ S 1 (M) (and called the localized symplectic cohomology. AninterestingaspectofTheorem1.0.3istheQ-coefficientrequirement. Infact, Zhao[41], alsoremarkablycomputedthatinthecaseof M =D 2 wehaveu − 1 SH ∗ S 1 (D 2 ;Z)= Q((u)) which suggests the following refinement of the localization property Conjecture 1.0.4. (Seidel [35]) Let M be a Liouville domain. If SH ∗ (M)=0, then u − 1 SH ∗ S 1(M;Z) ∼ = H ∗ (M;Q((u))). (1.12) We note here that SH ∗ (D 2 ;Z) = 0, another computation by Zhao for M =C ∗ reveals that the conjecture fails if SH ∗ (M)̸= 0. In section 4 we outline a proof of this conjecture using our main result as an important step as well as work in progress of Shelukhin-Zhang. 1.1 Conventions Welaydownheredefinitionsforcertainnon-standardtermswe’llusingthroughoutthepaper and notational conventions. 6 1. Asmall modelofanalgebra/bimodule/generatorreferstoanyfinite-dimensionalmodel with finitely many relations/operations. In various cases we’ll be interested in finding models with as few generators as possible satisfying certain constraints for ease of computation. 2. All singular chains are cohomologically, i.e they are negatively graded, so C − 1 (X) is the space of singular 1-chains on X. 3. We work over an arbitrary field K unless we specifically state otherwise. 7 Chapter 2 Modules on the algebra of singular chains on S 1 and Z/p as used in Floer theory and relationships between them The goals of this section are to review a small model for chain complexes which admit a C ∗ (S 1 ) action suitable for computations in Floer theory and to extend this formalism by showing how to compatibly incorporate the induced action of C ∗ (Z/p), where Z/p ⊂ S 1 denotes the subgroup of roots of unity. We begin by reviewing below, in §2.1, the notion of a S 1 -complex (i.e complex admitting a strictly unital A ∞ action of cellular chains on S 1 modelled concretely as K[Λ] /Λ 2 ). These complexes are introduced in [7],[34],[41],[17] generalizing the notion of a mixed complex as in [29], [27], [10]. Then in §2.3 we show how toincorporateC ∗ (Z/p)actionsatthelevelofthesmallmodels. Theseareapriorilostbythe passagefromC ∗ (S 1 )tothecellularsmallmodelK[Λ] /Λ 2 (whichisnotZ/pequivariant), but can be recovered in a minimal way suitable for later Floer theory constructions. Proposition 8 2.3.3 in §2.3 (one of the key technical results of this section) gives us a concrete way to do this. It computes a small model of the restriction from C ∗ (S 1 ) modules to C ∗ (Z/p) modules asabimodulebetweensmallmodels. FromhereProposition2.3.3givesusawayofexplicitly computing the the restriction of a K[Λ] /Λ 2 module to K[Z/p] assuming the module comes from a module with a little more structure, one that admits an action of cellular chains on S 1 for a suitableZ/p-equivariant cell structure (see definition 2.3.1 in §2.3). Theotherkeyresultsofthissectionarein§2.2whichisdevotedtofindingsmallcomputable models for the homotopy fixed points and homotopy orbits of the associated C ∗ (Z/p) action (done in Propositions 2.2.3 and 2.2.4). Following that, Propositions 2.2.6 and 2.2.7 show that if we have a module carrying the extra structure described above then there’s a Gysin sequence between the S 1 homotopy fixed points (resp. orbits) of the underlying C ∗ (S 1 ) moduleandtheZ/phomotopyfixedpoints(resp. orbits)oftheunderlying C ∗ (Z/p)module. We shall for the most part of this section follow the notation laid down in Ganatra (i.e [17]) and present again the relevant definitions and concepts in order to make this self contained. 2.1 Algebraic preliminaries and C ∗ (S 1 )-modules In this section we establish the algebraic background for describing modules over chains of S 1 (we work with a weaker notion of module action than standard dg-actions, called A ∞ modules). We also describe the various equivariant homology complexes of such modules and various exact triangles that these complexes satisfy. Now to make things precise, let C ∗ (S 1 ), C ∗ (Z/p) be the space of singular chains on S 1 and Z/p (thought of as a discrete space) respectively with coefficients in K (graded according to ourconvention). ThesehavethestructureofadgalgebrawhenequippedwiththePontryagin product (see [9, Chapter V§5]). There are A ∞ equivalences of dg algebras C ∗ (S 1 )≃K[Λ] /Λ 2 , |Λ |=− 1 (2.1) 9 C ∗ (Z/p)≃K[Z/p] (2.2) whereK[Z/p] denotes the group ring onZ/p with generator in degree 0. These equivalences give us the small models for these algebras in the sense explained in§1.1 and also give us an equivalence of module categories. They are particular instances of the homological pertur- bation Lemma (see [13],[20]). Moreover, we shall explicitly construct the two equivalences (2.1) and (2.2) later in§3.2, specifically in Lemmas 2.3.5 and 2.3.6. Definition 2.1.1. A S 1 -complex is a strictly unital A ∞ module overK[Λ] /Λ 2 . Let (A,d A ) be a dgK-algebra. Recall (see [18],[28],[33] for a more detailed exposition) that a (left) A ∞ -module N over A is a graded module over the base field K endowed with maps µ d|1 :A ⊗ d ⊗ N →N ofdegree1− dforeachd≥ 0, satisfying, foreach k≥ 0, theA ∞ moduleequationsasfollows k X i=1 µ k− 1|1 (a 1 ,··· ,a i− 1 ,a i .a i+1 ,a i+2 ,··· ,a k ,− )+ k X i=0 µ (k− i)|1 (a 1 ,··· ,a k− i ,µ i|1 (a k− i+1 ,··· ,a k ,− ))+ k X i=1 µ k− 1|1 (a 1 ,··· ,a i− 1 ,d A (a i ),a i+1 ,··· ,a k ,− )=0 (2.3) For an A ∞ -module N over a strictly unital algebra A, N is said to be strictly unital if the multiplication maps satisfy µ 1|1 (1 A ,a) = a and µ k|1 (··· ,1 A ,··· ,a) = 0 for any input sequence of length k > 1 containing at least one term equal to 1 A . Right modules are defined in the same way as graded modules equipped with structure maps µ 1|d satisfying similar equations as in (2.3) or equivalently as a left module over A op (in our applications we’ll have A = A op ). There is also a similar notion of an A ∞ -bimodule over two algebras A,B (see again [18, definition 2 .12]). These are gradedK-modules equipped with structure 10 maps µ k|l :A ⊗ k ⊗ N⊗ B ⊗ l →N satisfying similar equations as in 2.3 (as described in [18], for example). We clearly have a map of algebras i : C ∗ (Z/p) → C ∗ (S 1 ) induced by the inclusion of Z/p in S 1 via roots of unity. Therefore any C ∗ (S 1 ) module admits a restriction to a C ∗ (Z/p) module. GivenaS 1 -complexM wecanstudytheassociatedC ∗ (S 1 )modulewhichisobtained using the equivalence (2.1) and restrict to C ∗ (Z/p). We shall call this restriction res Z/p (M). This is the pullback module associated with (2.1) and the map i. Such structures are discussed in detail in [18,§2.8]. For the purposes of our calculations here we recall an map of algebras f : A → B induces a pullback functor between the categories of A ∞ modules f ∗ : B− mod → A− mod , which given a B-module (X,{µ k|1 }) maps it to the A-module f ∗ (X) where f ∗ (X) = X as algebras and structure maps defined by ν k|1 (a 1 ,··· ,a k ,x) := µ k|1 (f(a 1 ),··· ,f(a k ),x). In the case where f is an A ∞ morphism (see discussion preceding (2.5)) rather than an algebra morphism the pullback structure maps are defined as ν k|1 (a 1 ,··· ,a k ,x) := X r,i 1 +··· +ir=k µ r|1 (f i 1 |1 (a 1 ,··· ,a i 1 ),··· ,f ir|1 (a ir ,··· ,a k ),x) (2.4) . Pullbacks can also be done on the category of bimodules. Given a pair of A ∞ morphisms of algebras g 1 : A → A ′ and g 2 : B → B ′ , any A ′ − B ′ bimodule N can be pulled back g 1 and g 2 to give us the bimodule (g 1 ,g 2 ) ∗ N with structure maps satisfying similar equations as in (2.4) (see [18]). It is now apparent from the previous discussion that the restriction operation res Z/p cannot becarriedoutnaivelyatthelevelofsmallmodelsdefinedby(2.1)and(2.2). Indeed, K[Λ] /Λ 2 does not admit aK[Z/p] module structure. Proposition 2.3.3, as mentioned above, gives us a way around that. Given a left and a right A ∞ modules there is an operation (−⊗ L C∗ (S 1 ) − ) called the derived 11 tensor product and given two left (or right) A ∞ modules we can define the space of derived homomorphismsdenotedasRHom(-.