Multiplicative structures on cones and duality
aa r X i v : . [ m a t h . S G ] A ug MULTIPLICATIVE STRUCTURES ON CONES ANDDUALITY
KAI CIELIEBAK AND ALEXANDRU OANCEA
Abstract.
We initiate the study of multiplicative structures oncones and show that cones of Floer continuation maps fit naturallyin this framework. We apply this to describe the multiplicativestructure on Rabinowitz Floer homology via cones and we reprovethe Poincar´e duality theorem from this perspective. We generalizethe splitting theorem previously proved in the context of cotan-gent bundles to large classes of Weinstein domains. We express themixed cap products in string topology in terms of loop productsand coproducts, and we describe extended loop homology entirelyin terms of Morse homology. The paper contains three appen-dices. The first one discusses multilinear shifts in a multilinearsetting. The second one, of operadic flavor, discusses sign conven-tions and the language of trees to describe algebraic operations ina dg setting. The third one contains a description of the producton extended loop homology in the language of Morse theory andcones.
Contents
1. Introduction 22. Products on cones 72.1. The point of view of split pairs of A -algebras 92.2. The point of view of -bimodules 102.3. Arity 2 122.4. Examples 193. Cones of Floer continuation maps 223.1. Closed symplectically aspherical manifolds 223.2. Completions of Liouville domains 244. Duality theorems in Floer theory using cones 344.1. Cone duality theorem 344.2. Cone description of the Rabinowitz Floer homology ring 364.3. Proof of the Poincar´e Duality Theorem using cones 395. Products on cones and splittings 475.1. Algebraic setting 47 Date : September 1, 2020. A ` -structures 74B.1. Fundamental conventions 74B.2. Consequences of the fundamental conventions 76B.3. The language of trees 78B.4. A ` -structures 80Appendix C. Extended loop homology product via Morse theory 86References 91 Introduction
This paper is a companion to [4] and, as such, its initial motivation isprovided by the Poincar´e Duality Theorem [4, Theorem 1.1]. We referto that paper for further motivation coming from the topology of loopspaces.Let W be a Liouville domain of dimension 2 n and assume for simplicitythat (i) 2 c p W q “
0, and (ii) the square canonical bundle is trivialized,so that we have canonical Z -gradings in Hamiltonian Floer theory. (Inthe absence of these assumptions the gradings are relative and must beunderstood modulo twice the minimal Chern number.) Following [5]we denote the Rabinowitz Floer homology and cohomology groups of W by SH ˚ pB W q and SH ˚ pB W q , also called symplectic (co)homologygroups of B W . It was shown in [5] that Rabinowitz Floer homology SH ˚ pB W q carries a unital product, of degree ´ n with respect to theConley-Zehnder index grading. The main result of [4] is the following. Theorem 1.1 ([4, Theorem 4.8]) . Let W be a Liouville domain ofdimension n .(i) Rabinowitz Floer cohomology carries a canonical unital degree n ´ product.(ii) There is a canonical “Poincar´e duality” isomorphism of unital rings P D : SH ˚ pB W q » SH ´˚` pB W q . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 3
From the perspective of [4] the proofs of (i) and (ii) are inextrica-bly related: the canonical isomorphism
P D induces a canonical unitalproduct on Rabinowitz Floer cohomology. A first contribution of thepresent paper is to disconnect these two sides of Theorem 1.1. Thus: ‚ We give an intrinsic description of the unit and of the product on SH ˚ pB W q ( §§ not the tautological form of Poincar´e duality in Floer theory. The latter al-lows to identify SH ´˚ pB W q , which is a Floer co homology group basedon “ _ -shaped” Hamiltonians, with a Floer homology group based on“ ^ -shaped” Hamiltonians, denoted y SH ˚ pB W q , see Figure 6 and § SH ˚ pB W qr n s and y SH ˚ pB W qr n ´ s (Theorem 4.7) involves a bit of new theory dealingwith multiplicative structures on cones. This brings us to the secondcontribution of this paper: ‚ We study multiplicative structures on cones of continuation mapsin Floer theory.The algebraic setup is the following. Let A Ă B be a pair of A -algebras over a principal ideal domain R . Assuming that the exactsequence 0 Ñ A Ñ B Ñ B { A Ñ R -modules,the chain complex underlying B is identified with the cone of a map c : B { A r s Ñ A . We spell out the structure induced on the triple p B { A r s , c, A q by the A -structure on B , and, with a general chaincomplex M playing the role of B { A r s , we reach the definition ofan A -triple p M , c, A q , where c : M Ñ A is a degree 0 chain map(Definition 2.2). Restricting to arity 2 operations gives an A -triple (Definition 2.7, Lemma 2.8).Equation (12) from Proposition 2.9 is key to this paper. It gives theformula for the product on the cone of a degree 0 map c : M Ñ A extending a given product µ on A , given the data of an A -triple for p M , c, A q . Such data consists of maps m L : A b M Ñ M , m R : M b A Ñ M of degree 0 , This is the same as SH ˚ pB W ˆ I, B W ˆ B I q from [5]. KAI CIELIEBAK AND ALEXANDRU OANCEA τ L : M b A Ñ A , τ R : A b M Ñ A , σ : M b M Ñ M of degree 1 ,β : M b M Ñ A of degree 2that satisfy the compatibility relations spelled out in Definition 2.7.Equation (12) for the product on Cone p c q “ A ‘ M r´ s reads m ` p a, ¯ x q , p a , ¯ x q ˘ “ ` µ p a, a q ` p´ q | a | τ R p a, x q ` τ L p x, a q ´ p´ q | ¯ x | β p x, x q , p´ q | a | m L p a, x q ` m R p x, a q ´ p´ q | ¯ x | σ p x, x q ˘ . Here ¯ x P M r´ s denotes the shift of an element x P M , of degree | ¯ x | “ | x | `
1. This setup is applicable to Floer theory because of thefollowing observation. ‚ Floer continuation maps give rise to A -triples which are canoni-cally defined up to isomorphism.We prove in this paper only the arity 2 version of the above statement,i.e., the existence of A -triples which are canonically defined up to ho-motopy (Proposition 3.1). The construction involves moduli spaces ofsolutions of Floer equations parametrized by simplices of dimension 0,1, and 2. The construction of the A -triple, which we will address ina sequel paper, involves moduli spaces parametrized by higher dimen-sional simplices, much in the manner of [7]. See also Remark B.8.Given a Liouville domain W , we apply this construction to varioussystems of continuation maps. The archetypal one consists of the con-tinuation maps c ν,ν : F C ˚ p K ν q Ñ F C ˚ p K ν q , ν ă ă ν , where K ν , K ν are Hamiltonians defined on the symplectic completion x W “ W Y r ,
8q ˆ B W , equal to 0 on W and linear of slopes ν , ν on r , W with respect to the coordinate r P r , . See Example 3.5.Sara Venkatesh defined in [25] Rabinowitz Floer homology in non-exactsettings as the cone of such a continuation map at additive level. Theperspective on multiplicative structures that we adopt here is robustand would carry over to such situations.We use various other systems of continuation maps in order to inter-polate between the product structures on the groups that are of maininterest to us, namely SH ˚ pB W qr n s and y SH ˚ pB W qr n ´ s , see Exam-ple 3.6. As a consequence: ‚ We give alternative proofs of the Poincar´e Duality Theorem 1.1using the formalism of cones (Theorems 4.1 and 4.7).As an outcome we obtain several equivalent descriptions of the sec-ondary degree n ´ SH ˚ pB W q ,or of the secondary degree ´ n ` y SH ˚ pB W q , which coincideunder the previous canonical isomorphisms. Among these, we mention: ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 5 — the “Poincar´e duality product” σ P D on SH ˚ pB W q (degree n ´ σ P D on y SH ˚ pB W q (degree ´ n ` σ c on y SH ˚ pB W q (degree ´ n `
1) de-fined in § σ c on SH ˚ pB W q (degree n ´ The products σ P D and σ c admit restrictions to SH ˚ą p W q , similarlythe products σ P D and σ c admit restrictions to y SH ă ˚ p W q , and theserestrictions coincide. It was already known that positive symplecticcohomology SH ˚ą p W q carries a product of secondary nature, the “vary-ing weights product” σ w (see [7, 5], but the construction goes back toSeidel and Abbondandolo-Schwarz [2]). ‚ We show in § σ P D “ σ c “ σ w between the Poincar´e duality product, the continuation product,and the varying weights product on SH ˚ą p W q .One important new ingredient in [4] was the extension of the loop coho-mology product to so-called “reduced loop cohomology”. In the contextof general Liouville domains we can easily generalize the definition asfollows. Consider the cone long exact sequence ¨ ¨ ¨ Ñ SH ´˚ p W q c ˚ Ñ SH ˚ p W q Ñ SH ˚ pB W q Ñ SH ´˚` p W q Ñ . . . and define reduced symplectic (co)homology groups SH ´˚ p W q “ ker c ˚ , SH ˚ p W q “ coker c ˚ , so that we have short exact sequences(1) 0 Ñ SH ˚ p W q Ñ SH ˚ pB W q Ñ SH ´˚` p W q Ñ . The point of [4] was that, in the case of cotangent bundles, these shortexact sequences admit a canonical splitting which is compatible withproducts. ‚ We define the class of “cotangent-like Weinstein domains” (Defini-tion 5.13), for which the short exact sequence (1) admits a canonicalsplitting compatible with products.Cotangent-like Weinstein domains are by definition domains W of di-mension 2 n ě H n p W q n -handles and such that the map H n p W q Ñ SH p W q is in-jective. See Remark 5.12 for examples which include, besides cotangentbundles, subcritical Stein domains and many Milnor fibres. In view ofthe canonical maps SH ˚ p W q Ñ SH ˚ p W q and SH ˚ą p W q Ñ SH ˚ p W q , The underscores in σ c and σ c just indicate some degree shifts, see § KAI CIELIEBAK AND ALEXANDRU OANCEA the above statement expresses in particular the fact that, for cotangent-like Weinstein domains, the pair-of-pants product on symplectic homol-ogy descends to reduced symplectic homology, and the varying weightsproduct on positive symplectic cohomology extends to reduced sym-plectic cohomology. This was seen to be true for cotangent bundlesin [4] and motivates our terminology “cotangent-like Weinstein do-mains”.Still in the case of cotangent-like Weinstein domains, the ring structureon(2) SH ˚ pB W qr n s “ SH ˚ p W qr n s ‘ SH ´˚ p W qr n s can be expressed by components m ``` , m `´` etc., with the meaningthat the superscripts stand for the inputs, the subscript stands forthe output, ` stands for SH ˚ p W qr n s “ SH ˚` n p W q and ´ standsfor SH ´˚ p W qr n s “ SH ´˚´ n p W q . For example m ´´´ is the product σ P D “ σ c “ σ w on reduced symplectic cohomology. It turns out thatthe secondary product on reduced symplectic cohomology SH ˚ p W q has its origin in a secondary co product λ on reduced symplectic ho-mology SH ˚ p W q . Moreover, all the components m ˚˚˚ are determinedby the product µ on reduced symplectic homology, the coproduct λ ,and the canonical degree 0 pairing x¨ , ¨y : SH ´˚ p W q b SH ˚ p W q Ñ R .This is formalized in Theorem 1.2. In the statement we use the fol-lowing notation: given ¯ f P SH ´˚ p W qr n s “ SH ´˚´ n p W q , we denote f P SH ´˚ p W qr n s “ SH ´˚´ n the same element with degree shifteddown by 1, i.e. | f | “ | ¯ f | ´ Theorem 1.2.
Let W be a cotangent-like Weinstein domain of di-mension n ě . The components of the product m on SH ˚ pB W qr n s with respect to the splitting (2) are, for a, b P SH ˚ p W qr n s , ¯ f , ¯ g P SH ´˚ p W qr n s , given by(1) m ``` “ µ ;(2) x m ´´´ p ¯ f , ¯ g q , a y “ p´ q | g | x f b g, λ p a qy ;(3) m ``´ “ m ´´` “ ;(4) m ´`` p ¯ f , a q “ ´p´ q | f | x f b , λ p a qy , and m `´` p b, ¯ f q “ p´ q | b | x λ p b q , b f y ;(5) x m ´`´ p ¯ f , a q , b y “ x f, µ p a, b qy , and x a, m `´´ p b, ¯ f qy “ p´ q | b | x µ p a, b q , f y . In the special case of cotangent bundles of manifolds of dimension ě ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 7 associativity of the product m and thus pertains to item (i) in our listof further developments below.Theorem 1.2 above has an algebraic origin. This is discussed in Appen-dix B. It relies ultimately on the fact that the A -triple which governsthe multiplicative structure for Rabinowitz Floer homology is of theform p A _ , c, A q , with A _ the graded dual of A . We conjecture thatour Poincar´e duality theorem 1.1 is an instantiation of Tate duality and we intend to explore this in future work. One relevant reference inthis direction is the paper [17] by Rivera and Wang, where the authorsprove an isomorphism between Rabinowitz Floer homology with ra-tional coefficients of cotangent bundles of simply connected manifoldsand singular Hochschild cohomology of the dga of cochains. SingularHochschild cohomology is only one of the many names of Tate coho-mology.Appendix A discusses signs for degree shifts in a multilinear setting.Finally, in Appendix C: ‚ We apply the formalism of multiplicative structure on cones in or-der to give a purely Morse theoretic description of the product onextended loop homology from [4].
Further developments.