-). ForA ∞ modulesrespectively. Foraprecisedefinition ofthesenotionssee[17]). Weshallrecalltheiressentialpropertieshere. LetM,N betwoleft A ∞ modules over an algebra A. A pre-morphism F of degree k from M to N is a collection of operators F d|1 : A ⊗ k ⊗ M → N of degree k− d. Let RHom k A (M,N) be the collection of pre-morphisms of degree k and RHom A (M,N) = L k RHom k A (M,N). This space is a chain complex with a differential defined as ∂(F)= k X r=0 F k− r|1 (a k ,··· ,µ r|1 M (a r ,··· ,a 1 ,m)) +(− 1) deg(F) k X r=0 µ k− r|1 N (a k ,··· ,F r|1 (a r ,··· ,a 1 ,m)) (2.5) An A-module homomorphism (or morphism) now is pre-morphism satisfying ∂(F) = 0. If [F 0|1 ] : H ∗ (M,µ 0|1 M )→ H ∗ (N,µ 0|1 N ) is a isomorphism on homology then F is called a quasi- isomorphism. The composition of two morphisms is defined as ( G◦ F) s = P i G s− i ◦ F i . For M,N, a right and a left A ∞ A-module respectively. The derived tensor product M⊗ L A N has as its underlying vector space L k M⊗ (A[1]) ⊗ d ⊗ N and a differential ∂ given by ∂(m⊗ x 1 ⊗···⊗ x r ⊗ n)= r X i=0 (− 1) |m| µ i|0 N (n⊗ x 1 ⊗··· x i )⊗ x i+1 ⊗···⊗ x r ⊗ m +n⊗ x 1 ⊗··· x r− i ⊗ µ i|0 M (x r− i+1 ⊗···⊗ x r ⊗ m + r X i=1 n⊗ x 1 ⊗···⊗ x i− 2 ⊗ x i− 1 .x i ⊗···⊗ x r ⊗ m=0 (2.6) InspecialcasewhereA=K[Λ] /Λ 2 andM,N areunitalmodules, thederivedtensorproduct (by considering M as a right module over A via the identification K[Λ] /Λ 2 ≃ K[Λ] /Λ 2 op ) can be equivalently computed by considering the vector space L k M ⊗ (A[1]) ⊗ d ⊗ N where A is the augmentation ideal for the natural map K[Λ] ϵ − → K generated as a vector 12 space by Λ. In this case, the differential simplifies to ∂(m⊗ Λ ⊗···⊗ Λ | {z } r copies ⊗ n)= r X i=0 (− 1) |m| µ i|1 N (Λ ,··· ,Λ ,n)⊗ Λ ⊗··· Λ | {z } r− i ⊗ m +n⊗ Λ ⊗··· Λ | {z } r− i ⊗ µ i|1 M (Λ ,··· ,Λ ,m) (2.7) Definition 2.1.2. Given an S 1 -complex M, the homotopy orbit complex of M is defined to be M hS 1 :=M⊗ L K[Λ] /Λ 2K (2.8) and the homotopy fixed point complex of M is defined as M hS 1 :=RHom K[Λ] /Λ 2(K,M) (2.9) whereKbydefinitionisthestrictlyunital K[Λ] /Λ 2 moduleequaltoKindegree0andzeroin all other degrees equipped with the trivial S 1 -action (i.e the multiplication by 1 determined by unitality and multiplication by Λ equal to 0). These notions make sense over arbitrary unital augmented algebras too. We, in this work, will only be using it forK[Z/p]-modules in addition toK[Λ] /Λ 2 , for which we present the relevant definitions below Definition2.1.3. GivenastrictlyunitalA ∞ K[Z/p]-moduleM,thehomotopy orbit complex of M is defined to be M hZ/p :=M⊗ L K[Z/p] K (2.10) 13 and the homotopy fixed point complex of M is defined as M hZ/p :=RHom K[Z/p] (K,M) (2.11) whereK, as before, is the strictly unitalK[Z/p] module equal toK in degree 0 and zero in allotherdegreesequippedwiththe trivialZ/p-action(i.ethemultiplicationby1determined by unitality and multiplication by all other generators equal to 0). The Gysin long exact sequence relating S 1 -equivariant symplectic cohomology and ordinary symplectic cohomology (proved in [8]) that was remarked to exist in the previous section, arisesasaspecialcaseofamoregeneralconstructioninthecategoryof A ∞ -C ∗ (S 1 )modules. The general theory of such modules are well studied; see for example [17], [7], [41], [14]. It is known (see for instance, [17]) that there exists a canonical exact triangle in the category of S 1 -complexes as follows K[Λ] /Λ 2 →K→K[2] [1] − → (2.12) Therefore pushing forward by the functors (M⊗ L K[Λ] /Λ 2 − ) and RHom K[Λ] /Λ 2(− ,M) yield the following Gysin type exact triangles of chain complexes M →M hS 1 →M hS 1[2] [1] − → (2.13) and M hS 1 [− 2]→M hS 1 →M [1] − → (2.14) In the following section we prove a generalization of equation (2.12) which intertwines the homotopy orbit (resp. homotopy fixed point) complexes of a C ∗ (S 1 ) module and the associ- ated C ∗ (Z/p) module that one gets by restricting a C ∗ (S 1 ) to the sub-algebra C ∗ (Z/p) when pushed forward by the same functors as above (see Propositions 2.2.6 and 2.2.7). 14 2.2 The algebraic Gysin sequence in the category of C ∗ (Z/p) modules There is a map of algebras i : C ∗ (Z/p)→ C ∗ (S 1 ) induced by the inclusion of Z/p into S 1 . Therefore given any S 1 -complex M, we can use equivalence (2.1) to think of M as a C ∗ (S 1 ) module and then we can look at the associated C ∗ (Z/p) module formed by restriction which we shall call, as before, res Z/p (M). As remarked in the beginning, the goal of this section is to find a small model for res Z/p (M) hZ/p and res Z/p (M) hZ/p ; see Propositions 2.2.3 and 2.2.4. These also follow from the small model that we discuss in Proposition 2.3.3, but deriving it directly was somehow simpler. Recall that restrictions of A ∞ modules can be modelled as derived tensor products in the samewayasdgmodulesandordinarytensorproducts(see[3,Lemma2.8],foraproofofthis fact). Specifically, an A ∞ quasi isomorphism h : A ∼ − → B induces an equivalence of module categories Mod− A ≃ Mod− B. The equivalence can written in terms of derived tensor products and pullbacks as follows h ∗ :Mod− A ∼ − → Mod− B, h ∗ (X)=(X)⊗ L A (Id,h) ∗ B (2.15) and h ∗ :Mod− B ∼ − → Mod− A, h ∗ (X)=(X)⊗ L B (h,Id) ∗ A (2.16) In case, i : A ,→ B is the inclusion of sub-algebras then we use the shorthand B A to denote the module i ∗ (B) and A B to denote the module i ∗ (A). Denotingf :K[Λ] /Λ 2 ∼ − → C ∗ (S 1 )andg :K[Z/p] ∼ − → C ∗ (Z/p),thequasi-isomorphismsin(2.1) and (2.2) we can write, for any S 1 -complex M, res Z/p (M)=g ∗ (i ∗ (f ∗ (M))) (2.17) 15 Therefore, theZ/p homotopy fixed points for any S 1 -complex Mis given by, res Z/p (M) hZ/p =g ∗ (i ∗ (f ∗ (M)))⊗ L K[Z/p] K (2.18) ≃g ∗ (i ∗ (f ∗ (M)))⊗ L K[Z/p] g ∗ (C ∗ ({pt})) (2.19) ≃ (i ∗ (f ∗ (M)))⊗ L C∗ (Z/p) (C ∗ ({pt})) (2.20) ≃f ∗ (M)⊗ L C∗ (S 1 ) (Id,i) ∗ C ∗ (S 1 )⊗ L C∗ (Z/p) (C ∗ ({pt})) (2.21) or, res Z/p (M) hZ/p =f ∗ (M)⊗ L C∗ (S 1 ) (Id,i) ∗ C ∗ (S 1 )⊗ L C∗ (Z/p) K (2.22) We want to find a small computable model for the RHS of (2.22). Towards that end, let us define C ∗ (S 1 ) to be the algebra of chains on S 1 treated as a module over over itself with a non standard multiplication induced by the map S 1 × S 1 →S 1 which sends (x,t)7−→ x p t. Proposition 2.2.1. There is a quasi isomorphism of C ∗ (S 1 )− K bimodules between the derived tensor product C ∗ (S 1 )⊗ L C∗ (Z/p) K and C ∗ (S 1 ). Proof. Let us look closely at the term which appears in the brackets in (2.22). C ∗ (S 1 ) C∗ (Z/p) ⊗ L C∗ (Z/p) C ∗ ({pt})≃C ∗ (S 1 × h Z/p {pt}) (2.23) ≃C ∗ (S 1 /(Z/p)) (2.24) Tojustifythissequenceofequivalenceshere: equation(2.23)followsfrom[30,Theorem7.27] (for a more detailed discussion as to how, look at [17, Remark 20]) and (2.24) holds because Z/p acts freely on S 1 . Next we notice that S 1 /(Z/p) is isomorphic to S 1 again via the map which sends [z] 7−→ z p . Therefore, the end result that we get is again C ∗ (S 1 ) as a chain complex but with a non-standard C ∗ (S 1 ) module action induced via the Pontryagin product by the map S 1 × S 1 →S 1 which sends (x,t)7−→ x p t. 16 C ∗ (S 1 ) is a C ∗ (S 1 )-module automatically by construction. We claim further that, Proposition 2.2.2. Under the correspondence of module categories induced by equation (2.1), we have C ∗ (S 1 ) ∼ =K[ϵ ]/ϵ 2 (2.25) asK[Λ] /Λ 2 modules with module action given by 1× ϵ =ϵ Λ × 1=pϵ Λ × ϵ =0 (2.26) and all higher multiplications equal to 0. Proof. We firstexplicitlycalculatethehomologylevelactionof H ∗ (C ∗ (S 1 ))onH ∗ (C ∗ (S 1 )). We have that H ∗ (C ∗ (S 1 )) = H ∗ (S 1 ) = K[ϵ ]/ϵ 2 , |ϵ | = − 1 (we use ϵ here this generator from Λ). We claim the action of H ∗ (C ∗ (S 1 )) = K[Λ] /Λ 2 on K[ϵ ]/ϵ 2 is exactly as claimed in equation (2.26). The first line of (2.26) directly follows from analyzing the product (x,t)7−→ x p t which if x=1 is indeed the identity. For the second equality we notice that Λ, in the description of C ∗ (S 1 ) which we had in the beginning§2.1, represents the fundamental chain on the circle i.e a loop which goes once round the circle. Therefore under the map (x,t)7−→ x p t it goes around p times. The third line of (2.26) is due to degree reasons seeing as Λ × ϵ must live in degree− 2 which is empty Next, it is well known (compare Theorem 1 in [26] or see [23], [24]) that any A ∞ module Y oversomealgebraisquasiisomorphictosomeA ∞ modulestructure e Y overthesamealgebra which as a group is H . q (Y) and has µ 1|1 e Y (− ,− )=[µ 1|1 Y ]. However e Y can, a priori, have higher A ∞ module structure maps µ 1|r : e Y ⊗ K[Λ] /Λ 2 ⊗ r → e Y[1− r] 17 But in this particular case at hand we get lucky. First note that sinceK[Λ] /Λ 2 is unital, we can WLOG assume our modules to be strictly unital too (see [23, Theorem 4.1]). Therefore, µ 1|1 e Y (− ,1) = ± Id and µ r|1 e Y (− ,··· ,1,··· ) = 0 for r > 1. So we only need to care about whether there are higher operations of the form µ 1|r e Y (− ,Λ ,Λ ,··· ,Λ) : e Y → e Y[1− 2r] ButK[ϵ ]/ϵ 2 is only concentrated in degrees 0 and− 1 and 1− 2r≤− 3 for r > 1. It follows that for degree reasons such higher operations as described above cannot exist. What we have therefore proven is, Proposition 2.2.3. For any S 1 -complex M, its Z/p homotopy orbits are given by the equation res Z/p (M) hZ/p ≃M⊗ L K[Λ] /Λ 2K[ϵ ]/ϵ 2 (2.27) Proof. Proposition 2.2.2 implies that f ∗ (K[ϵ ]/ϵ 2 ) = C ∗ (S 1 ). Next, from (2.22) and the discussion following that we get res Z/p (M) hZ/p = f ∗ (M) ⊗ L C∗ (S 1 ) C ∗ (S 1 ). Therefore, res Z/p (M) hZ/p =f ∗ (M)⊗ L C∗ (S 1 ) f ∗ (K[ϵ ]/ϵ 2 ), from which the conclusion readily follows. We also have, Proposition 2.2.4. For any S 1 -complex M, itsZ/p homotopy fixed points are given by the equation (res Z/p (M)) hZ/p ≃ RHom K[Λ] /Λ 2 K[ϵ ]/ϵ 2 ,M (2.28) We defer the proof of this Proposition until the next section. We are now in a position to state the main result of this section which is an exact triangle of K[Λ] /Λ 2 modules involving our model for the non-standard chains on S 1 in the spirit 18 of (2.12). Before moving into details, however, let us recall the following Lemma which describes the space of A ∞ morphisms from the trivial moduleK to itself. Lemma 2.2.5. As dg-algebras, RHom S 1(K,K)=K[u], |u| =2 (2.29) where u corresponds to unique morphism of degree 2 that satisfies u 1|1 (− ,Λ) = Id(− ) and u 1|r (− ,Λ ,··· ,Λ)=0 for r > 1. Proposition 2.2.6. There exists an exact triangle ofK[Λ] /Λ 2 modules as follows K pu − → K[2]→K[ϵ ]/ϵ 2 [1] − → (2.30) Proof of Proposition 2.2.6: The way we shall prove this is via first principles, i.e, we show that Cone(pu) ≃ K[ϵ ]/ϵ 2 as K[Λ] /Λ 2 modules. Before proceeding with the proof, however, let us recall (or see [21, §2.7]) that given g : M 0 → M 1 a degree zero A ∞ module homomorphism, the mapping cone of g, denoted as Cone(g) is Cone(g)=M 0 [1]⊕ M 1 (2.31) with the module structure maps given by µ 1|r C    x y    ,a 0 ,··· ,a r =    µ 1|r M 0 (x,a 0 ,··· ,a r ) µ 1|r M 1 (y,a 0 ,··· ,a r )+g 1|r (x,a 0 ,··· ,a r )    (2.32) Using the above definition we get that Cone(pu)=K[1]⊕ K[2] (2.33) 19 with structure maps µ 1|r C    x y    ,a 0 ,··· ,a r =    µ 1|r K (x,a 0 ,··· ,a r ) µ 1|r K (y,a 0 ,··· ,a r )+pu 1|r (x,a 0 ,··· ,a r )    (2.34) Therefore, we can define the following map ϕ :Cone(pu)−→ K[ϵ ]/ϵ 2    a b    7−→ a+bϵ which we verify to be a module isomorphism next. As remarked earlier, since Cone(pu) andK[ϵ ]/ϵ 2 are strictly unital modules we need to only compute the multiplication maps only in the case all of the a i ’s are equal to Λ. Plugging in a 1 =··· =a r =Λ gives us µ 1|r C    x y    ,Λ ,··· ,Λ =    µ 1|r K (x,Λ ,··· ,Λ) µ 1|r K (y,Λ ,··· ,Λ)+ pu 1|r (x,Λ ,··· ,Λ)    (2.35) Note that as remarked in the beginning of the section, u 1|r =0 for r > 1 and u 1|1 =Id, and since K is a trivial S 1 -complex we also have that µ 1|r K = 0 for all r ≥ 1. Plugging all this back into (2.35) gives us µ 1|1 C    x y    ,Λ =    0 py    (2.36) and, 20 µ 1|r C    x y    ,Λ ,··· ,Λ =    0 0    (2.37) for all r > 1. LetusnowrecallthemoduleactionofK[Λ] /Λ 2 onK[ϵ ]/ϵ 2 whichwecomputedintheprevious section. Noting that K[ϵ ]/ϵ 2 = a+bϵ a,b∈K as vector spaces we can write the action as follows µ 1|1 K[ϵ ]/ϵ 2 a+bϵ, Λ =pbϵ (2.38) and µ 1|r K[ϵ ]/ϵ 2 a+bϵ, Λ ,··· ,Λ =0 (2.39) We notice that the pair of equations (2.36) and (2.37) are the same as the pair (2.38) and (2.39), and also the fact that both complexes have trivial differential and hence the map ϕ is an isomorphism ofK[Λ] /Λ 2 -modules as desired. Therefore applying Propositions 2.2.3 and 2.2.4, we get that, Proposition 2.2.7. Given any S 1 -complex M, we have the following exact triangles M hS 1 →M hS 1[2]→(res Z/p (M)) hZ/p [1] − → (2.40) and (res Z/p (M)) hZ/p →M hS 1 [− 2]→M hS 1 [1] − → (2.41) 21 2.3 A small model for the restriction map from S 1 - complexes to Z/p complexes As was observed before that C ∗ (S 1 ) when treated as a module over itself can be thought of alsoasaC ∗ (Z/p)moduleafterapplyingtherestrictionoperation. Atthelevelofsmallmodels this does not hold anymore asK[Λ] /Λ 2 does not admit aK[Z/p] module structure. The goal of this section is to find a small model of that restriction operator which is compatible with small models of C ∗ (S 1 ) and C ∗ (Z/p). A priori, to restrict a K[Λ] /Λ 2 module M to itsK[Z/p] counterpart, we need to go via the following circuitous route, res Z/p (M)=M K[Λ] /Λ 2⊗ L K[Λ] /Λ 2 K[Λ] /Λ 2C ∗ (S 1 ) C∗ (S 1 ) ⊗ L C∗ (S 1 ) C∗ (S 1 ) C ∗ (S 1 ) C∗ (Z/p) ⊗ L C∗ (Z/p) C∗ (Z/p) K[Z/p] K[Z/p] (2.42) One sees from the above equation that the restriction at the level of the small models is abstractly induced by tensoring with K[Λ] /Λ 2 − K[Z/p] bimodule. Therefore, we need to find a small model for this bimodule. We shall call this model Cell p (S 1 ), define it as follows and then verify that it has the required properties. Definition 2.3.1. Define the dg-algebra Cell p (S 1 ) by: Cell p (S 1 )=K[τ,h ] τ p =1, dh =τ − 1, h 2 =0 , |τ | =0, |h| =− 1 (2.43) It is evident from the definition of Cell p (S 1 ) thatK[Z/p] sits inside it as a sub-algebra and therefore Cell p (S 1 ) is a strict dg module over K[Z/p]. We can also make Cell p (S 1 ) into a K[Λ] /Λ 2 module by defining Λ .1 = h(1+τ +τ 2 +··· τ p− 1 ). One can check that this also defines a strict dg action and that both the actions are compatible with each other. These 22 make Cell p (S 1 ) into aK[Λ] /Λ 2 − K[Z/p] bimodule. Now note that, K[Λ] /Λ 2C ∗ (S 1 ) C∗ (S 1 ) ⊗ L C∗ (S 1 ) C∗ (S 1 ) C ∗ (S 1 ) C∗ (Z/p) ⊗ L C∗ (Z/p) C∗ (Z/p) K[Z/p] K[Z/p] =(f,g) ∗ (1,i 0 ) ∗ (C ∗ (S 1 )) (2.44) where f :K[Λ] /Λ 2 → C ∗ (S 1 ) and g :K[Z/p]→ C ∗ (Z/p) are the equivalences inducing (2.1) and 2.2 respectively, as discussed before. So we need to prove, Proposition 2.3.2. Cell p (S 1 ) is quasi isomorphic to (f,g) ∗ (1,i 0 ) ∗ (C ∗ (S 1 )) in the category BiMod K[Λ] /Λ 2 − K[Z/p] (i.e, A ∞ K[Λ] /Λ 2 − K[Z/p] bimodules). We postpone the proof of Proposition 2.3.2 until the end of this section. Let us first deduce a few consequences of this result. Proposition 2.3.3. If M is the restriction to K[Λ] /Λ 2 of a Cell p (S 1 ) module f M, then res Z/p (M) is the same as the naive restriction of f M toK[Z/p]⊂ Cell p (S 1 ). Proof. Note that, res Z/p (M)= f M⊗ L Cellp(S 1 ) Cellp(S 1 ) Cell p (S 1 ) K[Λ] /Λ 2 ⊗ L K[Λ] /Λ 2C ∗ (S 1 ) C∗ (S 1 ) ⊗ L C∗ (S 1 ) C∗ (S 1 ) C ∗ (S 1 ) C∗ (Z/p) ⊗ L C∗ (Z/p) C∗ (Z/p) K[Z/p] K[Z/p] Therefore, res Z/p (M)= f M⊗ L Cellp(S 1 ) Cell p (S 1 ) K[Λ] /Λ 2 ⊗ L K[Λ] /Λ 2 K[Λ] /Λ 2Cell p (S 1 ) K[Z/p] 23 Next we note that Cellp(S 1 ) Cell p (S 1 ) K[Λ] /Λ 2 ⊗ L K[Λ] /Λ 2 K[Λ] /Λ 2Cell p (S 1 ) K[Z/p] ≃Cell p (S 1 ) K[Z/p] and therefore by (2.15) we get res Z/p (M)= f M K[Z/p] which is indeed the naive restriction. Next we prove Proposition 2.2.4. Proof of Proposition 2.2.4. Since restrictions can be written as both tensor product with a bi-module and as hom from the same bi-module, so (2.42) can be written as res Z/p (M)=RHom C∗ (Z/p) C∗ (Z/p) K[Z/p] K[Z/p] ,RHom C∗ (S 1 ) C∗ (S 1 ) C ∗ (S 1 ) C∗ (Z/p) ,RHom K[Λ] /Λ 2 K[Λ] /Λ 2C ∗ (S 1 ) C∗ (S 1 ) ,M (2.45) using Hom-tensor adjunction and (2.44) we get therefore that res Z/p (M)=RHom K[Λ] /Λ 2 (f,g) ∗ (1,i 0 ) ∗ (C ∗ (S 1 )),M (2.46) which, in view of Proposition 2.3.3, implies that res Z/p (M)=RHom K[Λ] /Λ 2 Cell p (S 1 ),M . (2.47) Next we have, Lemma 2.3.4. Cell p (S 1 )⊗ L K[Z/p] K =C ∗ (S 1 ) asK[Λ] /Λ 2 modules, where C ∗ (S 1 )=K[ϵ ]/ϵ 2 (as defined in the previous section). 24 Proof. Define a map of algebras, F :Cell p (S 1 )× K→K[ϵ ]/ϵ 2 (h,1)7−→ ϵ (τ, 1)7−→ 1 ItisimmediatethatthisisaK[Λ] /Λ 2 modulemapanditK[Z/p]-balanced. Nowitiseasyto check that anyK[Z/p] balanced map g : Cell p (S 1 )× K→ X admits a lift g ′ :K[ϵ ]/ϵ 2 → X such that g ′ ◦ F =g. We simply set g ′ (ϵ )=g(h,1) and g ′ (1)=g(τ, 1) This verifies the universal property of tensor product and next we note that Cell p (S 1 ) is a dg free K[Z/p] module, therefore the map Cell p (S 1 )⊗ L K[Z/p] K → Cell p (S 1 )⊗ K[Z/p] K is a quasi-isomorphism [18, Remark 2.10] which proves the result. Using Lemma 2.3.4 we can compute that, (res Z/p (M)) hZ/p =RHom K[Z/p] K,RHom K[Λ] /Λ 2 Cell p (S 1 ),M =RHom K[Λ] /Λ 2 Cell p (S 1 )⊗ K[Z/p] K,M =RHom K[Λ] /Λ 2 K[ϵ ]/ϵ 2 ,M (2.48) as desired. We now prove Proposition 2.2.7. Proof of Proposition 2.2.7. Let us first explicitly construct the quasi equivalences in (2.1) and (2.2) which shall be an ingredient in the proof of the restriction. Lemma 2.3.5. There exists an A ∞ quasi-isomorphism g :C ∗ (Z/p) ∼ − → K[Z/p]. 