We indicate three directions:(i) construct A -structures on cones of Floer continuation maps;(ii) interpret the Poincar´e duality theorem 1.1 as Tate duality;(iii) incorporate BV-structures in the formalism of multiplicativestructures on cones.These questions are of very different caliber, with an obvious order(iii) ă (i) ă (ii). Acknowledgements.
This paper is a split-off from our collaborationwith Nancy Hingston on Poincar´e duality. Without her far reachingvision this could not have come to being. The second author acknowl-edges the hospitality of Helmut Hofer and IAS in 2017 and 2019, whenthis project took shape. The authors benefited from discussions withM. Abouzaid, B. Chantraine, P. Ghiggini, and S. Venkatesh. The sec-ond author acknowledges financial support via the ANR grants MI-CROLOCAL ANR-15-CE40-0007 and ENUMGEOM ANR-18-CE40-0009. 2.
Products on cones
Throughout this section we use homological conventions and coeffi-cients in a principal ideal domain R . Recall that, given a chain map c : M ˚ Ñ A ˚ between chain complexes whose differentials have degree KAI CIELIEBAK AND ALEXANDRU OANCEA ´
1, the cone of c is Cone p c q “ A ‘ M r´ s with differential B Cone p c q p a, x q “ pB A a ` c p x q , ´B M x q and M r´ s ˚ “ M ˚´ , B M r´ s “ ´B M . The inclusion A ã Ñ Cone p c q , a ÞÑ p a, q isa chain map. We shall freely refer in this section to the notation ofAppendix A regarding degree shifts in the multilinear setting. Definition 2.1. An A -algebra A ˚ is a Z -graded R -module endowedwith a collection of maps µ d A : A r´ s b d Ñ A r´ s , d ě of degree ´ satisfying the relations (3) ÿ i ` j “ k ` i ÿ t “ µ i A p b t ´ b µ j A b b i ´ t q “ , k ě . In the absence of the shift the structure maps µ d A : A b d Ñ A havedegree d ´
2. The functional relation translates by evaluation into therelation ÿ i ` j “ k ` i ÿ t “ p´ q } a }`¨¨¨`} a t ´ } µ i A p a , . . . , a t ´ , µ j A p a t , . . . , a t ` j ´ q , a t ` j , . . . a i q “ , k ě , where } a } “ | a | ` A -algebra is equivalent to the data of a square-zero coderivation on T c p A r´ sq “ ‘ k ě A r´ s b k , the reduced tensor coalgebra on A r´ s .This is because each µ d A can be uniquely extended as a coderivation µ d A : T c p A r´ sq Ñ T c p A r´ sq and, setting µ A “ ř d µ d A , the A -relations are equivalent to µ A ˝ µ A “ µ d A “ d ą A -structure determines a dga structure on A by the formulas B A “ ´ µ A “ µ A r s ,a ¨ a “ p´ q | a | µ A p a, a q “ µ A r ,
1; 1 sp a, a q . (4)See Appendix A for an explanation of the notation µ A r ,
1; 1 s . Inthe case where µ d A “ d ą A .More precisely, with f p a , a , a q “ p´ q | a | µ A p a , a , a q , i.e. f “ ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 9 µ A r , ,
1; 1 s in the notation of Appendix A, we have (5) p a ¨ a q ¨ a ´ a ¨ p a ¨ a q “ rB A , f s . Our discussion evolves around the following refinement of the notionof an A -algebra. Definition 2.2. An A -triple p M , c, A q consists of an A -algebra A ˚ ,a chain complex M ˚ and a degree chain map c : M ˚ Ñ A ˚ , together with an A -algebra structure on Cone p c q extending the A -algebra structure on A via the inclusion A ã Ñ Cone p c q “ A ‘ M r´ s ,and such that µ Cone p c q “ B Cone p c q r´ s . The point of view of split pairs of A -algebras. The dataof an A -triple is equivalent to the data of an R -split pair of A -algebras , or simply split pair of A -algebras , meaning an inclusion of A -algebras A Ă B together with a splitting (as graded R -modules) s : B { A Ñ B of the short exact sequence 0 Ñ A Ñ B Ñ B { A Ñ B in upper triangularform with respect to the decomposition B “ A ‘ p B { A q induced bythe splitting, we set M “ p B { A qr s and we define c : M Ñ A to bethe p B { A , A q -component of the differential. Then there is an inducedstructure of A -triple on p M , c, A q such that Cone p c q “ B . Conversely,given an A -triple p M , c, A q the inclusion A Ă Cone p c q is obviouslysplit. The conventions for the definition of an A -algebra vary greatly throughout theliterature, and the difference stems mainly from the point of view adopted: eithersquare-zero coderivation on T c p A r´ sq , or homotopy relaxation of dga structure.All conventions are equivalent to one another by suitable sign changes, and theshort note by Polishchuk [16] contains a useful comparison. Our convention de-rives the functional relation (3), which involves no signs, from the condition thatthe associated coderivation on the reduced tensor coalgebra on A r´ s squares tozero. This is the point of view of Fukaya-Oh-Ohta-Ono [9, Definition 3.2.3] and Sei-del [(2.1)][19]. Seidel’s convention in [20, (1.2)] is essentially the same except thatthe maps µ d A are viewed as acting from the right, so one passes from one conventionto the other by defining ˜ µ d A p a d , . . . , a q “ µ d A p a , . . . , a d q .The conventions of Lef`evre-Hasegawa [12, 1.2.1.1] and Markl [14, (2)], whichcoincide, are such that the first three structure maps directly define on A anassociative-up-to-homotopy differential graded algebra structure. They are thesame as the original one of Stasheff [21], cf. [16]. Lef`evre-Hasegawa writes downin [12, Lemma 1.2.2.1] a transformation through which this point of view is equiva-lent to ours, and that transformation inspired our treatment of shifts in Appendix A.The conventions by which one associates to an A -algebra in the sense of Defi-nition 2.1 an associative-up-to-homotopy dga structure vary greatly as well. Oursis different from both the one of Fukaya-Oh-Ohta-Ono [9, (3.2.5)] and the one ofSeidel [20, (1.3)]. We favour it because it fits into a systematic procedure of shiftingmultilinear maps, cf. Appendix A, and also because it realizes p A , B A q as the shiftof p A r´ s , µ A q . The key property of A -structures is that they obey the HomotopyTransfer Theorem, see [14] for its most general form and [13, § A -algebras A Ă B the homotopy transfer theorem adapts in an obvious way byconsidering only maps which are upper triangular with respect to thedecomposition B “ A ‘ B { A provided by the splitting. Definition 2.3.
Let A Ă B and A Ă B be split pairs of chain com-plexes. We say that the pair A Ă B is a homotopy retract of the pair A Ă B if there are maps H (cid:24) (cid:24) B P / / B I o o which are upper triangular with respect to the decompositions B “ A ‘ B { A and B “ A ‘ B { A provided by the splittings and such that rB , H s “ ´ IP.
The proof of the next theorem is the same as that of [14, Theorem 5].It specifically uses the upper triangular form of the maps
P, I, H , andalso the explicit formulas provided by Markl in [14].
Theorem 2.4 (Homotopy transfer for split pairs) . Given a homotopyretraction of pairs as above, and given an A -algebra structure on B such that A is a subalgebra, there is an A -structure on B such that A is a subalgebra, and there are extensions of P, I to -morphismsof pairs ˜ P , ˜ I and of H to an -homotopy ˜ H between ˜ I ˜ P and whichpreserves A . (cid:3) The transferred A -structure and the extensions ˜ P , ˜ I, ˜ H are describedvery explicitly in terms of summation over trees whose vertices are atleast trivalent, see [14] and also [11].One particularly relevant situation is that in which the maps P, I defin-ing a homotopy retract are actually homotopy inverses, with the ho-motopy H : B Ñ B such that rB , H s “ ´ P I being also in uppertriangular form. The next result is the analogue of [14, Proposition 12].
Proposition 2.5.
In the preceding situation, the homotopy H can beextended to an -homotopy ˜ H between ˜ P ˜ I and which preserves A if r P H ´ H P s “ P H p Hom p B , B qq , and P H ´ H P admits a primitive which is upper triangular. (cid:3) The point of view of -bimodules. Recall that an A -algebra B is a graded R -module with operations µ d B : B r´ s b d Ñ B r´ s , d ě ´ ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 11
The data of an A -triple p M , c, A q can then be explicitly encoded intwo collections of operations m i | j | ... | i k | j k : A r´ s b i b M r´ s b j b¨ ¨ ¨b A r´ s b i k b M r´ s b j k Ñ M r´ s , and τ i | j | ... | i k | j k : A r´ s b i b M r´ s b j b¨ ¨ ¨b A r´ s b i k b M r´ s b j k Ñ A r´ s of degree ´
1, indexed by tuples of non-negative integers i , j , . . . , i k , j k such that the intermediate indices j , i , j , . . . , i k are nonzero—a con-vention which we adopt for non-redundancy of the notation— and suchthat the following conditions hold: ‚ m d | “ τ d | “ µ d A for all d ě
1, where µ d A , d ě A -operations for A ; ‚ τ | “ ´ c r´ ´ s : M r´ s Ñ A r´ s and m | “ B M r´ s ; ‚ the operations µ d : Cone p c qr´ s b d Ñ Cone p c qr´ s , d ě µ d | A r´ s b i b M r´ s b j b ¨ ¨ ¨ b A r´ s b i k b M r´ s b j k “ τ i | j | ... | i k | j k ‘ m i | j | ... | i k | j k define an A -algebra structure on Cone p c q .The collections of operations t m i | j | ... | i k | j k u and t τ i | j | ... | i k | j k u can befurther partitioned according to the value of j “ j ` j ` ¨ ¨ ¨ ` j k .It is instructive to spell out the meaning of the sub-collections whichcorrespond to the first two values of j . ‚ The case j “ A is an A -subalgebra of Cone p c q . ‚ The case j “ m i | | i : A r´ s b i b M r´ s b A r´ s b i Ñ M r´ s for i , i ě M r´ s as an A -bimodule over A . Herewe slightly deviate from the above notational convention by allow-ing i “
0. The second sub-collection is τ i | | i : A r´ s b i b M r´ s b A r´ s b i Ñ A r´ s for i , i ě
0. This describes an A -bimodule -homomorphism M r´ s Ñ A r´ s , whose first term is ´ c r´ s : M r´ s Ñ A r´ s .From this perspective, the data of an A -triple can be equivalentlyrephrased as consisting of an A -algebra A , of an A -bimodule M r´ s ,and of an A -bimodule -homomorphism M r´ s Ñ A r´ s whose firstterm is ´ c r´ s : M r´ s Ñ A r´ s , together with collections of maps t m i | j | ... | i k | j k u and t τ i | j | ... | i k | j k u as above which extend the given dataand define an A -structure on Cone p c q . The discussion of the Homotopy Transfer Theorem from § Definition 2.6.
A triple p M , c , A q is a homotopy retract of a triple p M , c, A q if the pair A Ă Cone p c q is a homotopy retract of the pair A Ă Cone p c q in the sense of Definition 2.3. Let us write in upper triangular form the maps
P, I, H involved in thedefinition as P “ ˆ p K π ˙ , I “ ˆ i H ι ˙ , H “ ˆ h a ´ χ ˙ , so that we obtain the diagram(6) χ (cid:24) (cid:24) M c (cid:15) (cid:15) π / / K ❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆ M c (cid:15) (cid:15) ι o o H ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ h (cid:24) (cid:24) A p / / A i o o It is straightforward to check that the homotopy retract condition isequivalent to the following: ‚ ( P , I are chain maps) rB , H s “ ic ´ cι, rB , K s “ pc ´ c π. ‚ ( H is a homotopy between IP and 1l, i.e. 1l ´ IP “ rB , H s )1l ´ ιπ “ rB M , χ s , ´ ip “ rB A , h s , rB , a s “ cχ ´ hc ´ i K ´ H π. Arity . Of particular interest for us will be the operations ofarity d “
2. The above discussion provides degree ´ τ | “ µ A : A r´ s b A r´ s Ñ A r´ s ,m | | : A r´ s b M r´ s Ñ M r´ s ,m | | : M r´ s b A r´ s Ñ M r´ s (7) ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 13 which, after an appropriate shift discussed below, induce an algebrastructure on H p A q and also an H p A q -bimodule structure on H p M q .We also have the degree ´ τ | | : A r´ s b M r´ s Ñ A r´ s ,τ | | : M r´ s b A r´ s Ñ A r´ s (8)which, also after an appropriate shift, provide homotopies ensuring that c induces in homology a bimodule map H p M q Ñ H p A q . There are twomore degree ´ m | | : M r´ s b M r´ s Ñ M r´ s and(10) τ | | : M r´ s b M r´ s Ñ A r´ s . Definition 2.7. An A -triple p M , c, A q consists of an associative upto homotopy dg algebra p A , µ q , of a chain complex M and of a degree chain map c : M Ñ A , together with bilinear maps m L : A b M Ñ M , m R : M b A Ñ M of degree , τ L : M b A Ñ A , τ R : A b M Ñ A ,σ : M b M Ñ M of degree , and β : M b M Ñ A of degree , subject to the following conditions: rB , µ s “ , rB , m L s “ , rB , m R s “ , rB , τ L s “ µ p c b q ´ cm R , rB , τ R s “ µ p b c q ´ cm L , rB , σ s “ m R p b c q ´ m L p c b q , and rB , β s “ ´ cσ ` τ R p c b q ´ τ L p b c q . The brackets are understood with respect to the indicated degrees,e.g. rB , m L s “ B M m L ´ m L pB A b M q ´ m L p A b B M q and rB , τ L s “B A τ L ` τ L pB M b A q ` τ L p M b B A q .The definition is motivated by Lemma 2.8 and Proposition 2.9 below. Lemma 2.8.