25 Proof of Lemma 2.3.5. Define g as follows g 1 :K[Z/p]−→ C ∗ (Z/p) τ k 7−→ τ k (2.49) where τ k denotes the zero simplex that maps to τ k inZ/p. This is indeed an isomorphism on homology. Also, since the product of two zero simplices is again a zero simplex we see that g is a map of algebras so we can set all the higher order g k s to zero. Lemma 2.3.6. There exists an A ∞ quasi-isomorphism f :C ∗ (S 1 ) ∼ − → K[Λ] /Λ 2 . Proof of Lemma 2.3.6. We define f 1 :K[Λ] /Λ 2 −→ C ∗ (S 1 ) 17−→ 1 Λ 7−→ Λ sing (2.50) where Λ sing denotes some choice of the fundamental chain on S 1 and 1, as before, denotes the singular 0-simplex that maps to 1∈S 1 . Again, that this is an isomorphism on homology follows directly from definition. However we note that this is not a map of algebras because Λ sing · Λ sing need not equal zero in C ∗ (S 1 ) (though it must be exact). So we must add non trivial higher order terms in order to get a well defined A ∞ morphism. Inductively, assume all f k s up to k≤ j has been defined. The A ∞ equation for (j +1) inputs reads as follows X n;i 1 +i 2 +··· +in=j+1 µ n (f i 1 (a 1 ,··· a i 1 ),··· ,f in (a ((j+1)− i n− 1 +1) ,··· ,a j+1 ))= X t,s≤ j+1 f (j+1)− s+1 (a 1 ,··· ,a t ,µ s (a t+1 ,··· ,a t+s ),a t+s+1 ,··· a j+1 ) (2.51) 26 (here by conflating notation we have called the product operation on both sides as µ , hoping that the context will be enough to differentiate them). NotenowthatsinceK[Λ] /Λ 2 andC ∗ (S 1 )arebothstrictlyunital,wecanseekastrictlyunital f which is completely determined on any sequence of inputs involving 1. So in studying (2.51) we can set all a i ’s to be equal to Λ. Also note that µ ≥ 3 = 0 on both sides and µ 1 (Λ) = µ 2 (Λ ,Λ) = 0 in K[Λ] /Λ 2 . These observations then simplify (2.51) so that we are seeking an element f j+1 (Λ ,··· ,Λ) satisfying: µ 1 (f j+1 (Λ ,··· ,Λ))+ µ 2 (f j (Λ ,··· ,Λ) ,f 1 (Λ))+ µ 2 (f 1 (Λ) ,f j (Λ ,··· ,Λ))=0 (2.52) Next one recalls that f j has degree (1− j) therefore the terms µ 2 (f j (Λ ,··· ,Λ) ,f 1 (Λ)) and µ 2 (f 1 (Λ) ,f j (Λ ,··· ,Λ)) live in degree (1 − 2j− 1) = − 2j. For j > 1 then, these terms already have degree less than− 2. (2.51) also implies that the sum of terms not involving a µ 1 is a cycle and in degree less than− 2 cycles in C ∗ (S 1 ) are automatically boundaries. This observation guarantees that a solution to (2.52) exists. This completes our inductive proof regarding the existence of f. For ease of notation we shall abbreviate (f,g) ∗ (1,i 0 ) ∗ (C ∗ (S 1 )) as B 0 and Cell p (S 1 ) as B 1 . Using this new notation, our goal is to construct a bi-module quasi-isomorphism between B 0 and B 1 . Recall that a bimodule homomorphism between B 0 and B 1 is a collection of maps F m|n : K[Λ] /Λ 2 ⊗ m ⊗B 1 ⊗ K[Z/p] ⊗ n −→B 0 satisfying the equations of an A ∞ bimodule homomorphism(as in [18, definition 2 .12]). As before we shall first define F 0|0 and inductively establish the existence of higher terms. 27 Define, F 0|0 :B 1 −→B 0 τ k 7−→ τ k h7−→ h sing (2.53) whereτ k isasbeforeandh sing denotesachoiceofthe1-simplexthatmapstothearcjoining 1 and τ . Clearly, this is an isomorphism on the level of homology as generators get mapped to generators. Fix some m,n and let us assume that all lower multiplications have been constructed from which we shall construct F m|n inductively. To that end, study the bimodule equation for m,n and for an arbitrary b∈B 1 (as in Lemma 2.3.6 it will suffice to take all inputs to be Λ on the left hand side as we are aiming to construct an unital homomorphism) µ B 0 ( b F(Λ ⊗···⊗ Λ ⊗ b⊗ τ i 1 ⊗···⊗ τ in ))+F(b µ B 1 (Λ ⊗···⊗ Λ ⊗ b⊗ τ i 1 ⊗···⊗ τ in ))=0 (2.54) where µ ’s denote the module action maps corresponding to each subscript and the hat ex- tensions are as defined in [18, Remark 2 .5]. We now appeal to some special properties of the structures at hand to reduce this further. Recall that since g was a strict algebra map, we have that µ k|l B 0 = 0 for all l≥ 2. Also note that since B 1 is a dg bi-module we have that µ k|l B 1 = 0 whenever both l,k ≥ 1. Applying these niceties to (2.54) yields 28 X r<m µ 2 f m− r (Λ ,··· ,Λ) ,F r|n (Λ ,··· ,b,τ i 1 ,··· ) +µ 1 F m|n (Λ ,··· ,b,τ i 1 ,··· ) + µ 2 F m|n− 1 (Λ ,··· ,b,τ i 1 ,··· ),τ in +F m− 1|n Λ ,··· ,Λ · b,τ i 1 ,··· + F m|n− 1 Λ ,··· ,b· τ i 1 ,··· + X k F m|n− 1 Λ ,··· ,b,τ i 1 ,··· ,τ i k +i k+1 ,··· + F m|n (Λ ,··· ,µ (b),τ i 1 ,··· )=0 (2.55) (again we have abused notation here, the µ on the LHS refers to that of C ∗ (S 1 ) and that of Cell p (S 1 ) which will be differentiated depending upon the input). Let’s call F m|n (Λ ,··· ,− ,τ i 1 ,··· ,τ in ) as G m|n i 1 ,··· ,in (− ). Then (2.55) can be rearranged to a simpler form µ ◦ G m|n i 1 ,··· ,in (− )− G m|n i 1 ,··· ,in ◦ µ (− )=F(− ) (2.56) whereF issomeelementofHom(C ∗ (S 1 ),Cell p (S 1 ))obtainedbycollectingrestoftheterms together and is defined and is a cycle by inductive hypothesis. Note that F has degree (− 2m− n + 1), and since we are working over a field H · (Hom Ch K (C ∗ (S 1 ),Cell p (S 1 ))) = Hom grVect K (H · (S 1 ),H · (S 1 ))whichisnontrivialonlyindegrees− 1,0,1. Thereforewhenever (− 2m− n+1) <− 1, a solution G m|n i 1 ,··· ,in to (2.56) exists just for formal reasons. The only values of (m,n) that this argument doesn’t cover are (1,0),(0,1) and (0,2). We shall tackle thesebasecasesofourinductionindividuallynextwhichshallcompletetheproofofexistence. We can set F 0|1 and F 0|2 both equal to zero again for formal reasons. Note that, forgetting the left module action, bothB 0 andB 1 are dgK[Z/p] right modules and F 0|0 defines a strict dg module map between them. So we can set all F 0|r to be identically zero and therefore the same can be done for the two special cases at hand. 29 Coming to the last case, where (m,n)=(1,0), the A ∞ relation reads as follows µ 1 (F 1|0 (Λ ,b)+µ 2 (f 1 (Λ) ,F 0|0 (b))=F 0|0 (Λ · b)+F 1|0 (Λ ,µ (b)) (2.57) Setting b=τ k , and noticing µ (τ k )=0 this reduces to µ 1 (F 1|0 (Λ ,τ k ))=F 0|0 (Λ · τ k )− µ 2 (f 1 (Λ) ,τ k ) (2.58) Now note that both the terms on the right hand side represent co-homologous chains and therefore F 1|0 (Λ ,τ k ) exists. Next setting b = h in (2.57), and using the fact that Λ · h = 0, we have µ 1 (F 1|0 (Λ ,h))+µ 2 (f 1 (Λ) ,h sing )=F 1|0 (Λ ,τ )− F 1|0 (Λ ,1) (2.59) Again we notice that the chains (F 1|0 (Λ ,τ )− F 1|0 (Λ ,1)) and µ 2 (f 1 (Λ) ,h sing ) are cohomolo- gous implying F 1|0 (Λ ,h) exists, completing our proof. 30 Chapter 3 S 1 and Z/p actions in Symplectic cohomology and their compatibility In this section we construct geometric actions on the Floer co-chain complex of the algebras of chains on S 1 and chains onZ/p. We do it via the small models described in the previous chapter. We begin by reviewing symplectic cohomology in 3.1, followed by a review of geometric construction of the S 1 -action on symplectic cohomology. In 3.3 we construct the Z/p-action and in 3.4 we establish that these actions are compatible with small model of restriction that we constructed at the end of the last chapter. 3.1 Review of Symplectic cohomology We begin by recalling the class of symplectic manifolds we will study using symplectic co- homology called Liouville domains. Definition 3.1.1. A Liouville domain is a compact manifold with boundary M 2n equipped with a 1-form θ such that ω = dθ should be symplectic and the vector field Z defined by ι Z ω =θ should point strictly outwards along the boundary ∂M. 31 (Note that in this case α =θ | ∂M is a contact one form on the boundary making it a contact manifold and comes equipped with a vector called the Reeb vector field (denoted R), which is the unique vector field satisfying dα (R,− )=0 and α (R)=1.) The Liouville manifold associated to a Liouville domain is by definition, a certain non- compact manifold obtained by attaching an infinite cone to the boundary using the flow of Z near it. More precisely, one can construct a non compact manifold ˆ M (the associated Liouville manifold) defined as: ˆ M =M∪ ∂M ([0,∞]× ∂M) (3.1) where, ˆ θ |([0,∞]× ∂M)=e r α, ˆ Z|([0,∞]× ∂M)=∂ r , ˆ ω =d ˆ θ (3.2) Symplectic cohomology, defined on a Liouville manifold M which comes equipped with the extra structure of (3.2), is a generalization of Hamiltonian Floer cohomology for compact symplecticmanifolds. RecallthatHamiltonianFloercohomologyforaHamiltonianfunction H on a compact symplectic manifold (W,ω) and some choice of almost complex structure J, denoted HF ∗ (W,H,J), is defined similarly as the Morse cohomology for the following action functional A H :LW →R (3.3) γ 7−→− Z D 2 γ ∗ ω− Z S 1 H(t,γ (t))dt (3.4) so the underlying chain complex CF ∗ (W,H,J) is generated by critical points of the ac- tion functional (which are 1-periodic orbits of H), and the differential δ given by counting solutions of Floer’s equation, i.e, maps u :R× S 1 →W satisfying ∂ s u+J t (u)(∂ t u− X Ht )=0 (3.5) 32 with suitable boundary conditions. AfundamentalresultinFloertheorysaysthatonacompactmanifoldW,theaboveconstruc- tionisindependentofthechoiceoftheHamiltonianH andthealmostcomplexstructure(see [32]). In case of a Liouville manifolds M, the corresponding result fails and in fact the con- struction only makes sense for special kinds of Hamiltonians (for example, Hamiltonians which are linear near the boundary) and almost complex structures. More precisely, fix a τ > 0 which is generic (i.e, not equal to the period of any orbit of the Reeb vector field R) and consider a time dependent Hamiltonian function H τ : M → R which is linear of slope τ on the cylindrical end. Denote the set of such Hamiltonians which are linear on the cylindrical end with generic slopes asH(M). Consider time dependent almost complex structures of contact type which means that near the cylindrical end J must satisfy ˆ θ ◦ J =f(r)dr for some function f satisfying f(r) > 0 and f ′ (r) ≥ 0. Denote the set of such admissible almost complex structures asJ(M). GivenachoiceofadmissiblelinearHamiltonianH τ ∈H(M)andanalmostcomplexstructure J ∈J(M), denote by CF ∗ (M,H τ ,J) the chain complex generated by the 1-periodic orbits of the Hamiltonian vector field X H tau associated to H τ . The differential δ is computed by counting Floer cylinders. More precisely, let x + and x − be two 1-periodic orbits of X H tau, define M(x + ,x − )= u| u :R× S 1 →M,u(s,·)− −−− → s→±∞ x ± , (du− X H τ ⊗ dt) (0,1) =0 (3.6) Lemma3.1.2. Undersuitablegenericityconditionsandundertheconditionsnearthecylin- drical end of M imposed on the Hamiltonian and almost complex structure,M(x + ,x − ) can be shown to have the structure of a manifold with corners. 33 The proof that these are manifolds follow from linearizing the Floer equation at an input cylinder and suitably perturbing our H and J to achieve transversality (see [41],[1] for instance). The moduli space of such cylinders can be compactified by traditional Gromov compactness methods [1, §1.3.3]. Two pathological cases can occur in the limit: we might have sphere bubbling i.e, a sphere might be the result of a sequence of such Floer cylinders. This is prevented by the exactness assumption. The other pathology can be that sequences of curves can escape to escape to ∞. That this is impossible follows from a result akin to a maximum principle which provide conditions so that the curves do not escape to∞ (see [34, §3b,3c]). The dimension of this compactified moduli space is equal to deg(x + )− deg(x − ) where the degree is the Conley-Zehnder index as defined in [1, §1.4.5] If deg(x + ) = deg(x − ), counting the points (with sign) of the compactified manifolds (com- pactness guarantees a finite count) gives us the matrix coefficient of x − in δ (x + ) and a basic theorem in Floer theory which comes from analyzing the boundary components of 1-dimensional compactified moduli spaces states that δ 2 =0 (as in [1, Proposition 1.5.10]). We now define, SH τ (M) = HF(M,H τ ,J), to be the homology of the above complex. It can be shown that for different values of τ the resulting groups are not always the same. However for two different generic choices τ 1 < τ 2 , there are construct continuation maps SH τ 1 (M)→SH τ 2 (M)[1,§1.6.2] from which we define SH ∗ (M)= lim − → τ →∞ SH τ (M) (3.7) with respect to this directed system. This group, it can be shown, is an invariant of the underlying manifold up to Liouville isomorphisms [6, Theorem 1.1],[34]. We shall sometimes refer to the spaces M(x + ,x − ) as M 0 (x + ,x − ) and, the differential δ as δ 0 in the following sections of this chapter to achieve consistency of notation. 34 It is worthwhile to remark here that symplectic cohomology can be defined without direct limits based upon a single Hamiltonian which has quadratic growth at infinity (see for in- stance, [34, § 3d, 3e] for a proof that both approaches give isomorphic results). We shall construct equivariant refinements of both in the following sections. 3.2 The circle action As was noted in the introduction, one can define S 1 -equivariant symplectic cohomology through geometrically defining a (homotopical) action of C ∗ (S 1 ) or rather, its small model K[Λ] /Lambda 2 on the symplectic cochain complex HF(M,H τ ). The circle action on sym- plectic cohomology then can be succinctly packaged as a sequence of operators that act on HF(M,H τ ). This has been investigated in detail in other works before. For example see [17], [41], [34], [7]. Our discussion will essentially follow and summarize the discussion of Ganatra [17,§4.3]. Weconsiderspacesofcylindersasin[17, section4.3]. LetC r bethespaceofcylindersR× S 1 with r marked points p 1 ,··· ,p r satisfying (p 1 ) s <··· <(p r ) s with (x) s denoting theR-coordinate of a point in C r which shall be referred to as height in subsequent discussions. This space (called angle decorated cylinders in [17]) has a naturalR action which acts by translating all the points by a given amount. Let M r =C r /R be the resulting space after quotienting by theR-action. GivenarepresentativeC ofthisspaceofcylinders,leth 1 ,··· ,h r betheheightsofthemarked points and let θ 1 ,··· ,θ r be the angles of the marked points (that is, the S 1 coordinates of the points calculated by identifying the cylinder C withR× S 1 ). The total rotation of the cylinder C is, by definition, the angle θ 1 and the successive differences ( θ i − θ i+1 ) (with 35 θ r+1 = 0) will be the called the incremental angles of the cylinder. We also fix cylindrical ends on C denoted by ϵ + and ϵ − as follows ϵ + : [0,∞)× S 1 →C (s,t)7→(s+h r +δ,t ) (3.8) ϵ − : (−∞ ,0]× S 1 →C (s,t)7→(s− δ − h 1 ,tθ 1 ) (3.9) These ends are disjoint from the r marked points and represent the asymptotic top half and bottom half (rotated by the total angle) of the cylinder and they also vary smoothly with C. There is a natural compactification of the space M r by adding in broken cylinders. The codimension-1 boundary components of the compactification M r are given by two types of strata. The first is a 1-fold broken configuration of the form M r− k × M k and the other arisingfromthecompactificationofthesetofcylinderswheretwoofthemarkedpointshave coincident heights. The set where the heights of the ith and (i+1)th marked points are coincident will be denoted by M i,i+1 r . There is a forgetful map π i : M i,i+1 r → M r− 1 that remembers only first of the angles with coincident heights. That is π i : M i,i+1 r →M r− 1 (h 1 ,··· ,h i ,h i+1 =h i ,h i+2 ,··· ,h r− 1 )7→(h 1 ,··· ,h i ,h i+2 ,··· ,h r− 1 ) (θ 1 ,··· ,θ i ,θ i+1 ,··· ,θ r )7→(θ 1 ,··· ,θ i ,θ i+2 ,··· ,θ r ) (3.10) π i is compatible with the positive and negative ends described above so it extends to com- pactifications M i,i+1 r →M r . We will equip each representative C ofM r with a choice of Floer data as follows: 1. A choice of cylindrical ends ϵ ± as mentioned earlier. 36 2. An one form on C given by α =dt near the cylindrical ends. 3. A surface dependent Hamiltonian H C :C→H(M) satisfying (ϵ ± ) ∗ (H C )=H τ . 4. A surface dependent almost complex structure J C : C → J(M) of contact type on the cylindrical end of our manifold satisfying (ϵ ± ) ∗ (J C )=J, where J is a choice of an admissible time dependent almost complex structure as explained in§3.1 . ThefullchoiceofFloerdatafortheS 1 actionnowconsistsofsmoothlyvaryingsetofchoices made inductively for all C inM r which satisfies the following consistency conditions: • At a boundary component of the type M r− k × M k the choice of data coincides with the product of data chosen on the lower dimensional components • AtaboundarycomponentofthetypeM i,i+1 r thechoiceofdatacoincideswiththepull back of our Floer data via the forgetful map π i . Inductively, since the space of choices at each level is non-empty and contractible, consistent choices of Floer data exist. Choosing such a consistent choice of Floer data as described above, we form the following moduli spaces of curves for two 1-periodic orbits x + and x − of H τ and for each k≥ 1. M r (x + ,x − )= C,u| C∈M r , u :C→M, (ϵ ± ) ∗ u(s,·)− −− → s→∞ x ± , (du− X H C ⊗ dt) (0,1) =0 (3.11) Gromov compactness arguments and the consistency conditions tells us that this moduli space has co-dimension 1 components given by M j (x + ,y)× M r− j (y,x − ), M j,j+1 r (x + ,x − ) (followingfromthediscussionabove(3.10))alongwithcomponentsoftheformM r (x+,y)× M(y,x − ) andM(x + ,y)× M r (y,x − ) (where an unmarked cylinder breaks off from the top or the bottom). 