Given an A -triple p M , c, A q , the arity operations from(7–10) induce canonically the structure of an A -triple on p M , c, A q .Proof. We define µ , m L etc. by suitable shifts of the arity 2 operationsof the A -triple. As explained in Appendix A, the order of successiveshifts matters. Since our goal is to define an algebra structure on thecone, we first shift uniformly all the arity 2 operations by r ,
1; 1 s sothat all inputs and outputs are tensor products of A and M r´ s ; we then further shift by ` M r´ s -factor in order to obtain arity2 operations whose inputs and outputs are tensor products of A and M . This means that we define (with µ A “ τ | ) µ “ µ A r ,
1; 1 s ,m L “ m | | r ,
1; 1 sr ,
1; 1 s , m R “ m | | r ,
1; 1 sr ,
0; 1 s ,τ R “ τ | | r ,
1; 1 sr ,
1; 0 s , τ L “ τ | | r ,
1; 1 sr ,
0; 0 s ,σ “ m | | r ,
1; 1 sr ,
1; 1 s , β “ τ | | r ,
1; 1 sr ,
1; 0 s . For further use, it is also convenient to define the maps (of degree 0) µ “ µ “ µ A r ,
1; 1 s ,m L “ m | | r ,
1; 1 s , m R “ m | | r ,
1; 1 s ,τ R “ τ | | r ,
1; 1 s , τ L “ τ | | r ,
1; 1 s ,σ “ m | | r ,
1; 1 s , β “ τ | | r ,
1; 1 s . We claim that the maps µ, m L , m R , τ L , τ R , σ, β define the structure ofan A -triple on p M , c, A q . Denoting B “ A ‘ M r´ s , the proof consistsof a direct verification by decomposing the A -relation(11) µ B µ B ` µ B p µ B b q ` µ B p b µ B q “ µ B “ p τ | ` τ | | ` τ | | ` τ | | , m | | ` m | | ` m | | q and µ B “ B Cone p c q r´ s “ ˆ B A r´ s ´ c r´ ´ s B M r´ s ˙ . While the verification is straightforward, the signs are subtle and forthis reason we give the proof in detail.
Step 1. We prove that the maps µ “ µ, m L , m R , τ L , τ R , σ, β satisfy therelations rB , µ s “ , rB , m L s “ , rB , m R s “ , rB , τ L s “ µ p c r´
1; 0 s b q ´ c r´
1; 0 s m R , rB , τ R s “ µ p b c r´
1; 0 sq ´ c r´
1; 0 s m L , rB , σ s “ m R p b c r´
1; 0 sq ` m L p c r´
1; 0 s b q , rB , β s “ ´ c r´
1; 0 s σ ` τ R p c r´
1; 0 s b q ` τ L p b c r´
1; 0 sq . The relation rB , µ s “ µ A is a chain map, so is its shift. ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 15
The relations for m L and τ R are obtained by restricting equation (11)to A r´ s b M r´ s , where it becomes µ B p τ | | , m | | q ` p τ | | , m | | qp µ A b q` p τ | | ` τ | , m | | qp b µ B q “ ô prB , τ | | s ´ c r´ ´ s m | | ´ τ | p b c r´ ´ sq , rB , m | | sq “ ô rB , τ | | s ´ c r´ ´ s m | | ´ τ | p b c r´ ´ sq “ , rB , m | | s “ ô prB , τ | | s ´ c r´ ´ s m | | ´ τ | p b c r´ ´ sqqr ,
1; 1 s “ , rB , m | | sr ,
1; 1 s “ ô ´ rB , τ R s ´ c r´
1; 0 s m L ` µ p b c r´
1; 0 sq “ , ´ rB , m L s “ . The relations for m R and τ L are obtained by restricting equation (11)to M r´ s b A r´ s , where it becomes µ B p τ | | , m | | q ` p τ | | ` τ | , m | | qp µ B b q` p τ | | , m | | qp b µ B q “ ô prB , τ | | s ´ c r´ ´ s m | | ´ τ | p c r´ ´ s b q , rB , m | | sq “ ô rB , τ | | s ´ c r´ ´ s m | | ´ τ | p c r´ ´ s b q “ , rB , m | | s “ ô prB , τ | | s ´ c r´ ´ s m | | ´ τ | p c r´ ´ s b qqr ,
1; 1 s “ , rB , m | | sr ,
1; 1 s “ ô ´ rB , τ L s ´ c r´
1; 0 s m R ` µ p c r´
1; 0 s b q “ , ´ rB , m R s “ . The relations for σ and β are obtained by restricting equation (11) to M r´ s b M r´ s , where it becomes µ B p τ | | , m | | q ` p τ | | ` τ | | , m | | ` m | | qp µ B b q` p τ | | ` τ | | , m | | ` m | | qp b µ B q “ ô rB , τ | | s ´ c r´ ´ s m | | ´ τ | | p c r´ ´ s b q ´ τ | | p b c r´ ´ sq “ , rB , m | | s ´ m | | p c r´ ´ s b q ´ m | | p b c r´ ´ sq “ ô prB , τ | | s ´ c r´ ´ s m | | ´ τ | | p c r´ ´ s b q ´ τ | | p b c r´ ´ sqqr ,
1; 1 s “ prB , m | | s ´ m | | p c r´ ´ s b q ´ m | | p b c r´ ´ sqqr ,
1; 1 s “ ô ´ rB , β s ´ c r´
1; 0 s σ ` τ R p c r´
1; 0 s b q ` τ L p b c r´
1; 0 sq “ , ´ rB , σ s ` m L p c r´
1; 0 s b q ` m R p b c r´
1; 0 sq “ . Step 2. We prove the relations for µ, m L , m R , τ L , τ R , σ, β . Recall that we have m L “ m L r ,
1; 1 s , m R “ m R r ,
0; 1 s ,τ R “ τ R r ,
1; 0 s , τ L “ τ L r ,
0; 0 s ,σ “ σ r ,
1; 1 s , β “ β r ,
1; 0 s . We already proved rB , µ s “
0, i.e. µ is a chain map. That m L , m R arealso chain maps follows from the fact that they are shifts of the chainmaps m L , m R .To derive the equation for rB , τ L s we use the equation for rB , τ L s : rB , τ L s ´ µ p c r´
1; 0 s b q ` c r´
1; 0 s m R “ ô prB , τ L s ´ µ p c r´
1; 0 s b q ` c r´
1; 0 s m R qr ,
0; 0 s “ ô rB , τ L s ´ µ p c b q ` cm R “ . In the last equivalence we use c r´
1; 0 s ω “ c , where ω : M Ñ M r´ s is the shift.To derive the equation for rB , τ R s we use the equation for rB , τ R s : rB , τ R s ´ µ p b c r´
1; 0 sq ` c r´
1; 0 s m L “ ô prB , τ R s ´ µ p b c r´
1; 0 sq ` c r´
1; 0 s m L qr ,
1; 0 s “ ô rB , τ R s ´ µ p b c q ` cm L “ . To derive the equation for rB , σ s we use the equation for rB , σ s : rB , σ s ´ m R p b c r´
1; 0 sq ´ m L p c r´
1; 0 s b q “ ô prB , σ s ´ m R p b c r´
1; 0 sq ´ m L p c r´
1; 0 s b qqr ,
1; 1 s “ ô ´ rB , σ s ` m R p b c q ´ m L p c b q “ . Finally, to derive the equation for rB , β s we use the equation for rB , β s : rB , β s ` c r´
1; 0 s σ ´ τ R p c r´
1; 0 s b q ´ τ L p b c r´
1; 0 sq “ ô prB , β s ` c r´
1; 0 s σ ´ τ R p c r´
1; 0 s b q ´ τ L p b c r´
1; 0 sqqr ,
1; 0 s “ ô rB , β s ` cσ ´ τ R p c b q ` τ L p b c q “ . (cid:3) In the next statement we denote an element of
Cone p c q “ A ‘ M r´ s by p a, ¯ x q , meaning that ¯ x P M r´ s is the shift of an element x P M .In particular | ¯ x | “ | x | ` Proposition 2.9.
Let p M , c, A q be an A -triple with operations µ, m L , m R , τ L , τ R , σ, β as above. The formula m ` p a, ¯ x q , p a , ¯ x q ˘ “ ` µ p a, a q ` p´ q | a | τ R p a, x q ` τ L p x, a q ´ p´ q | ¯ x | β p x, x q , p´ q | a | m L p a, x q ` m R p x, a q ´ p´ q | ¯ x | σ p x, x q ˘ (12) ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 17 defines a degree bilinear product m : Cone p c q b Cone p c q Ñ Cone p c q which is a chain map. This product coincides with the one from (4) ifthe A -triple is induced from an A -triple as in Lemma 2.8.Proof. That the formula defines a chain map can be checked directly.Assume now that the A -triple is induced from an A -triple. Theproduct induced on Cone p c q by the A -structure is m “ p µ ` τ R ` τ L ` β, m L ` m R ` σ q . We therefore merely need to express the maps τ R , τ L , β, m L , m R , σ interms of τ R , τ L , β, m L , m R , σ .We work out in full detail the case of σ . We claim that(13) σ “ ´ σ r´ , ´ ´ s ,σ p ¯ x, ¯ x q “ ´p´ q | ¯ x | σ p x, x q . Indeed, we start with σ “ σ r ,
1; 1 s and then compute σ r´ , ´ ´ s “ σ r ,
1; 1 sr´ , ´ ´ s “ ´ σ . The explicit formula in terms of elementsis a consequence of the definition of the shift by r´ , ´ ´ s .Similarly, we compute: ‚ (14) m L “ m L r , ´ ´ s ,m L p a, ¯ x q “ p´ q | a | m L p a, x q since m L r , ´ ´ s “ m L r ,
1; 1 sr , ´ ´ s “ m L . ‚ (15) m R “ m R r´ , ´ s ,m R p ¯ x, a q “ m R p x, a q since m R r´ , ´ s “ m R r ,
0; 1 sr´ , ´ s “ m R . ‚ (16) τ L “ τ L r´ ,
0; 0 s ,τ L p ¯ x, a q “ τ L p x, a q since τ L r´ ,
0; 0 s “ τ L r ,
0; 0 sr´ ,
0; 0 s “ τ L . ‚ (17) τ R “ τ R r , ´
1; 0 s ,τ R p a, ¯ x q “ p´ q | a | τ R p a, x q since τ R r , ´
1; 0 s “ τ R r ,
1; 0 sr , ´
1; 0 s “ τ R . ‚ β “ ´ β r´ , ´
1; 0 s ,β p ¯ x, ¯ x q “ ´p´ q | ¯ x | β p x, x q since β r´ , ´
1; 0 s “ β r ,
1; 0 sr´ , ´
1; 0 s “ ´ β . (cid:3) Remark 2.10.
We chose to infer the equations for the maps m L , m R , τ R , τ L , σ , β in Definition 2.7 from the A -equations. As such, theyassemble canonically into the product structure on the cone inducedfrom the A -structure. But even so, there is a small amount of choiceinvolved: we could have defined σ as m | | r ,
2; 2 s , which would havechanged its sign. We settled for our convention for the reasons men-tioned in the preamble of the proof of Lemma 2.8.The existence of this potential change of sign can also be understoodfrom the following perspective. Assume one wishes to determine equa-tions for such a collection of maps so that they assemble into some product structure on the cone. A close inspection of the formula defin-ing the product m shows that, once we require that it restricts to theproduct µ on A , the equations for m L , m R , τ R , τ L , σ , β are uniquelydetermined up to obvious multiplications by ˘ rB Cone p c q , m s “
0, translates into functional equations for the var-ious maps involved. Our procedure to define the maps from the A -structure can be seen as one convenient way to fix the signs.We call the product m the canonical product on the cone defined bythe A -structure . Associativity up to homotopy for the product m isnot a priori guaranteed. For this, one would need to enhance the dataof an A -triple precisely with the operations of arity 3 involved in thedefinition of an A -triple, see (5).The A -triples used in this paper will always be arity 2 restrictionsof genuine A -triples canonically defined up to homotopy. While wewill not construct nor make use of the full A -structure, it is impor-tant to acknowledge its existence. In particular, the homotopy transferand homotopy invariance statements for A -triples discussed below areavatars of the homotopy transfer and homotopy invariance statementsfor A -structures. Proposition 2.11.
Let p M , c , A q be a triple which is a homotopyretract of the triple p M , c, A q as in Definition 2.6. Given the structureof an A -triple on p M , c, A q , there is an induced structure of an A -triple on p M , c , A q such that the maps Cone p c q P / / Cone p c q I o o ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 19 involved in the homotopy retract are compatible with the products on
Cone p c q and Cone p c q . (cid:3) Very explicitly, and because we are in arity 2, the transfer of structureis obtained by summing over the different ways of labeling the inputsand output of the unique binary rooted tree with two leaves by A and M , and inserting accordingly at the inputs the maps i, ι, H , and atthe output the maps p, π, K (compare to [14]). For example, the map σ : M b M Ñ M is given by σ “ ˘ πσι b ι ˘ πm L H b ι ˘ πm R ι b H . See Figure 1, in which we see 3 different labelings contributing to thetransfer. In contrast, the map µ : A b A Ñ A is given by µ “ pµi b i since there is only one labeling contributing to the transfer because ofthe upper triangular form of the maps P and I . The formula for thetransferred map β involves 7 terms. π M M M MMM σ ιι π A M M MMM m L ι H π M M M AMM m R H ι Figure 1.