37 Applying usual transversality results (see [41,§4.1],[1, Chapter 2]) give us that, Lemma 3.2.1. For generic choices of the Floer data (as discussed in the preceding bullet points) the moduli spacesM r (x + ,x − ) are manifolds with boundary of dimension deg(x + )− deg(x − )+(2r− 1) (and they are smooth in the case of dimension≤ 1). Signed counts of the rigid elements of these moduli spaces, gives us a sequence of linear operators for each r≥ 1 δ r : CF ∗ (M,H τ )→CF ∗ (M,H τ )[2r− 1] For r =0, we set δ 0 to be the usual Floer differential. Lemma 3.2.2. The operators δ r , for each r satisfy r X i=0 δ i δ r− i =0 . Proof. The discussion of boundary components following (3.11) readily implies that X i δ i δ r− i + X i δ i,i+1 r =0 Next we note that the forgetful map π i has 1-dimensional fibers, therefore given an element (S,u)∈M i,i+1 r (x + ,x − ), we have (S ′ ,u)∈M i,i+1 r (x + ,x − ) for all S ′ ∈π − 1 i (π i (S)) which is an 1-dimensionalspace. SoweconcludethatelementsofM i,i+1 r (x + ,x − )areneverrigid. Hence, we must have that δ i,i+1 r =0 for each i. Observing that, C ∗ (S 1 ) ∼ = K[Λ] /Λ 2 , we can set µ k|1 (Λ ,··· ,Λ) | {z } k times ,− ) = δ k and therefore we have, 38 Proposition 3.2.3. There exists an A ∞ module action ofK[Λ] /Λ 2 on CF ∗ (M,H τ ). That is there exists, for all k≥ 0, maps µ k|1 : K[Λ] /Λ 2 ⊗ k ⊗ CF ∗ (M,H τ )→CF ∗ (M,H τ ) satisfying the A ∞ module equations as in (2.3) We now have to construct equivariant continuation maps between two Hamiltonians as also done in [41, §4.1]. We consider cylinders as before decorated with r points satisfying an order condition on their heights. To that we now add a freely moving (vertically) point β at S 1 coordinate equal to 0. Let M ′ r be the space of such cylinders up to overall translation. As before, for a representative cylinder C of this space, let h 1 ,··· ,h r be the heights of the r points. Theanglesoftheθ 1 ,··· ,θ r arealsodefinedasbeforewith θ 1 beingthetotalrotation of the cylinder. WLOG we can assume that h 1 = 0 and on such a representation of C, we now fix cylindrical ends ϵ + : [0,∞)× S 1 →C ′ (s,t)7→(s+h r− 1 +M(β 1 − h r− 1 )+δ,t ) and ϵ − : (−∞ ,0]× S 1 →C ′ (s,t)7→(s+m(β 1 )− δ,tθ 1 ) These as before represent the asymptotic top half and rotated bottom half of the cylinder disjoint from all the chosen points. Here δ is a fixed positive constant and M(− ), m(− ) are smoothenings of the functions max(− ,0) and min(− ,0) respective so that the ends vary smoothly as we vary C. Compactifying this set yields as its codimension-1 strata 1-fold broken cylinders of the form M ′ k− r × M r ,M k− r × M ′ r and cylindersM ′ j,j+1 k where heights h i =h i+1 . A forgetful map 39 π :M ′ i,i+1 r →M ′ r− 1 also exists as before. A choice of Floer data for such cylinders is similar to what we had in the definition of the circle action and consists of the following data for each C (varying smoothly in C) 1. A choice of cylindrical ends ϵ ± 2. An one form on C given by α =dt near the cylindrical ends. 3. A surface dependent Hamiltonian H C : C → H(M) satisfying (ϵ + ) ∗ (H C ) = H τ 1 and (ϵ − ) ∗ (H C )=H τ 2 . 4. A surface dependent almost complex structure J C :C→J(M) of contact type on the cylindrical end of our manifold satisfying (ϵ ± ) ∗ (J C )=J. To ensure that our operators are well defined we need our choice of Floer data to satisfy consistency conditions as below • AtaboundarystratumoftheformM ′ k− r × M r theFloerdataagreeswiththeproduct of the data chosen forM ′ k− r andM r with Hamiltonian H τ 2 • AtaboundarystratumoftheformM k− r × M ′ r theFloerdataagreeswiththeproduct of the data chosen forM k− r with Hamiltonian H τ 1 and the data chosen forM ′ r . • At a boundary stratum of the formM ′ j,j+1 k the choice of data coincides with the pull back of our Floer data via the forgetful map π i . Let x be an 1-periodic orbit for H τ 1 , y be an 1-periodic orbit for H τ 2 we define the following moduli spaces for each k≥ 0 M ′ k (x,y)= C,u| C∈M ′ k , u :C→M, (ϵ − ) ∗ u(s,·)− −− → s→∞ y (ϵ + ) ∗ u(s,·)− −− → s→∞ x, (du− X H C ⊗ dt) (0,1) =0 (3.12) Compactifying, we observe due to the consistency conditions laid down before, that the 40 moduli space has codimension 1 boundary components of the formM ′ r (x,z)× M k− r (z,y), M k− r (x,z)× M ′ r (z,y) for 0≤ r < k (where a cylinder breaks along the middle taking the continuation zone in the bottom one or in the top one withM ′ 0 denoting the usual moduli spaceofnonequivariantFloercylindersthatdefinecontinuationmaps), M ′ j,j+1 k (x,y)(where two of the heights are coincident) and components of the form M ′ k (x,z)× M(z,y) and M(x,z)× M ′ k (z,y) (where a cylinder breaks off from the top or the bottom). The same transversality results cited as above give us that, Lemma 3.2.4. For generic choices of the Floer data (as discussed in the preceding bullet points) the moduli spaces M ′ k (x,y) are manifolds with boundary of dimension deg(x)− deg(y)+2k (and they are smooth in the case of dimension≤ 1). Signed counts of the rigid elements of such moduli spaces gives us a series of operators f k :CF ∗ (M,H τ 1 )→CF ∗ (M,H τ 2 )[2k] We set, F k|1 τ 1 ,τ 2 (Λ ,··· ,Λ | {z } k times − ) := f k . Analyzing the boundary components of the space of our Floer cylinders which were laid down following (3.12) and observing here too that the map π i has 1-dimensional fibers, we get that X r F k− r|1 τ 1 ,τ 2 (Λ ,··· ,µ r|1 (Λ ,··· ,Λ ,− ))+ X r µ k− r|1 (Λ ,··· ,F r|1 τ 1 ,τ 2 (Λ ,··· ,Λ ,− ))+ δ H τ 1 0 F k τ 1 ,τ 2 +F k τ 1 ,τ 2 δ H τ 2 0 =0 (3.13) where δ H τ i 0 are the usual Floer differential for the respective Hamiltonians. This proves Proposition 3.2.5. Let τ 1 < τ 2 . There exists a well defined degree zero A ∞ K[Λ /Λ 2 ] module homomorphism from CF ∗ (M,H τ 1 ) to CF ∗ (M,H τ 2 ). In other words, there exists for 41 all k≥ 0, a sequence of maps F k τ 1 ,τ 2 : K[Λ /Λ 2 ] ⊗ k ⊗ CF ∗ (M,H τ 1 )→CF ∗ (M,H τ 2 ) satisfying the A ∞ module equations as in (3.13) . A similar argument using two freely moving points instead of one, gives us the following Proposition guaranteeing that these maps form a directed system Proposition 3.2.6. For τ 1 < τ 2 < τ 3 such that none of them are periods of Reeb orbits on ∂M we have that the composition F τ 1 ,τ 2 ◦ F τ 2 ,τ 3 is chain homotopic to F τ 1 ,τ 3 in the category of A ∞ -K[Λ /Λ 2 ] module homomorphisms. 3.3 The Z/p action It was remarked in section 3.1 that there are two equivalent ways of defining Symplectic cohomology. Therefore, as with the circle action one can define Z/p-equivariant symplectic cohomology in two ways. One can define a C ∗ (Z/p) action directly on the symplectic co- chain complex (i.e, the direct limit complex) or one can define a C ∗ (Z/p) action on each of theFloerco-chaincomplexesforthelinearHamiltoniansalongwithequivariantcontinuation maps and then take a direct limit (this is in keeping with Zhao’s construction in [41]). As remarked before, we shall concern ourselves with the second case primarily. But similar constructions can be carried out in the first case too Consider a cylinder R× S 1 with n many marked points {p 1 ,p 2 ,··· ,p n } such that the S 1 - coordinate of each point is fixed and let L 1 <L 2 <··· <L n be the successive heights of the marked points. We assign to each of the p i ’s angles θ i such that the successive differences of the angles or the incremental angles ω i satisfy ω i := (θ i − θ i+1 )∈Z/p⊂ S 1 . LetC n be the spaceofsuchcylinderswheretheL i ’sareallowedtovarybutthemarkersω i ∈Z/parefixed. 