Trees for the transferred map σ .In the situation in which the maps P, I defining a homotopy retract arehomotopy inverses, with the homotopy H : B Ñ B such that rB , H s “ ´ P I being also in upper triangular form, we obtain in particularthat the (non-associative) algebras p Cone p c q , m q and p Cone p c q , m q arehomotopy equivalent. In contrast to Proposition 2.5 above, this factis immediate and needs no additional assumption (because we do notask for any higher compatibilities of the homotopies). Thus, homotopyinvariance in the context of A -triples is automatic.2.4. Examples.Example 2.12 ( Ideals).
Let p A , B A , µ q be a dga and M Ă A be a dgideal. Let c “ incl : M ã Ñ A be the inclusion. The triple p M , incl, A q has a canonical structure of an A -triple defined as follows: the opera-tions m L : A b M Ñ M and m R : M b A Ñ M are given by multi-plication in A and endow M with the structure of a strict A -bimodule, whereas the operations τ R , τ L , σ, β are all zero. The corresponding prod-uct m on Cone p incl q “ A ‘ M r´ s is given by m ` p a, ¯ x q , p a , ¯ x q ˘ “ ` µ p a, a q , p´ q | a | µ p a, x q ` µ p x, a q ˘ . The projection proj : Cone p incl q Ñ A { M , p a, x q ÞÑ r a s is clearly a ring map. Assuming that the short exact sequence of R -modules Ñ M Ñ A Ñ A { M Ñ is split, it is a general factthat proj is a homotopy equivalence with an explicit homotopy inversedefined in terms of the splitting (see for example [5, Lemma 4.3] ). Example 2.13 ( Quotients).
Let now p M , c, A q be an A -triple withoperations µ, m L , m R , τ L , τ R , σ, β . Let m be the corresponding productstructure on Cone p c q .Assume c : M Ñ A to be surjective, denote K “ ker c and assumethat the short exact sequence Ñ K Ñ M c Ñ A Ñ is split. Writingthe differential B M in upper triangular form with respect to the split-ting, denote f : A Ñ K r´ s its p A , K q -component. We then have ahomotopy equivalence H % % Cone p c q “ A ‘ K r´ s ‘ A r´ s Σ / / K r´ s T o o where T is the inclusion on the K r´ s -factor, Σ p a, ¯ k, ¯ a q “ f p a q` ¯ k , and Σ T “ . The homotopy H acts by H p a, ¯ k, ¯ a q “ p , , a q . The shiftedkernel K r´ s inherits the product structure ˜ σ “ Σ mT b T . Explicitly ˜ σ p ¯ x K , ¯ x K q “ ´p´ q | ¯ x K | f β p x K , x K q ´ p´ q | ¯ x K | pr K σ p x K , x K q , where pr K : M Ñ K is the projection determined by the splitting.In the presence of arity 3 data on the triple p M , c, A q , this product isassociative up to homotopy and the maps Σ , T interchange ˜ σ and m up to homotopy.Under the additional assumptions β | K b K “ and σ p K b K q Ă K, In [10, Lemma 2.1] the authors call it “Nagata product”. However, this termi-nology is potentially misleading. Nagata defined in [15, p. 2] a product on the directsum between a module and its base ring in the context of his general procedure of“idealization”, i.e. of turning a module into an ideal, which signed the beginningof the theory of square zero extensions. In the dg setting, a square zero extensionis a surjective map of dga’s whose kernel squares to zero. In contrast, our setupis concerned with injective maps of dga’s, i.e. with pairs consisting of an algebraand a subalgebra. In the current setup, if M were zero then A would be a squarezero extension of A { M , inheriting a “Nagata product”. In contrast, the cone ofthe inclusion M ã Ñ A always carries a dga structure. ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 21 we have ˜ σ p ¯ x K , ¯ x K q “ ´p´ q | ¯ x K | σ p x K , x K q , i.e. ˜ σ “ ´ σ r´ , ´ ´ s . In case the A -triple is induced by an A -structure as before, we obtain ˜ σ “ σ , the product on M r´ s induced by the A -operations.In any case, the map T is an algebra map (and a homotopy equiva-lence), so that p Cone p c q , m q and p K r´ s , ˜ σ q are homotopy equivalentas algebras. Example 2.14 ( Duality M “ A _ ). Let p A , Bq be a dg R -module, A _˚ “ Hom R p A ´˚ , R q its graded dual, and ev : A _ b A Ñ R thecanonical evaluation map. In Appendix B we describe a structure on A which gives rise to an A -triple p A _ , c, A q . It is called an A ` -structure and consists of the following maps: ‚ the continuation quadratic vector c : R Ñ A b A , of degree ; ‚ the product µ : A b A Ñ A , of degree ; ‚ the secondary coproduct λ : A Ñ A b A , of degree ; ‚ the cubic vector B : R Ñ A b A b A , of degree .The continuation quadratic vector c gives rise to the continuation map c : “ p ev b qp b c q “ p ev b qp b τ c q : A _ Ñ A . These maps are subject to the following conditions:(1) the continuation quadratic vector is symmetric, i.e. τ c “ c ;(2) the product µ commutes with B and is associative up to chainhomotopy;(3) the image of c lies in the center of µ , i.e. µ p c b q “ µ p b c q τ ;(4) rB , λ s “ p µ b qp b c q ´ p b µ qp c b q .(5) B B “ p b λ q c ` p λ b q c ´ τ p λ b q c .The A -triple maps m L , m R , τ L , τ R , σ, β can be given by explicit formu-las in terms of µ , λ , B . The correct way to understand these is interms of TQFT-like pictures, for which we refer to Appendix B.Given an A ` -structure p A , B , c , µ, λ, B q , dualizing all the maps yieldsa similar structure p A _ , B _ , c _ , µ _ , λ _ , β _ q on the dual complex. Itsmaps are ‚ the continuation quadratic covector c _ : A _ b A _ Ñ R , of degree ; ‚ the coproduct µ _ : A _ Ñ A _ b A _ , of degree ; ‚ the secondary product λ _ : A _ b A _ Ñ A _ , of degree ; ‚ the cubic covector B _ : A _ b A _ b A _ Ñ R , of degree , and the relations are dual to the ones for A . This structure inducesa co product on the cone of c _ : A _ Ñ A __ , so one may call it a co- A ` -structure .Assume now that A is free and finite dimensional over R . Then we havea canonical isomorphism A – A __ under which c _ “ c : A _ Ñ A . Inthis case the co- A ` -structure on A _ determines the structure of an A -triple on p A _ , c _ , A q , and we get canonical homotopy equivalences of dgrings Cone p c q » Cone p c _ q » Cone p c q _ r´ s . On the level of homologywe thus obtain the following algebraic counterpart of our main Poincar´eduality theorem: Theorem 2.15.
Given an A ` -structure on A as above such that A isfree and finite dimensional, we have a canonical isomorphism of rings H ˚ p Cone p c qq » H ˚´ p Cone p c q _ q . (cid:3) Cones of Floer continuation maps
Floer continuation maps give naturally rise to A -triples, and indeed A -triples.3.1. Closed symplectically aspherical manifolds.
Although main-ly interested in the noncompact case, we start with a discussion of thecompact symplectically aspherical case. In this situation the continua-tion maps are homotopy equivalences and their cones are acyclic, butit is interesting to see how the A -triple arises. We work on a closedsymplectically aspherical manifold W of dimension 2 n with trivial firstChern class and a fixed choice of trivialization of its canonical bundle.All our Floer chain complexes are graded by the Conley-Zehnder index,computed in this trivialization. Whenever we write Floer chain com-plexes, continuation maps, and more generally equations for pseudo-holomorphic maps defined on Riemann surfaces, we tacitly mean thatthere are choices of compatible almost complex structures involved. Inorder not to burden the notation we will not make further reference tothese unless absolutely necessary. Proposition 3.1.
Let M ˚ “ F C ˚` n p H q , A ˚ “ F C ˚` n p K q be shifted Floer chain complexes determined by two nondegenerate Hamil-tonians H, K on a closed symplectically aspherical manifold W of di-mension n . Let t H s : s P R u be a homotopy such that H s “ K for s ! and H s “ H for s " , and denote c : M ˚ Ñ A ˚ ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 23 the corresponding continuation map. Then p M , c, A q carries the struc-ture of an A -triple, canonically defined up to homotopy.Proof. We need to define operations µ , m L , m R , τ L , τ R , σ and β .The operations µ : A b A Ñ A , m L : A b M Ñ M , m R : M b A Ñ M are pair-of-pants products. They are defined by counts of index 0 pairs-of-pants with 2 inputs (positive punctures) and 1 output (negativepuncture). See Figure 2, in which we depict pairs-of-pants with twoinputs and one output schematically as binary rooted trees with twoleaves. The two inputs are ordered, the first one is depicted on the leftand the second one is depicted on the right. µ K HH m L H KH m R K KK
Figure 2.
Curves defining the maps µ , m L , m R in Floer theory.The operations σ : M b M Ñ M , τ L : M b A Ñ A , τ R : A b M Ñ A are defined by counts of index ´ r , s . See Figure 3.The operation β : M b M Ñ A is defined by the count of index ´ A -triple is straightforward. Thatthe resulting structure is canonically defined up to homotopy is also astraightforward—though combinatorially involved—argument. (cid:3) Remark 3.2 ( A -triple) . The above A -triple is the arity 2 part of an A -triple. However, we will not construct the latter here. Remark 3.3 (filtration on the cone) . Assume H ď K and the ho-motopy is monotone. Then all the maps defining the A -triple can beconstructed such that they decrease the action. The cone then has acanonical R -filtration Cone p c q ď a “ F C ď a ˚ p K q ‘ F C ď a ˚´ p H q , a P R and the product structure m defined by the A -triple preserves thisfiltration, meaning that m ` Cone p c q ď a b Cone p c q ď b ˘ Ď Cone p c q ď a ` b . Denote
Cone p c q p a,b q “ Cone p c q ă b { Cone p c q ď a . We obtain induced par-tial products m : Cone p c q p a,b q b Cone p c q p a ,b q Ñ Cone p c q p max t a ` b ,a ` b u ,b ` b q . t H s u K HH cm L t H s u K K H KK µ p b c qt H s u H KK τ L K HK τ R H HH σ H H KH m R p b c qt H s u K H HH m L p c b q t H s u H KH cm R t H s u K K H KK µ p c b q Figure 3.
Curves defining the maps σ , τ L , τ R in Floer theory.3.2. Completions of Liouville domains.
Consider now the noncom-pact case where the underlying manifold is the symplectic completion x W of a Liouville domain W . One additional complication is added bythe fact that solutions of the relevant Cauchy-Riemann equations with0-order Hamiltonian perturbation are required to obey a maximum ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 25 H HH cσ t H s u K K H HK τ R p c b qt H s u H H KK ´ τ L p b c qt H s u H H KH cm R p b c qt H s ut H s u KK H HH cm L p c b qt H s ut H s u K K H KK µ p c b c qt H s u t H s u HH HK β
Figure 4.
Curves defining the map β in Floer theory.principle. More precisely, given a map u : Σ Ñ x W defined on a (punc-tured) Riemann surface Σ and solving the perturbed Cauchy-Riemannequation p du ´ X H b β q , “ , where β P Ω p Σ , R q and H “ t H z u , z P Σ is a Σ-dependent familyof Hamiltonians, one requires that, outside a compact set, we have d z p Hβ q ď z P Σ. When restricting to admissible Hamiltonians,i.e. Hamiltonians which are linear in the region t r ě u Ă x W , itis enough to impose this condition in that region. If this condition holds on the entire completion x W , then in addition the relevant mapsdecrease the action.The resulting structure is the following. Given admissible Hamiltoni-ans H ď H and H ď H together with non-increasing homotopies t H s u and t H s u connecting H and H , respectively H and H , denote c : F C ˚ p H qr n s Ñ F C ˚ p H qr n s and c : F C ˚ p H qr n s Ñ F C ˚ p H qr n s thecorresponding continuation maps. Denote H H p t, x q “ H p t, x q ` H p t, p ϕ tH q ´ p x qq , where φ tH denotes the Hamiltonian flow of H . As-sume further the conditions(18) H ě H H , H ě H “ H H . The equality 2 H “ H H is a condition which holds for time-independent Hamiltonians, hence for the Hamiltonians that we usein the sequel. By modifying the moduli spaces considered in the proofof Proposition 3.1 into moduli spaces with inputs 1-periodic orbits of H , H and outputs 1-periodic orbits of H , H , we construct operations µ , m L , m R , τ L , τ R , σ , β which assemble into a product m : Cone p c q b Cone p c q Ñ Cone p c q . Moreover, this product respects the canonical filtrations on the conesand induces m : Cone p c q p a,b q b Cone p c q p a ,b q Ñ Cone p c q p max t a ` b ,a ` b u ,b ` b q . Thus, in the noncompact case there is strictly speaking no pre-subal-gebra extension structure for a fixed pair of Hamiltonians. However,there is a “directed system” of such, constructed as above.
Remark 3.4.