42 LetM n (⃗ ω)=C n /Rwhere⃗ ω ={ω 1 ,··· ,ω n }andtheRactionisgivenbytranslatingthefirst marked point. Given a representative C of this moduli space we consider a representative whereh 1 =0 and on that we a fix positive end of the cylinder ϵ + : [0,∞)× S 1 →C mapping (s,t)→(s+L n +δ,t ) and a negative end ϵ − : [0,∞)× S 1 →C mapping (s,t)→(s− δ,tθ 1 ), where δ is a fixed positive constant. The moduli space M n (⃗ ω) admits a compactification M n (⃗ ω) by adding in broken cylinders and cylinders where the ith and the (i+1)th heights are coincident (and the corresponding markersinZ/paremultiplied). Therearethefollowinginclusionmapsfromthecodimension- 1 boundary ofM n (⃗ ω): M n− k (⃗ ω 1 )× M k (⃗ ω 2 )→M n (⃗ ω) (3.14) and M i,i+1 n (⃗ ω)→M n (⃗ ω) (3.15) where {⃗ ω 1 , ⃗ ω 2 } is a partition of ⃗ ω. M i,i+1 n (⃗ ω ′ ) consists of the boundary cylinders with the heights of the ith and the (i+1)th marked points being equal. We will equip each represen- tative C ofM n (⃗ ω) with a choice of Floer data as follows: 1. A choice of cylindrical ends ϵ ± as mentioned earlier. 2. An one form on C given by α =dt near the cylindrical ends. 3. A surface dependent Hamiltonian H C :C→H(M) satisfying (ϵ ± ) ∗ (H C )=H τ . 4. A surface dependent almost complex structure J C :C→J(M) of contact type on the cylindrical satisfying (ϵ ± ) ∗ (J C )=J. ThefullchoiceofFloerdatafortheZ/pactionnowconsistsofsmoothlyvaryingsetofchoices made inductively for all C inM n (⃗ ω) which satisfies the following consistency conditions: • At a boundary component of the type (3.14) the choice of data coincides with the product of data chosen on the lower dimensional components 43 • At a boundary component of the type (3.15) the choice of data coincides with the pull back of our Floer data via the projection map π i :M i,i+1 n (⃗ ω)→M n− 1 ( ⃗ ω ′ ) which forgets one of the coincident heights and ⃗ ω ′ ={ω 1 ,··· ,ω i− 1 ,ω i ω i+1 ,··· ,ω n } • The choice of Hamiltonian for a representative ofM 1 ({1}) is translation invariant. • If any ω i ∈ ⃗ ω equals 1, then the Floer data must agree with the pullback of the Floer data on the space of lower dimensional cylinders via the forgetful map ψ i :M n (⃗ ω)→ M n− 1 ( ˆ ⃗ ω) which forgets ω i . Inductively, since the space of choices at each level is non-empty and contractible, consistent choices of Floer data exist. Choosing such a consistent choice of Floer data as described above, we form the following moduli spaces of curves for two 1-periodic orbits x + and x − of H τ and for each k≥ 1. M k (⃗ ω,x + ,x − )= C,u| C∈M k , u :C→M, (ϵ ± ) ∗ u(s,·)− −− → s→∞ x ± , (du− X H C ⊗ dt) (0,1) =0 (3.16) Gromov compactness arguments tells us that this moduli space has co-dimension 1 com- ponents given by M j (⃗ ω 1 ,x + ,y)× M k− j (⃗ ω 2 ,y,x − ) (where one of the heights L j → ∞), M j,j+1 k (⃗ ω ′ ,x + ,x − ) (where the height L j = L j+1 ) along with components of the form M k (⃗ ω,x+,y)× M 0 (y,x − ) andM 0 (x + ,y)× M k (⃗ ω,y,x − ) (where a cylinder breaks off from the top or the bottom). Similar transversality results give us that, Lemma 3.3.1. For generic choices of the Floer data (as discussed in the preceding bul- let points) the moduli spaces M k (⃗ ω,x + ,x − ) are manifolds with boundary of dimension deg(x + )− deg(x − )+(k− 1) (and they are smooth in the case of dimension≤ 1). Signed counts of the rigid elements of these moduli spaces, gives us a sequence of differential 44 operators µ k|1 (ω 1 ,··· ,ω k ,− ) :CF ∗ (M,H τ )→CF ∗ (M,H τ ) where{ω 1 ,··· ,ω k }=⃗ ω. Varying over all choices of theZ/p markers gives us the following action for all k≥ 0 µ k|1 : K[Z/p] ⊗ k ⊗ CF ∗ (M,H τ )→CF ∗ (M,H τ ) (3.17) Proposition 3.3.2. There exists a well defined strictly unital A ∞ module action of the associative algebraK[Z/p] on CF ∗ (M,H τ ). Proof. The discussion preceding Lemma 3.3.1 directly give us that X r µ k− r|1 (ω 1 ,··· ,ω k− r− 1 (µ r|1 (ω k− r ,··· ,ω k ,− ))+ X r µ k− 1|1 (ω 1 ,··· ,ω r ω r+1 ,··· ,ω k ,− )+δ 0 µ k|1 +µ k|1 δ 0 =0 (3.18) which are precisely the equations we need themto satisfy. Tocheck that the action is unital, we need to establish that for k > 1, µ k|1 (ω 1 ,··· ,ω k ,− )=0 whenever any of the ω i ’s is equal to 1 and that µ 1|1 (1,− ) is the identity operator. To tackle the first implication, without loss of generality, let’s look at µ k|1 (1,··· ,ω k ,− ), the definition of the moduli space then tells us thatM k =M k− 1 × R (due to the consistency conditions laid down) which implies that there can be no rigid components. Similarly, if we look at the moduli space associated with µ 1|1 (1,a)forsomeorbita, weseethatitisjusttheusualmodulispaceofFloercylindersand due to the fact that these admit a freeR action (again because of the consistency conditions we have laid down) we again conclude that there are no rigid solutions except constant solutions which imply that µ 1|1 (1,− ) is indeed the identity operator. NextwedefinecontinuationmapsforbetweentwoHamiltonianwithslopes τ 1 <τ 2 . Consider 45 cylinders C ′ k for k ≥ 0 as before decorated with k points on a straight line at heights L 1 < ··· < L k and angles θ 1 ,··· ,θ k with fixed incremental angles ω i = θ i − θ i+1 inZ/p assigned to them. Also assign a freely moving point β = (β 1 ,1) (in cylindrical coordinates) on the cylinder. LetV k (⃗ ω) be the collection of all such marked cylinders where the heights L i and the height β of the marked points are allowed to vary on a vertical line but the incremental anglemarkersarefixedquotientedbyoveralltranslation. Asbeforeanyelementofthespace of cylinders has a representative C ′ where h 1 =0 on which we fix cylindrical ends ϵ ± where ϵ + : [0,∞)× S 1 →C ′ (s,t)7→(s+L r +M(β 1 − L r )+δ,t ) and ϵ − : (−∞ ,0]× S 1 →C ′ (s,t)7→(s+m(β 1 )− δ,tθ 1 ) Here δ is a fixed positive constant and M(− ), m(− ) are smoothenings of the functions max(− ,0) and min(− ,0) respectively. Thecompactification V k (⃗ ω)ofthissetyields,incodimension1,boundarystrataconsistingof broken cylinders of the formV k− r (⃗ ω 1 )× M r (⃗ ω 2 ),M k− r (⃗ ω 1 )× V r (⃗ ω 2 ) and cylindersV j,j+1 k (⃗ ω) where the heights of the jth and (j +1)th marked points are coincident. A choice of Floer data now for a given cylinder C ′ in this compactification consists of (as before): • A choice of cylindrical ends ϵ ± • A smoothly varying surface dependent Hamiltonian H C ′ which satisfies ( ϵ + ) ∗ (H C ′) = H τ 1 , (ϵ − ) ∗ (H C ′)=H τ 2 . • A surface dependent almost complex structure J C :C→J(M) of contact type on the cylindrical satisfying (ϵ ± ) ∗ (J C )=J • An 1-form α satisfying α =dt near the cylindrical ends 46 Consistency conditions for the data require that • At a boundary stratum of the formV k− r (⃗ ω 1 )× M r (⃗ ω 2 ) the Floer data agrees with the product of the data chosen forV k− r (⃗ ω 1 ) andM r (⃗ ω 2 ) with Hamiltonian H τ 2 • At a boundary stratum of the formM k− r (⃗ ω 1 )× V r (⃗ ω 2 ) the Floer data agrees with the product of the data chosen for M k− r (⃗ ω 1 ) with Hamiltonian H τ 1 and the data chosen forV r (⃗ ω 2 ). • At a boundary stratum of the formV j,j+1 k (⃗ ω) the choice of data coincides with the pull back of our Floer data via the projection map π i :V i,i+1 n (⃗ ω)→V n− 1 (⃗ ω ′ ) which forgets one of the coincident heights and sets ⃗ ω ′ ={ω 1 ,··· ,ω i− 1 ,ω i ω i+1 ,··· ,ω n }. • If any ω i ∈ ⃗ ω equals 1, then the Floer data must agree with the pullback of the Floer data on the space of lower dimensional cylinders via the forgetful map ψ i : V n (⃗ ω)→ V n− 1 ( ˆ ⃗ ω) which forgets ω i . As before, due to contractibility of the space of choices such data can be chosen inductively and consistently. Let x be an 1-periodic orbit for H τ 1 , y be an 1-periodic orbit for H τ 2 we define the following moduli spaces for each k≥ 0 V k (⃗ ω,x,y)= C,u| C∈V k (⃗ ω), u :C→M, (ϵ − ) ∗ u(s,·)− −− → s→∞ y (ϵ + ) ∗ u(s,·)− −− → s→∞ x, (du− X H C ⊗ dt) (0,1) =0 (3.19) Compactifying,weobserveduetotheconsistencyconditionslaiddownbefore,thatthemod- uli space has codimension 1 boundary components of the formV r (⃗ ω 1 ,x,z)× M k− r (⃗ ω 2 ,z,y), M k− r (⃗ ω 1 ,x,z)× V r (⃗ ω 2 ,z,y) for 0≤ r <k (where a cylinder breaks along the middle taking thecontinuationzoneinthebottomoneorinthetoponewithV 0 denotingtheusualmoduli spaceofnonequivariantFloercylindersthatdefinecontinuationmaps), V j,j+1 k (⃗ ω,x,y)(where 47 two of the marked points collide) along with components of the formV k (⃗ ω,x,z)× M 0 (z,y) andM 0 (x,z)× V k (⃗ ω,z,y) (where a cylinder breaks off from the top or the bottom). Transversality considerations give us Lemma 3.3.3. For generic choices of the Floer data (as discussed in the preceding bullet points) the moduli spaces V k (⃗ ω,x,y) are manifolds with boundary of dimension deg(x)− deg(y)+k (and they are smooth in the case of dimension≤ 1). Signed counts of the rigid elements of these moduli spaces, gives us a sequence of operators F k τ 1 ,τ 2 (ω 1 ,··· ,ω k ,− ) :CF ∗ (M,H τ )→CF ∗ (M,H τ ) where the ω i ’s are the fixed markers defined at the