That conditions (18) are indeed sufficient for the exis-tence of continuation maps is a consequence of the following construc-tion, originally due to Matthias Schwarz [18, Proposition 4.1 sqq.].Given Hamiltonians H and K , it is possible to construct a perturbedFloer equation on a pair-of-pants with two positive punctures and onenegative puncture such that: near the positive punctures it special-izes to the Floer equation for H and K , near the negative punctureit specializes to the Floer equation for the Hamiltonian H K p t, x q “ H p t, x q` K p t, p ϕ tH q ´ x q , and the solutions of this Floer equation satisfythe maximum principle and do not increase the Hamiltonian action. Example 3.5 (The family of Hamiltonians t K λ u , λ P R ) . Given λ P R denote by K λ the Hamiltonian which is on W and is linear of slope λ on t r ě u , with a convex smoothing if λ ą , respectively a concavesmoothing if λ ă . See Figure 5.Given parameters λ ´ ď λ ` we denote c λ ´ ,λ ` : F C ˚ p K λ ´ qr n s Ñ F C ˚ p K λ ` qr n s ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 27 K λ r Figure 5.
The family of Hamiltonians t K λ u , λ P R . the continuation map induced by a monotone homotopy. Given alsoparameters λ ď λ such that λ ´ ` λ ` ď λ and λ ` ď λ , and givenaction bounds ´8 ă a ă b ă 8 , we obtain bilinear maps (19) m : Cone p c λ ´ ,λ ` q p a,b q b Cone p c λ ´ ,λ ` q p a,b q Ñ Cone p c λ ,λ q p a ` b, b q . Let now p a, b q be fixed. Given λ ´´ ď λ ´ ď λ ` we have a canonicalcontinuation map cont : Cone p c λ ´´ ,λ ` q Ñ Cone p c λ ´ ,λ ` q . This map iscompatible in homology with the bilinear maps m defined above, mean-ing that we have commutative diagrams for allowable values of the pa-rameters H ˚ p Cone p c λ ´ ,λ ` q p a,b q q b H ˚ p Cone p c λ ´ ,λ ` q p a,b q q m / / H ˚ p Cone p c λ ,λ q p a ` b, b q q H ˚ p Cone p c λ ´´ ,λ ` q p a,b q q b H ˚ p Cone p c λ ´´ ,λ ` q p a,b q q m / / cont b cont O O H ˚ p Cone p c λ ,λ q p a ` b, b q q cont O O Similarly, given λ ´ ď λ ` ď λ `` we have a canonical continuationmap cont : Cone p c λ ´ ,λ ` q Ñ Cone p C λ ´ ,λ `` q which is also compatible inhomology with the bilinear products m , with a similar meaning.Note moreover that the canonical maps lim ÐÝ λ ´ Ñ´8 H ˚ p Cone p c λ ´ ,λ ` q p a,b q q » ÝÑ H ˚ p Cone p c a,λ ` q p a,b q q are isomorphisms for all a ď λ ` , and similarly the maps H ˚ p Cone p c λ ´ ,b q p a,b q q » ÝÑ lim ÝÑ λ ` Ñ8 H ˚ p Cone p c λ ´ ,λ ` q p a,b q q are isomorphisms for all λ ´ ď b . Define SH p a,b q˚ pt K λ uq : “ lim ÝÑ λ ` Ñ8 lim ÐÝ λ ´ Ñ´8 H ˚ p Cone p c λ ´ ,λ ` q p a,b q q . We obtain in the first-inverse-then-direct limit bilinear maps m actingas SH p a,b q˚ pt K λ uq b SH p a,b q˚ pt K λ uq Ñ SH p a ` b, b q˚ pt K λ uq . These maps are compatible with the canonical action truncation mapsin Floer theory. We therefore define SH ˚ pt K λ uq : “ lim ÝÑ b Ñ8 lim ÐÝ a Ñ´8 SH p a,b q˚ pt K λ uq , and obtain a product SH ˚ pt K λ uq b SH ˚ pt K λ uq Ñ SH ˚ pt K λ uq . The associativity of this product can be proved directly by incorporatingarity 3 operations in the above discussion. However, associativity alsofollows a posteriori from Theorem 4.2 below, according to which we havea natural isomorphism SH ˚ pt K λ uq » SH ˚ pB W qr n s compatible with theproducts. In view of this, we will refer to SH ˚ pt K λ uq as being a ring. Example 3.6 (The family of Hamiltonians H λ,µ , λ, µ P R ) . Consideragain a Liouville domain W with Liouville completion x W . Given realparameters λ, µ which do not belong to the action spectrum of B W wedefine a Hamiltonian H λ,µ on x W to be a smoothing of the Hamiltonianwhich is constant equal to ´ λ { on t r ď { u , equal to at r “ ,linear of slope λ on t { ď r ď u , and linear of slope µ on t r ě u .See Figure 6. We denote L λ “ H λ,λ .We have H λ ,µ ě H λ,µ for λ ď λ and µ ě µ . Also, we have (20) H λ ` λ ,µ ` µ “ H λ,µ ` H λ ,µ for all λ, λ , µ, µ . See Figure 6.As already seen before, the -periodic orbits of H λ,µ fall into two groups:orbits of type F which are located in a neighborhood of the region t r ď { u , and orbits of type I which are located in a neighborhood of theregion t r “ u . We will distinguish the following two cases:Case (i): λ ă ă µ . In this situation the orbits of type F form a sub-complex, cf. [5, § , and we denote i H λ,µ : F C F ˚ p H λ,µ qr n s Ñ F C ˚ p H λ,µ qr n s the inclusion of this subcomplex, with quotient complex F C I ˚ p H λ,µ qr n s .Case (ii): λ ą ą µ . In this situation the orbits of type I form a sub-complex, cf. [5, § , and we denote (21) p H λ,µ : F C ˚ p H λ,µ qr n s Ñ F C F ˚ p H λ,µ qr n s ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 29 ´ λ { rµ λ λµ µλλ ´ λ { { ´ µ { Figure 6.
The family of Hamiltonians t H λ,µ u , λ, µ P R . the projection onto the quotient complex consisting of orbits of type F ,with kernel the subcomplex generated by orbits of type I .In both cases, we have canonical identifications F C F ˚ p H λ,λ q – F C ˚ p L λ q . Remark 3.7 (Subcomplexes from action vs. geometry) . That orbits oftype F in case (i), or of type I in case (ii), form subcomplexes followsfrom the geometric Lemmas 2.2 and 2.3 in [5]. Alternatively, we couldargue by Hamiltonian action: choosing the Hamiltonian H λ,µ to beconstant on r , δ λ s rather than r , { s , with δ λ ą F have smaller actionthat orbits of type I (and similarly in case (ii)). However, such a shapeof Hamiltonian would not satisfy equality (20) which we need in thesequel, so we prefer the geometric approach. We now consider separately the families H _ “ t H λ,µ : λ ă ă µ u and H ^ “ t H λ,µ : λ ą ą µ u , and define from each of them a certain symplectic homology ring.Case (i). The family H _ “ t H λ,µ : λ ă ă µ u .Given parameters λ ă ă µ we denote c p λ,λ q , p λ,µ q : F C ˚ p L λ qr n s Ñ F C ˚ p H λ,µ qr n s the continuation map induced by a monotone homotopy. Note that wecan choose the homotopy to be constant in the region t r ď u , and withthis choice the continuation map c p λ,λ q , p λ,µ q is canonically identified withthe inclusion i H λ,µ from above. We obtain bilinear maps (22) m “ m ` p λ,λ q , p λ,µ q ˘ , ` p λ,λ ` µ q , p λ, µ q ˘ : Cone p c p λ,λ q , p λ,µ q q p a,b q b Cone p c p λ,λ q , p λ,µ q q p a,b q Ñ Cone p c p λ,λ ` µ q , p λ, µ q q p a ` b, b q . We claim that, given p a, b q , the above maps canonically stabilize in ho-mology for µ fixed and λ negative enough. To prove the claim, consider λ ď λ ă ă µ . We then have homotopy commutative diagrams ofcontinuation maps F C ˚ p L λ qr n s / / (cid:15) (cid:15) F C ˚ p H λ ,µ qr n s F C ˚ p H λ ,λ qr n s / / F C ˚ p H λ ,µ qr n s F C ˚ p L λ qr n s O O / / F C ˚ p H λ,µ qr n s O O and F C ˚ p H λ ,λ ` µ qr n s / / (cid:15) (cid:15) F C ˚ p H λ , µ qr n s F C ˚ p H λ ,λ ` µ qr n s / / F C ˚ p H λ , µ qr n s F C ˚ p H λ,λ ` µ qr n s O O / / F C ˚ p H λ, µ qr n s O O The vertical maps induce maps between the cones of the horizontalmaps, and we obtain homology commutative diagrams for the prod-uct structures involving the horizontal continuation maps. We denote
ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 31 symbolically the resulting diagram m ` p λ ,λ q , p λ ,µ q ˘ , ` p λ ,λ ` µ q , p λ , µ q ˘ (cid:15) (cid:15) m ` p λ ,λ q , p λ ,µ q ˘ , ` p λ ,λ ` µ q , p λ , µ q ˘ m ` p λ,λ q , p λ,µ q ˘ , ` p λ,λ ` µ q , p λ, µ q ˘ . O O For p a, b q fixed, µ fixed and λ ď λ ! , all the above maps are isomor-phisms. This shows that the products stabilize.Similarly, we claim that, for p a, b q fixed, diagram (22) stabilizes inhomology for λ ! and b ď µ ď | λ | . The argument is similar, basedon the homotopy commutative diagrams of continuation maps in which b ď µ ď µ ď | λ | : F C ˚ p L λ qr n s / / F C ˚ p H λ,µ qr n s (cid:15) (cid:15) F C ˚ p L λ qr n s / / F C ˚ p H λ,µ qr n s and F C ˚ p H λ,λ ` µ qr n s / / (cid:15) (cid:15) F C ˚ p H λ, µ qr n s (cid:15) (cid:15) F C ˚ p H λ,λ ` µ qr n s / / F C ˚ p H λ, µ qr n s These induce homology commutative diagrams for the product struc-tures involving the horizontal continuation maps, which we denote sym-bolically m ` p λ,λ q , p λ,µ q ˘ , ` p λ,λ ` µ q , p λ, µ q ˘ (cid:15) (cid:15) m ` p λ,λ q , p λ,µ q ˘ , ` p λ,λ ` µ q , p λ, µ q ˘ The outcome of this discussion is that the groups defined by SH p a,b q˚ p H _ q : “ lim ÝÑ µ Ñ8 lim ÐÝ λ Ñ´8 H ˚ p Cone p c p λ,λ q , p λ,µ q q p a,b q q inherit a product m : SH p a,b q˚ p H _ q b SH p a,b q˚ p H _ q Ñ SH p a ` b, b q˚ p H _ q . We define SH ˚ p H _ q : “ lim ÝÑ b Ñ8 lim ÐÝ a Ñ´8 SH p a,b q˚ p H _ q and this inherits a bilinear product m : SH ˚ p H _ q b SH ˚ p H _ q Ñ SH ˚ p H _ q . Case (ii). The family H ^ “ t H λ,µ : λ ą ą µ u .Given parameters λ ą ą µ we denote c p λ,µ q , p λ,λ q : F C ˚ p H λ,µ qr n s Ñ F C ˚ p L λ qr n s the continuation map induced by a monotone homotopy. We can choosethe homotopy to be constant in the region t r ď u , and with this choicethe continuation map c p λ,µ q , p λ,λ q is canonically identified with the pro-jection p H λ,µ from (21) . We obtain bilinear maps (23) m “ m ` p λ,µ q , p λ,λ q ˘ , ` p λ,λ ` µ q , p λ, λ q ˘ : Cone p c p λ,µ q , p λ,λ q q p a,b q b Cone p c p λ,µ q , p λ,λ q q p a,b q Ñ Cone p c p λ,λ ` µ q , p λ, λ q q p a ` b, b q . Given p a, b q , the above maps canonically stabilize in homology for λ fixed and µ negative enough. To prove this, consider λ ą ą µ ě µ .We then have homotopy commutative diagrams of continuation maps F C ˚ p H λ,µ qr n s / / F C ˚ p H λ,λ qr n s F C ˚ p H λ,µ qr n s / / O O F C ˚ p H λ,λ qr n s and F C ˚ p H λ,λ ` µ qr n s / / F C ˚ p H λ, λ qr n s F C ˚ p H λ,λ ` µ qr n s / / O O F C ˚ p H λ, λ qr n s The vertical maps induce maps between the cones of the horizontalmaps, and we obtain homology commutative diagrams for the prod-uct structures involving the horizontal continuation maps. We denotesymbolically the resulting diagram m ` p λ,µ q , p λ,λ q ˘ , ` p λ,λ ` µ q , p λ, λ q ˘ m ` p λ,µ q , p λ,λ q ˘ , ` p λ,λ ` µ q , p λ, λ q ˘ O O For p a, b q fixed, λ fixed and " µ ě µ , these maps are isomorphisms.This shows that the products stabilize.Similarly, for p a, b q fixed, diagram (23) stabilizes in homology for " µ and b ď λ ď | µ | . The argument is similar, based on the homotopy ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 33 commutative diagrams of continuation maps for parameters | µ | ě λ ě λ ą ą µ : F C ˚ p H λ,µ qr n s / / F C ˚ p H λ,λ qr n s (cid:15) (cid:15) F C ˚ p H λ,µ qr n s / / F C ˚ p H λ,λ qr n s F C ˚ p H λ ,µ qr n s O O / / F C ˚ p H λ ,λ qr n s O O and F C ˚ p H λ,λ ` µ qr n s / / (cid:15) (cid:15) F C ˚ p H λ, λ qr n s (cid:15) (cid:15) F C ˚ p H λ,λ ` µ qr n s / / F C ˚ p H λ, λ qr n s F C ˚ p H λ ,λ ` µ qr n s O O / / F C ˚ p H λ , λ qr n s O O The corresponding homology commutative diagram between products oncones is m ` p λ,µ q , p λ,λ q ˘ , ` p λ,λ ` µ q , p λ, λ q ˘ (cid:15) (cid:15) m ` p λ,µ q , p λ,λ q ˘ , ` p λ,λ ` µ q , p λ, λ q ˘ m ` p λ ,µ q , p λ ,λ q ˘ , ` p λ ,λ ` µ q , p λ , λ q ˘ . O O The outcome of the discussion is that the groups defined by SH p a,b q˚ p H ^ q : “ lim ÝÑ λ Ñ8 lim ÐÝ µ Ñ´8 H ˚ p Cone p c p λ,µ q , p λ,λ q q p a,b q q inherit a product m : SH p a,b q˚ p H ^ q b SH p a,b q˚ p H ^ q Ñ SH p a ` b, b q˚ p H ^ q . We define SH ˚ p H ^ q : “ lim ÝÑ b Ñ8 lim ÐÝ a Ñ´8 SH p a,b q˚ p H ^ q and this inherits a product m : SH ˚ p H ^ q b SH ˚ p H ^ q Ñ SH ˚ p H ^ q . As in Example 3.5, the associativity of the products on SH ˚ p H _ q and SH ˚ p H ^ q can be proved directly by incorporating arity 3 operations inthe discussion. However, associativity also follows a posteriori from Theorems 4.1 and 4.2 below, according to which we have natural iso-morphisms SH ˚ p H _ q » SH ˚ p H ^ q » SH ˚ pB W qr n s compatible with theproducts. In view of this, we will refer to SH ˚ p H _ q and SH ˚ p H ^ q asbeing rings. Duality theorems in Floer theory using cones
Cone duality theorem.Theorem 4.1 (Duality theorem) . There is a canonical isomorphismwhich respects the products SH ˚ p H _ q » SH ˚ p H ^ q . Proof.
Consider λ ă ă µ . Then we have a homotopy commutativediagram of continuation maps F C ˚ p H µ,λ qr n s c ^ (cid:15) (cid:15) π „ / / F C ˚ p L λ qr n s c _ (cid:15) (cid:15) ι o o F C ˚ p L µ qr n s p „ / / F C ˚ p H λ,µ qr n s i o o in which the horizontal maps are chain homotopy equivalences. (Themaps π and p preserve the filtration, but the maps ι and i do not.However, in the proof below we will use the total complexes for suitablechoices of the parameters, which will palliate to this ailment.) The leftvertical map c ^ “ c p µ,λ q , p µ,µ q is involved in the definition of SH ˚ p H ^ q .The right vertical map c _ “ c p λ,λ q , p λ,µ q is involved in the definition of SH ˚ p H _ q . See Figure 7. c ^ λ µ H λ,µ »» λ L λ λ H µ,λ µ µ L µ c _ Figure 7.
Duality theorem via cones: continuation di-agram at the source.
ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 35
Consider also the homotopy commutative diagram of continuation maps(Figure 8)
F C ˚ p H µ,λ ` µ qr n s c ^ (cid:15) (cid:15) π „ / / F C ˚ p H λ,λ ` µ qr n s c _ (cid:15) (cid:15) ι o o F C ˚ p L µ qr n s p „ / / F C ˚ p H λ, µ qr n s i o o λ ` µ »» µL µ c _ c ^ λ λ ` µH λ,λ ` µ µ H µ,λ ` µ µH λ, µ λ Figure 8.
Duality theorem via cones: continuation di-agram at the target.By Proposition 2.11 and the subsequent discussion on homotopy in-variance for A -triples, we obtain an isomorphism between productstructures(24) m ` p µ,λ q , p µ,µ q ˘ , ` p µ,λ ` µ q , p µ, µ q ˘ » m ` p λ,λ q , p λ,µ q ˘ , ` p λ,λ ` µ q , p λ, µ q ˘ . In order to conclude the proof, we fix an action interval p a, b q with a ă ă b and consider the parameter values λ “ a , µ “ b in theprevious setup. A stabilization argument as in Example 3.6 shows thatwe have isomorphisms H ˚ p Cone p c p a,a q , p a,b q q b H ˚ p Cone p c p a,a q , p a,b q q m / / » (cid:15) (cid:15) H ˚ p Cone p c p a,a ` b q , p a, b q q » (cid:15) (cid:15) SH p a,b q˚ p H _ q b SH p a,b q˚ p H _ q m / / SH p a ` b, b q˚ p H _ q and H ˚ p Cone p c p b,a q , p b,b q q b H ˚ p Cone p c p b,a q , p b,b q q m / / » (cid:15) (cid:15) H ˚ p Cone p c p b,a ` b q , p b, b q q » (cid:15) (cid:15) SH p a,b q˚ p H ^ q b SH p a,b q˚ p H ^ q m / / SH p a ` b, b q˚ p H ^ q The top lines in the above two diagrams are isomorphic by (24), andwe infer the isomorphism of the bottom lines. This isomorphism iscompatible with action truncation maps and yields an isomorphism ofrings SH ˚ p H _ q » SH ˚ p H ^ q . (cid:3) Cone description of the Rabinowitz Floer homology ring.
Let W be a Liouville domain with symplectic completion x W . Let usrecall the definition of the Rabinowitz Floer homology ring SH ˚ pB W q from [5] in terms of the family of Hamiltonians H λ,µ , λ ă ă µ fromExample 3.6. For a fixed finite action interval p a, b q we set SH p a,b q˚ pB W q “ lim ÝÑ µ Ñ8 lim ÐÝ λ Ñ´8
F H p a,b q˚ p H λ,µ q , and further SH ˚ pB W q “ lim ÝÑ b Ñ8 lim ÐÝ a Ñ´8 SH p a,b q˚ pB W q . The product is defined from the pair-of-pants product
F H p a,b q˚ p H λ,µ qr n s b F H p a,b q˚ p H λ,µ qr n s Ñ F H p a ` b, b q˚ p H λ, µ qr n s , which induces SH p a,b q˚ pB W qr n s b SH p a,b q˚ pB W qr n s Ñ SH p a ` b, b q˚ pB W qr n s and further SH ˚ pB W qr n s b SH ˚ pB W qr n s Ñ SH ˚ pB W qr n s . Theorem 4.2.
We have canonical isomorphisms which respect theproducts SH ˚ pt K λ uq » SH ˚ p H _ q » SH ˚ pB W qr n s . Proof. Step 1. We prove the first isomorphism SH ˚ pt K λ uq » SH ˚ p H _ q . The proof is essentially the same as that of Theorem 4.1. Denote forconvenience H ,λ “ K λ . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 37
Consider λ ă ă µ . Then we have a homotopy commutative diagramof continuation maps F C ˚ p K λ qr n s c (cid:15) (cid:15) π „ / / F C ˚ p L λ qr n s c _ (cid:15) (cid:15) ι o o F C ˚ p K µ qr n s p „ / / F C ˚ p H λ,µ qr n s i o o in which the horizontal maps are chain homotopy equivalences, andwhere the maps π and p preserve the filtration, but the maps ι and i do not. The left vertical map c “ c p ,λ q , p ,µ q is involved in the definitionof SH ˚ pt K λ uq . The right vertical map c _ “ c p λ,λ q , p λ,µ q is involved in thedefinition of SH ˚ p H _ q . See Figure 9. H λ,µ »» λ L λ “ H λ,λ λ K λ “ H ,λ µ K µ “ H ,µ c _ c λ µ Figure 9.
Isomorphism SH ˚ pt K λ uq » SH ˚ p H _ q : con-tinuation diagram at the source.Consider also the homotopy commutative diagram of continuation maps(Figure 10) F C ˚ p K λ ` µ qr n s c ^ (cid:15) (cid:15) π „ / / F C ˚ p H λ,λ ` µ qr n s c _ (cid:15) (cid:15) ι o o F C ˚ p K µ qr n s p „ / / F C ˚ p H λ, µ qr n s i o o By Proposition 2.11 and the subsequent discussion on homotopy in-variance for A -triples, we obtain an isomorphism between productstructures(25) m ` p ,λ q , p ,µ q ˘ , ` p ,λ ` µ q , p , µ q ˘ » m ` p λ,λ q , p λ,µ q ˘ , ` p λ,λ ` µ q , p λ, µ q ˘ . λ »» L λ “ H λ,λ K λ ` µ µK µ c _ c µH λ, µ λ ` µλ ` µ λ Figure 10.
Isomorphism SH ˚ pt K λ uq » SH ˚ p H _ q :continuation diagram at the target.In order to conclude the proof, we fix an action interval p a, b q with a ă ă b and consider the parameter values λ “ a , µ “ b in theprevious setup. A stabilization argument as in Example 3.6 shows thatwe have isomorphisms H ˚ p Cone p c p a,a q , p a,b q qq b H ˚ p Cone p c p a,a q , p a,b q qq m / / » (cid:15) (cid:15) H ˚ p Cone p c p a,a ` b q , p a, b q qq » (cid:15) (cid:15) SH p a,b q˚ p H _ q b SH p a,b q˚ p H _ q m / / SH p a ` b, b q˚ p H _ q and H ˚ p Cone p c p ,a q , p ,b q qq b H ˚ p Cone p c p ,a q , p ,b q qq m / / » (cid:15) (cid:15) H ˚ p Cone p c p ,a ` b q , p , b q qq » (cid:15) (cid:15) SH p a,b q˚ pt K λ uq b SH p a,b q˚ pt K λ uq m / / SH p a ` b, b q˚ pt K λ uq The top lines in the above two diagrams are isomorphic by (25), andwe infer the isomorphism of the bottom lines. This isomorphism iscompatible with action truncation maps and yields an isomorphism ofrings SH ˚ p H _ q » SH ˚ pt K λ uq . Step 2. We prove the second isomorphism SH ˚ p H _ q » SH ˚ pB W qr n s . Consider parameters λ ă ă µ . The group and product structure on SH ˚ p H _ q are built from the cone of the continuation map c _ “ c p λ,λ q , p λ,µ q : F C ˚ p L λ qr n s Ñ F C ˚ p H λ,µ qr n s . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 39
Recall that the 1-periodic orbits of H λ,µ fall into two classes, F consist-ing of orbits located in a neighborhood of t r ď { u and I consistingof orbits located in a neighborhood of t r “ u , giving rise to a sub-complex F C F ˚ p H λ,µ q and to a quotient complex F C I ˚ p H λ,µ q . We havea canonical identification F C ˚ p L λ q ” F C F ˚ p H λ,µ q and c _ r´ n s ” i H λ,µ ,the inclusion of F C F ˚ p H λ,µ q into F C ˚ p H λ,µ q .The projection π : Cone p i H λ,µ q Ñ F C I ˚ p H λ,µ q is a homotopy equiva-lence (see for example [5, Lemma 4.3]). Moreover, the map π does notincrease the action, and also its homotopy inverse ¨˝ ´B I,F ˛‚ : F C I ˚ p H λ,µ q Ñ F C F ˚ p H λ,µ q ‘ F C I ˚ p H λ,µ q ‘ F C F ˚´ p H λ,µ q does not increase the action. As a consequence, the induced maps π p a,b q : Cone p i H λ,µ q p a,b q Ñ F C I, p a,b q˚ p H λ,µ q are also homotopy equivalences for any action interval p a, b q .Let us now fix such a finite action interval p a, b q . For λ ! F falls below the action window. Thus theonly elements in Cone p i H λ,µ q p a,b q are of the form p A, q , where A P F C ˚ p H λ,µ q , and actually A P F C I ˚ p H λ,µ q . The product of two suchelements in Cone p i H λ, µ q is considered modulo action ď a ` b , and assuch is also represented for λ ! F C ˚ p H λ, µ q , andactually in F C I ˚ p H λ, µ q . We then have π p a ` b, b q m pp A, q , p A , qq “ π p a ` b, b q p µ p A, A q , q “ µ p A, A q mod ď a ` b. Thus π interchanges in the relevant action window the product m on Cone p i H λ,µ q p a,b q with the pair-of-pants product µ on F H p a,b q˚ p H λ,µ q .These identifications are compatible with the limits involved in thedefinitions of SH ˚ p H _ q and SH ˚ pB W q . The desired isomorphism ofrings follows. (cid:3) Proof of the Poincar´e Duality Theorem using cones.
Alternative definition of the secondary product on cohomology.