beginning of the section. Varying the markers over all possible choices inZ/p, gives us the following Proposition Proposition 3.3.4. Let τ 1 <τ 2 . There exists a well defined degree zero A ∞ K[Z/p] module homomorphism from CF ∗ (M,H τ 1 ) to CF ∗ (M,H τ 2 ). In other words, there exists for all k≥ 0, a sequence of maps F k τ 1 ,τ 2 : K[Z/p] ⊗ k ⊗ CF ∗ (M,H τ 1 )→CF ∗ (M,H τ 2 ) satisfying the A ∞ module equations X r F k− r|1 τ 1 ,τ 2 (ω 1 ,··· ,µ r|1 (ω k− r ,··· ,ω k ,− ))+ X r µ k− r|1 (ω 1 ,··· ,F r|1 τ 1 ,τ 2 (ω k− r ,··· ,ω k ,− ))+ δ H τ 1 0 F k τ 1 ,τ 2 +F k τ 1 ,τ 2 δ H τ 2 0 =0 (3.20) where δ H τ i 0 are the usual Floer differentials for the respective Hamiltonians. Also, F k τ 1 ,τ 2 (··· ,1,··· ,− )=0 for k≥ 1 48 Proof. the proof of this follows directly from analyzing the boundary components that we havelistedfollowingthedefinitionof themodulispaces V k (⃗ ω)and unitalityfollowsas before from the special choice of the consistency conditions that we wanted the Floer data to satisfy. A similar argument using two freely moving points instead of one, gives us the following Proposition which guarantees that these maps form a directed system Proposition 3.3.5. For τ 1 < τ 2 < τ 3 such that none of them are periods of Reeb orbits on ∂M we have that the composition F τ 1 ,τ 2 ◦ F τ 2 ,τ 3 is chain homotopic to F τ 1 ,τ 3 in the category of A ∞ -K[Z/p] module homomorphisms. The resulting equivariant homology of this complex will be denoted by SH Z/p (M). TheZ/2 case can be defined precisely as in (1.6). The general case follows the definition in [37, §6]. Before moving on, let us state a result that constructs the equivariant complex in the case of a quadratic Hamiltonian too, as was remarked in the first paragraph of this section. It was noted in §3.1 that the symplectic co-chain complex as in the direct limit definition is equivalenttotheFloercomplexgenerated bya suitable suitable quadratic Hamiltonian. Let H q be this Hamiltonian as before, then the arguments of this section imply Proposition 3.3.6. There exists an A ∞ module action ofK[Z/p] on CF ∗ (M,H q ). That is there exists, for all k≥ 0, maps µ k : K[Z/p] ⊗ k ⊗ CF ∗ (M,H q )→CF ∗ (M,H q ) satisfying the A ∞ equations as in (3.18). 49 3.4 Compatibility of the S 1 action and the Z/p action NowweneedtoshowthattherestrictionmapwhenappliedtotheK[Λ] /Λ 2 modulestructure in§3.2inducestheK[Z/p]modulestructureof§3.3. FollowingProposition2.3.2itisenough to show that there exists a strictly unital Cell p (S 1 ) module structure on CF ∗ (M,H τ ) which whenrestrictedtothesub-algebrasK[Λ] /Λ 2 andK[Z/p]inducestheactionsdescribedabove. Therefore, we need to construct operators µ k (⃗ x,− ) where ⃗ x ={x 1 ,··· ,x k } is a sequence of length k with each x i ∈ {1,ω,··· ,ω p− 1 ,h,hω,··· ,hω p− 1 }. We shall go about this induc- tively by establishing a partial order over all possible sequence of inputs of varying lengths. The ordering will be as follows • A sequence of inputs ⃗ x 1 ≺ ⃗ x 2 , if the length of ⃗ x 1 is less than the length of ⃗ x 2 . • If two sequences ⃗ x 1 and ⃗ x 2 have the same length, then ⃗ x 1 ≺ ⃗ x 2 if ⃗ x 2 has more terms involving the variable h. We start by setting µ 1 (ω k ,− ), for 0 ≤ k < p, to be the same as the µ 1|1 (ω k ,− ) defined in the section where we defined the Z/p action. Next we go about defining operators of the form µ 1 (hω k ,− ), for 0≤ k < p. For that we consider spaces of cylinders R× S 1 with one marked point at height h 1 and angle θ 1 ∈ (2kπ/p, 2(k+1)π/p ). LetM 1 (hω k ) be the space of all such cylinders quotiented by theR-action of translation. Given a representative C ofM 1 (hω k ), we choose an equivalent normalised representation of C so that h 1 =0, and define cylindrical ends ϵ + : [0,∞)× S 1 →C (s,t)7→(s+δ,t ) (3.21) ϵ − : (−∞ ,0]× S 1 →C (s,t)7→(s− δ,tθ 1 ) (3.22) 50 Clearly, these ends are disjoint from the marked point and vary smoothly in C. CompactifyingM 1 (hω k )leadstocodimension1boundarystrataoftheformM 1 (ω k )(where θ 1 =2kπ/p ) and of the formM 1 (ω k+1 ) (where θ 1 =2(k+1)π/p ). For a normalised element ofM 1 we choose Floer data as before specifying 1. A choice of cylindrical ends ϵ ± as mentioned earlier. 2. An one form on C given by α =dt near the cylindrical ends. 3. A surface dependent Hamiltonian H C :C→H(M) satisfying (ϵ ± ) ∗ (H C )=H τ . 4. A surface dependent almost complex structure J C :C→J(M) of contact type on the cylindrical satisfying (ϵ ± ) ∗ (J C )=J. These choices must be compatible with the codimension 1 boundary components in the sense that the choice restricted to the boundary component M 1 (ω k ) must match with the data chosen while defining µ 1 (ω k ,− ) and the choice restricted to the boundary component M 1 (ω k+1 ) must match with the data chosen while defining µ 1 (ω k+1 ,− ). Since the total spaces of choices is contractible we can make universal and consistent choices satisfying the consistency condition. Choosing such a consistent choice of Floer data as described above, we form the following moduli spaces of curves for two 1-periodic orbits x + and x − of H τ M 1 (hω k ,x + ,x − )= C,u| C∈M 1 (hω k ), u :C→M, (ϵ ± ) ∗ u(s,·)− −− → s→∞ x ± , (du− X H C ⊗ dt) (0,1) =0 (3.23) The consistency conditions guarantee that the compactified moduli space M 1 (hω k ,x + ,x − ) has codimension 1 boundary components given byM 1 (ω k ,x + ,x − ),M 1 (ω k+1 ,x + ,x − ) along with semi stable strip breaking boundaries of the form M 0 (x + ,y)× M 1 (hω k ,y,x − ) and M 1 (hω k ,x + ,y)× M 0 (y,x − ). 51 UsingusualtransversalityresultswegetthatthemodulispaceM 1 (hω k ,x + ,x − )isamanifold with boundary of dimension deg(x + )− deg(x − ) + 1 (and they are smooth in the case of dimension≤ 1). A signed count of it’s rigid components gives us the operator µ 1 (hω k ,− ) : CF ∗ (M,H τ )→CF ∗ (M,H τ ). Analyzing the boundary components described earlier, we conclude that this operator satis- fies the relationship δ 0 µ 1 (hω k ,− )+µ 1 (hω k ,− )δ 0 +µ 1 (ω k+1 ,− )− µ 1 (ω k ,− )=0 or δ 0 µ 1 (hω k ,− )+µ 1 (hω k ,− )δ 0 +µ 1 (d(hω k ),− )=0 whereδ 0 istheusualFloerdifferential. This,werealize,ispreciselythe A ∞ relationthatthese operatorsaresupposedtosatisfy. Thisgivesusthebasecasesfortheinductiveconstruction. Now we move on to construct more general operators of the form µ k (⃗ x,− ) where ⃗ x = {x 1 ,··· ,x k } is a fixed input sequence of length k with each of its components x i in the set {1,ω,··· ,ω p− 1 ,h,hω,··· ,hω p− 1 }. We shall assume as our inductive step that all operators corresponding input sequences ⃗ y satisfying ⃗ y≺ ⃗ x has been constructed before. For this we consider cylinders C k with k marked points. Let h 1 < h 2 < ··· < h k be the height coordinate of the points and let θ 1 ,··· ,θ k be the angle coordinates satisfying θ i =θ i− 1 +2bπ/p if x i =ω b or θ i =θ i− 1 +ϕ i where ϕ i ∈(2bπ/p, 2(b+1)π/p ) if x i =hω b , for 0≤ b

Abstract (if available)

Abstract Symplectic cohomology is used as an invariant of a class of exact symplectic manifolds with boundary. There’s also an S^1-equivariant refinement of the theory that has been defined in literature. Equivariant theories with respect to the subgroup Z/p of roots of unity in S^1 are also of interest in Floer Theory. In my thesis, I define Z/p-equivariant symplectic cohomology for any prime p extending work done in the literature so far in the case of p = 2. The main result of this thesis is the existence of a Gysin type long exact sequence relating the S^1 and Z/p equivariant versions of symplectic cohomology. We use this result to answer affirmatively a conjecture made by Seidel regarding the structure of localized S^1-equivariant symplectic cohomology.

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Creator Sen, Debtanu (author)

Core Title The Z/pZ Gysin sequence in symplectic cohomology

Contributor Electronically uploaded by the author (provenance)

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Degree Conferral Date 2023-05

Publication Date 02/02/2023

Defense Date 01/18/2023

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag differential geometry,equivariant,Gysin sequence,OAI-PMH Harvest,symplectic cohomology,symplectic geometry

Format theses (aat)

Language English

Advisor Ganatra, Sheel (committee chair), Bars, Itzhak (committee member), Helfer, Joseph (committee member), Siegel, Kyler (committee member)

Creator Email dsen@usc.edu,sendebtanu@gmail.com

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