Our original definition of the secondary product on SH ˚ pB W q was inti-mately tied to the proof of Poincar´e duality. We give here an alternativedefinition which is independent of that proof. We prove the equivalenceof the two definitions in Proposition 4.10 and its Corollary 4.11.We consider the family of Hamiltonians H ^ “ t H µ,λ : µ ą ą λ u .Define(26) y SH ˚ pB W q : “ lim ÝÑ b Ñ8 lim ÐÝ a Ñ´8 lim ÐÝ µ Ñ8 , λ Ñ´8
F H p a,b q˚ p H µ,λ q . (This is the same as SH ˚ pB W ˆ I, B W ˆ B I q from [5].) We consider the continuation map c ^ “ c p µ,λ q , p µ,µ q : F C ˚ p H µ,λ q Ñ F C ˚ p L µ q and the associated map σ involved in the definition of themultiplication m : Cone p c p µ,λ q , p µ,µ q q b Ñ Cone p c p µ,λ ` µ q , p µ, µ q q . Thus σ : F C ˚ p H µ,λ qr n s b F C ˚ p H µ,λ qr n s Ñ F C ˚` p H µ,λ ` µ qr n s . (We could have taken the target of σ to be also F C ˚` p H µ, λ qr n s , butthe above choice is in line with the previous discussion.)Recall that the map σ satisfies the relation rB , σ s “ m R p b c ^ q ´ m L p c ^ b q . The relevant observation now is that, for a fixed finiteaction interval p a, b q , the filtered map σ p a,b q : F C p a,b q˚ p H µ,λ qr n s b F C p a,b q˚ p H µ,λ qr n s Ñ F C p a ` b, b q˚` p H µ,λ ` µ qr n s is actually a chain map as soon as µ ą b . Indeed, for µ ą b the 1-periodic orbits of the Hamiltonian L µ have action larger than b , hencethe continuation map c ^ vanishes on F C ď b ˚ p H µ,λ q . We obtain a degree1 product σ p a,b q : F H p a,b q˚ p H µ,λ qr n s b Ñ F H p a ` b, b q˚` p H µ,λ ` µ qr n s for µ ą b . This product stabilizes for p a, b q fixed as λ Ñ ´8 and µ Ñ 8 , and it is compatible with the tautological maps given byenlarging the action window. As such, it induces a degree 1 product σ : y SH ˚ pB W qr n s b y SH ˚ pB W qr n s Ñ y SH ˚` pB W qr n s . In view of the canonical isomorphism y SH ˚ pB W q » SH ´˚ pB W q , weinfer a degree ´ σ _ : SH ˚ pB W qr´ n s b SH ˚ pB W qr´ n s Ñ SH ˚ pB W qr´ n s . It is useful to recast σ and σ _ as degree 0 products. Our convention isto use the shift σ “ ´ σ r´ , ´ ´ s , i.e. σ “ σ r ,
1; 1 s , which definesa degree 0 product σ : y SH ˚ pB W qr n ´ s b y SH ˚ pB W qr n ´ s Ñ y SH ˚` pB W qr n ´ s . Dually, we have a degree 0 product in cohomology σ _ : SH ˚ pB W qr ´ n s b SH ˚ pB W qr ´ n s Ñ SH ˚ pB W qr ´ n s . This is our alternative definition of the product on SH ˚ pB W qr ´ n s . Definition 4.3.
We call the products σ and σ _ the continuation sec-ondary products on y SH ˚ pB W qr n ´ s and SH ˚ pB W qr ´ n s . In orderto emphasize the role played by continuation maps, we denote them by σ c and σ c in the Introduction and in §
7. Their unshifted versions arethen denoted σ c and σ c . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 41
The unit.
The ring p y SH ˚ pB W qr n ´ s , σ q is unital and we give inthis section a description of the unit. As before, we consider the familyof Hamiltonians H ^ “ t H µ,λ : µ ą ą λ u . The starting point of theconstruction is to consider the family of cycles U µ,λ P F C p H µ,λ qr n ´ s defined as follows.Consider a Morse perturbation of H µ,λ in the region t r ď { u with asingle minimum denoted 1 µ . (We can assume w.l.o.g. that the pertur-bation is independent of λ and also independent of µ up to translatingthe values of the function. Also, for further use, we can assume w.l.o.g.that the actions of all the constant orbits are slightly larger than µ { µ is equal to n , and we define U µ,λ “ Bp µ q , where B is the Floer differential for the complex F C ˚ p H µ,λ qr n ´ s .The reason for denoting the minimum 1 µ is that it is a Floer cycle in F C ˚ p L µ qr n s which defines the unit-up-to-continuation for F H ˚ p L µ qr n s .Note also that U µ,λ P F C I ˚ p H µ,λ q , the subcomplex generated by orbitslocated in a neighborhood of t r “ u . Moreover, since U µ,λ is thedifferential of an element of action (slightly larger than) µ {
2, we have U µ,λ P F C ď µ p H µ,λ qr n ´ s . Given a ă µ { U aµ,λ P F C p a, µ q p H µ,λ qr n ´ s the truncation of the cycle U µ,λ in action ą a .The following Lemma is a variant of [5, Lemma 7.4]. Lemma 4.4.
The group y SH ˚ pB W q defined by (26) coincides with lim ÝÑ b Ñ8 lim ÐÝ a Ñ´8 lim ÐÝ λ Ñ´8
F H p a,b q˚ p H b,λ q . (cid:3) Lemma 4.5.
The collection of classes r U a b,λ s P F H p a,b q p H b,λ qr n ´ s , a ă ă b , λ ă defines a class U “ lim ÝÑ b Ñ8 lim ÐÝ a Ñ´8 lim ÐÝ λ Ñ´8 r U a b,λ s P y SH pB W qr n ´ s . Proof.
The key observation is the following. Let µ ě µ ą ą λ ě λ and consider the shifted continuation map c “ c p µ ,λ q , p µ,λ q : F C ˚ p H µ ,λ qr n ´ s Ñ F C ˚ p H µ,λ qr n ´ s . Then(27) c ˚ r U µ ,λ s “ r U µ,λ s . To prove this we choose the homotopy from H µ ,λ to H µ,λ to be constant-up-to-translation in a small neighborhood of the region t r ď { u that contains the constant 1-periodic orbits and the nonconstant 1-periodicorbits of H µ ,λ that correspond to Reeb orbits with period ď µ . Then c p µ q “ µ ` β with β P F C I ˚ p H µ,λ qr n ´ s , and we obtain c p U µ ,λ q “ c Bp µ q “ B c p µ q “ Bp µ ` β q “ U µ,λ ` B β. Equation (27) has filtered variants. For a finite action window p a, b q anda choice of parameters 0 ą λ ě λ , the continuation map c “ c p b,λ q , p b,λ q satisfies c ˚ r U a b,λ s “ r U a b,λ s , so that we can define the limit U p a,b q “ lim ÐÝ λ Ñ´8 r U a b,λ s P lim ÐÝ λ Ñ´8
F H p a,b q p H b,λ qr n ´ s “ y SH p a,b q˚ pB W qr n ´ s . The rest of the proof is formal. The classes r U p a,b q s are compatible withthe morphisms given by enlarging the action windows, hence the class U is well-defined. (cid:3) Proposition 4.6.
The class U from Lemma 4.5 is the unit of the ring p y SH ˚ pB W qr n ´ s , σ q .Proof. Recall the fundamental relation rB , σ s “ m R p b c ^ q´ m L p c ^ b q ,which translates into rB , σ s “ m R p b c ^ q ` m L p c ^ b q in the notationof Lemma 2.8, where c ^ “ c ^ r´
1; 0 s . Let us evaluate both sides at 1 b at the first entry and denote ζ “ σ p b b q : F C ˚ p H b,λ qr n ´ s Ñ F C ˚ p H b,λ ` b qr n ´ s . This is a linear map of degree 1, the shifted degree of 1 b . The relationfor rB , σ s becomes(28) rB , ζ s ` σ p U b,λ b q “ m R p b b c ^ q ` m L p c ^ p b q b q . This is a relation between degree 0 maps defined on
F C ˚ p H b,λ qr n ´ s and taking values in F C ˚ p H b,λ ` b qr n ´ s .The filtered version of relation (28) is rB , ζ s ` σ p U a b,λ b q “ m R p b b c ^ q ` m L p c ^ p b q b q and holds at the level of filtered maps acting as F C p a,b q˚ p H b,λ qr n ´ s Ñ F C p a ` b, b q˚ p H b,λ ` b qr n ´ s . The term m R p b b c ^ q vanishes because c ^ acts as c ^ : F C p a,b q˚ p H b,λ qr n ´ s Ñ F C p a,b q˚ p L b qr n ´ s and the latter complex is zero because all theorbits of L b have action larger than b . On the other hand c ^ p b q “ b and the second term on the right hand side is therefore equal to m L p b b q . This is precisely the continuation map F C p a,b q˚ p H b,λ qr n ´ s Ñ F C p a ` b, b q˚ p H b,λ ` b qr n ´ s induced by a homotopy which is non-increasing on t r ě u and non-decreasing with gap equal to b on Note the sign change!
ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 43 t r ď u . The outcome of the discussion is that σ p U a b,λ b q induces inhomology the continuation map c ˚ : F H p a,b q˚ p H b,λ qr n ´ s Ñ F H p a ` b, b q˚ p H b,λ ` b qr n ´ s . As a consequence of Lemma 4.4, the limitlim ÝÑ b Ñ8 lim ÐÝ a Ñ´8 lim ÐÝ λ Ñ´8 ´ c ˚ : F H p a,b q˚ p H b,λ qr n ´ s Ñ F H p a ` b, b q˚ p H b,λ ` b qr n ´ s ¯ is equal to Id y SH ˚ pB W qr n ´ s . However, the previous discussion shows thatthe above limit is also equal to σ p U b q . This shows that U is the unitfor the ring p y SH ˚ pB W qr n ´ s , σ q . (cid:3) Alternative proof of the Poincar´e Duality Theorem in terms ofthe Duality Theorem 4.1.
Theorem 4.7 (Poincar´e duality redux) . We have a canonical isomor-phism of rings p SH ˚ pB W qr n s , µ q » p y SH ˚ pB W qr n ´ s , σ q . Remark 4.8.
The unitality of the ring p y SH ˚ pB W qr n ´ s , σ q also fol-lows from the above isomorphism. However, this point of view is round-about and the direct description of the unit given in § SH ˚ p H ^ q » y SH ˚ pB W qr n ´ s below—from a discussion ofunitality for products on cones. More specifically, given an A -triple p M , c, A q and assuming that the algebra A is unital, one can writedown conditions under which the ring p Cone p c q , m q is unital with unitequal to p , q , 1 P A . More generally, this is related to the notion ofhomological unitality for A -algebras. Proof of Theorem 4.7.
In view of Theorems 4.1 and 4.2, it is enoughto prove that we have a canonical isomorphism of rings SH ˚ p H ^ q » y SH ˚ pB W qr n ´ s . The proof follows exactly the same lines as those of Theorem 4.2. Givena Hamiltonian H µ,λ as in the definition of y SH ˚ pB W q , its 1-periodicorbits are of two types: type F located in a neighborhood of the re-gion t r ď { u or type I located in a neighborhood of the region t r “ u . Accordingly, the free module F C ˚ p H µ,λ q splits as a directsum F C I ˚ p H µ,λ q ‘ F C F ˚ p H µ,λ q , and F C I ˚ p H µ,λ q is a subcomplex, while F C F ˚ p H µ,λ q is a quotient complex. There is a canonical identification F C F ˚ p H µ,λ q ” F C ˚ p L µ q .Denote c ^ “ c p µ,λ q , p µ,µ q : F C ˚ p H µ,λ qr n s Ñ F C ˚ p L µ qr n s the continu-ation map. We choose the homotopy from H µ,λ to be constant in the region t r ď { u , so that c ^ coincides with the projection p H µ,λ : F C ˚ p H µ,λ qr n s Ñ F C F ˚ p H µ,λ qr n s . It is then a general fact that the in-clusion ι : F C I ˚ p H µ,λ qr n ´ s “ ker p H µ,λ r´ s ã Ñ Cone p p H µ,λ q is a chainhomotopy equivalence which preserves the action filtration, and so doesits explicit homotopy inverse (see [5, Lemma 4.3] and Example 2.13).We therefore obtain chain homotopy equivalences ι p a,b q : F C I, p a,b q˚ p H µ,λ qr n ´ s „ ÝÑ Cone p p H µ,λ q p a,b q “ Cone p c ^ q p a,b q for all action intervals p a, b q .Let us now fix such a finite action interval p a, b q . For µ " F rises above the action window.Thus the only elements in Cone p p H µ,λ q p a,b q are of the form p , ¯ x q , where¯ x P F C ˚ p H µ,λ qr n ´ s , and actually ¯ x P F C I ˚ p H µ,λ qr n ´ s . The prod-uct of two such elements in Cone p p H µ,λ ` µ q is therefore also repre-sented for µ " F C ˚ p H µ,λ ` µ qr n ´ s , and actuallyin F C I ˚ p H µ,λ ` µ qr n ´ s . We thus have ι p a ` b, b q σ p ¯ x, ¯ x q “ p , σ p ¯ x, ¯ x qq “ m pp , ¯ x q , p , ¯ x qq . Thus ι interchanges in the relevant action window the product σ on F H p a,b q˚ p H λ,µ qr n ´ s with the product m on Cone p p H µ,λ q p a,b q .These identifications and products are compatible with the limits in-volved in the definitions of SH ˚ p H ^ q and y SH ˚ pB W qr n ´ s . The desiredisomorphism of rings follows. (cid:3) The products µ and σ preserve the action filtration at chain level. Asa consequence, the homology groups truncated in negative values ofthe action SH ă ˚ pB W q and y SH ă ˚ pB W q inherit products still denoted µ and σ . We refer to [5] for the formal definitions of SH ă ˚ pB W q and y SH ă ˚ pB W q . The following statement is a direct consequence of the factthat the isomorphism from Theorem 4.7 preserves the action filtration. Corollary 4.9.
We have a canonical isomorphism of rings p SH ă ˚ pB W qr n s , µ q » p y SH ă ˚ pB W qr n ´ s , σ q . (cid:3) Recall that y SH ˚ pB W q “ SH ˚ pB W ˆ I, B W ˆ B I q » SH ´˚ pB W q . Weproved in the Duality Theorem 1.1(a) that SH ˚ pB W q carries a productof degree n ´
1, or alternatively that y SH ˚ pB W qr n ´ s carries a productof degree 0. Part (b) of the Duality Theorem 1.1 can then be rephrasedas an isomorphism of rings SH ˚ pB W qr n s » y SH ˚ pB W qr n ´ s . Onthe other hand, we constructed in § y SH ˚ pB W qr n ´ s and the Duality Theorem Redux 4.7 also provides anisomorphism of rings SH ˚ pB W qr n s » y SH ˚ pB W qr n ´ s . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 45
Proposition 4.10.
The isomorphisms SH ˚ pB W qr n s » y SH ˚ pB W qr n ´ s from Theorems 1.1 and 4.7 coincide. Corollary 4.11.
The two products on y SH ˚ pB W qr n ´ s , defined inTheorem 1.1 and in § (cid:3) Proof of Proposition 4.10.
Consider parameters µ ą ą λ and denoteby λ ´ a real number slightly smaller but very close to λ . Denote H λ ´ , p µ,λ q a Hamiltonian obtained from L λ by replacing the linear partof slope λ on the interval r { , s by a “dent” of slopes p λ ´ , µ q , i.e. acontinuous function which is linear of slope λ ´ on r { , r s and linearof slope µ on r r , s for a suitable value r “ r p λ ´ , µ q . See Figure 11. λ ´ µ λH λ ´ , p µ,λ q L λ r Figure 11.
The Hamiltonian H λ ´ , p µ,λ q .We use the graphical notation H λ ´ , p µ,λ q “ λ ´ µ λ , H λ ´ ,µ “ λ ´ µ , L λ “ λ . We also denote the corresponding Floer complexes
F C ˚ p λ ´ µ λ q etc.The module F C ˚ p λ ´ µ λ qr n s splits as a direct sum F C ˚ p µ λ qr n s ‘ F C ˚ p λ ´ qr n s ‘ F C ˚ p λ ´ µ qr n s and the differential is upper triangularwith respect to this splitting. Here the factors denote respectively theorbits appearing in the concave part, in the neighborhood of t r ď { u ,and in the convex part. We denote the diagonal terms of the differential B , B , B and the mixed terms B , etc. We use the same subscriptsfor the components of maps acting between complexes that are split inthis way.We prove the statement of the Proposition in an arbitrary finite actionwindow p a, b q . The statement in the limit a Ñ ´8 , b Ñ 8 follows byarguments similar to the ones encountered before. Also as before, it isenough to discuss the case of a single set of parameters µ " " λ . Wefirst claim that the isomorphism from the Duality Theorem Redux 4.7is described at finite energy as the composition of the chain homotopy equivalences F C ˚ p µ λ qr n ´ s » Cone ´ proj : F C ˚ p λ ´ µ λ qr n s Ñ F C ˚ p λ ´ µ qr n s ¯ » Cone ´ incl : F C ˚ p λ qr n s Ñ F C ˚ p λ µ qr n s ¯ » F C ˚ p λ µ qr n s . Indeed, although the shapes of Hamiltonians used in that proof wereslightly different, their slopes at infinity were the same as the ones ofthe Hamiltonians used above, so that the claim follows by homotopy in-variance of the cone construction. On the other hand, the isomorphismfrom the Duality Theorem 1.1 was induced by the p , q -componentof the differential of the Floer complex F C ˚ p qr n s . We are thusleft to show that the above composition of chain homotopy equivalencesinduces in the action window p a, b q the same map in homology, denotedΦ p a,b q : F C p a,b q˚ p λ µ qr n s Ñ F C p a,b q˚´ p µ λ qr n s . We write
Cone ´ incl : F C ˚ p λ qr n s Ñ F C ˚ p λ µ qr n s ¯ “ F C ˚ p qr n s ‘ F C ˚ p qr n ´ s“ F C ˚ p qr n s ‘ F C ˚ p qr n s ‘ F C ˚´ p qr n ´ s ,Cone ´ proj : F C ˚ p λ ´ µ λ qr n s Ñ F C ˚ p λ ´ µ qr n s ¯ “ F C ˚ p qr n s ‘ F C ˚ p qr n ´ s“ F C ˚ p qr n s ‘ F C ˚ p qr n ´ s ‘ F C ˚ p qr n ´ s . The above composition is explicitly expressed in matrix form as follows(for the middle map we only write a 2 ˆ “ ` B ,
1l 0 ˘ ˆ H , c , ˙ ¨˝ ´ B , ˛‚ . Here H , denotes a homotopy between two possible continuationmaps from to as in the discussion following Definition 2.6,and c , is a continuation map induced by a small homotopy. Themap H , decreases the action, and the map c , distorts the actionby an arbitrarily small amount.Given an element A P F C ˚ p λ µ q its image under Φ isΦ p A q “ B , A ´ B , H , B , A ´ c , B , A. ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 47
The point now is that we work in a finite action window p a, b q . Fora choice of the parameter λ such that λ { ă a , all the generatorsof the complex F C ˚ p λ q have action ă a . Since c , distorts theaction by an arbitrarily small amount and all the other maps involvedin the expression of Φ p A q decrease the action, it follows that, given A P F C p a,b q˚ p µ λ q , the truncation of Φ p A q in action p a, b q isΦ p a,b q p A q “ B p a,b q , A. This proves that the isomorphisms from Theorems 4.7 and 1.1 coincidein the finite action range p a, b q . As already indicated, the statement inthe limit a Ñ ´8 , b Ñ 8 follows by standard arguments which werealready seen before. (cid:3) Products on cones and splittings
Algebraic setting.
Let p M , c, A q be an A -triple as in Defini-tion 2.7. The short exact sequence 0 Ñ A Ñ Cone p c q Ñ M r´ s Ñ . . . H ˚ p M q c ˚ Ñ H ˚ p A q Ñ H ˚ p Cone p c qq Ñ H ˚´ p M q c ˚´ Ñ H ˚´ p A q . . . and thus to short exact sequences(30) 0 Ñ coker c ˚ Ñ H ˚ p Cone p c qq Ñ ker c ˚´ Ñ . Definition 5.1.
We call c homologically split if the above exact se-quence of R -modules is split. We investigate in this section the way in which such homological split-tings interact with product structures.
Lemma 5.2.
Let c be homologically split and denote c ˚ : H ˚ p M q Ñ H ˚ p A q . A choice of splitting / / coker c ˚ / / H ˚ p Cone p c qq / / ker c ˚ r´ s S p p / / determines canonically a product on ker c ˚ r´ s .Proof. Let pr : A ‘ M r´ s Ñ M r´ s be the projection on the secondfactor. We define a product ˜ σ on ker c ˚ r´ s as˜ σ “ pr ˚ m p S b S q . (cid:3) Remark 5.3.
Given x P M we recall that we write ¯ x for its shiftedimage in M r´ s . If we write explicitly S pr ¯ x sq “ rp a x , ¯ x qs we have˜ σ pr ¯ x s , r ¯ y sq “ rp´ q | a | m L p a x , y q ` m R p x, a y q ´ p´ q | ¯ x | σ p x, x qs . The cycle condition on the pair p a x , ¯ x q is cx “ ´B A a x . The pair p a x , ¯ x q is well-defined up to im B Cone , with B Cone p b, ¯ y q “ pB A b ` cy, ´B M y q , butif we impose the condition that the second component is ¯ x , the element a x is well-defined up to im B A ` c p ker B M q . One can check directly thatthe above formula is independent of all choices, as expected.There is no a priori reason for a splitting S to be a ring map withrespect to the product that it induces on ker c ˚ r´ s . Here is a geometricexample where no splitting can be a ring map. Example 5.4.
Let V Ñ C P n be the -disc bundle whose Euler class e is twice the positive generator of H p C P n q , so its boundary B V isdiffeomorphic to R P n ` . Let c : C ˚ p V, B V q Ñ C ˚ p V q be the canonicalinclusion of singular cochains. Then Cone p c q » C ˚ pB V q and the longexact sequence (29) becomes the Gysin sequence ¨ ¨ ¨ Ñ H ˚´ p C P n q c ˚ Ñ H ˚ p C P n q Ñ H ˚ p R P n ` q Ñ H ˚´ p C P n q c ˚´ Ñ . . . where c ˚ is the cup product with the Euler class. Let us now takecoefficients in Z . Then c ˚ “ and the short exact sequence (30) reads Ñ H ˚ p C P n q p ˚ ÝÑ H ˚ p R P n ` q p ˚ ÝÑ H ˚´ p C P n q Ñ for the circle bundle projection p : R P n ` Ñ C P n . The cohomologyrings are H ˚ p R P n ` q “ Z r x s{x x n ` y and H ˚ p C P n q “ Z r y s{x y n ` y with deg x “ and deg y “ . The map p ˚ sends y to x , while p ˚ sends x i to and x i ` to y i . This shows that there exists no splitting H ˚ p R P n ` q “ im p ˚ ‘ B with B a subring, of equivalently, there existsno right inverse S of p ˚ which is a ring map. We are interested in conditions under which a splitting compatible withthe products exists. Our next result gives one answer in this direction(see also Remark 5.7). Recall that the inclusion j : ker c ã Ñ M isa chain map inducing on homology a map j ˚ : H p ker c q Ñ ker c ˚ Ă H ˚ p M q . Proposition 5.5 (Sufficient condition for product-compatible split-tings) . (i) Assume that the canonical map j ˚ : H p ker c q Ñ ker c ˚ is an isomorphism. Then the short exact sequence (30) admits a canon-ical splitting S : ker c ˚ r´ s Ñ H ˚ p Cone p c qq given by S pr ¯ x sq “ rp , ¯ x qs , ¯ x P ker c r´ s . The product ˜ σ induced on ker c ˚ r´ s by the splitting S is ˜ σ pr ¯ x s , r ¯ y sq “ r σ p ¯ x, ¯ y qs , ¯ x, ¯ y P ker c r´ s , ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 49 with σ “ ´ σ r´ , ´ ´ s : M r´ s b Ñ M r´ s , or σ “ σ r ,
1; 1 s .(ii) Assume further that im β Ď im c . Then the canonical splitting S isa ring map with respect to the product that it induces on ker c ˚ r´ s .Proof. (i) Two distinct representatives ¯ x, ¯ x P ker c r´ s of a given classin ker c ˚ r´ s differ by an element of B M r´ s p ker c r´ sq . If ¯ x “ ¯ x `B M r´ s ξ with ξ P ker c r´ s , then p , ¯ x q “ p , ¯ x q ` B Cone p , ξ q whichshows that S is well-defined. It is clearly a splitting, and the formulafor the induced product is a consequence of the explicit formula fromRemark 5.3.(ii) In what follows we consider classes r ¯ x s , r ¯ y s P ker c ˚ r´ s with x, y P ker c r´ s uniquely determined up to B M r´ s p ker c r´ sq . By definition m p S pr ¯ x sq , S pr ¯ y sqq “ rp β p ¯ x, ¯ y q , σ p ¯ x, ¯ y qqs with β “ ´ β r´ , ´
1; 0 s , i.e. β “ β r ,
1; 0 s . On the other hand S ˜ σ pr ¯ x s , r ¯ y sq “ rp , ¯ z qs , where σ p ¯ x, ¯ y q “ ¯ z ` B M r´ s ζ and z P ker c r´ s is uniquely determined up to adding anelement in B M r´ s p ker c r´ sq .Assume now im β Ď im c . Then β p ¯ x, ¯ y q “ c r´
1; 0 s ¯ η for some ¯ η P M ˚ r´ s and we can write p β p ¯ x, ¯ y q , σ p ¯ x, ¯ y qq “ B Cone p , ¯ η q ` p , σ p ¯ x, ¯ y qq ´ p , B M r´ s ¯ η q“ p , σ p ¯ x, ¯ y q ´ B M r´ s ¯ η q ` B Cone p , ¯ η q . Under our assumption that ¯ x, ¯ y P ker c r´ s and also, of course, that ¯ x ,¯ y are cycles for B M r´ s , the equation rB , β s “ ´ c r´
1; 0 s σ ` τ R p c r´
1; 0 sb q ` τ L p b c r´
1; 0 sq of an A -triple yields c r´
1; 0 s σ p ¯ x, ¯ y q “ ´B A β p ¯ x, ¯ y q “ ´B A c r´
1; 0 s ¯ η “ c r´
1; 0 sB M r´ s ¯ η, so that σ p ¯ x, ¯ y q´B M r´ s ¯ η P ker c r´
1; 0 s . Thus we can take ¯ z “ σ p ¯ x, ¯ y q´B M r´ s ¯ η and ¯ ζ “ ¯ η in the equality σ p ¯ x, ¯ y q “ ¯ z ` B M r´ s ¯ ζ , and this showsthat S is a ring map. (cid:3) Lemma 5.6 (Transport of splittings through homotopy equivalences) . Let p M , c , A q be a triple which is homotopy equivalent to the triple p M , c, A q . Assume p M , c, A q carries the structure of an A -triple, andendow p M , c , A q with the transported structure. If Cone p c q admits ahomological splitting which is a ring map, then so does Cone p c q .Proof. In the notation of § I , P theupper triangular maps between the cones and i , ι , p , π their diagonalentries, so that we have a commutative diagram0 / / coker c ˚ / / p ˚ (cid:15) (cid:15) H ˚ p Cone p c qq pr ˚ / / P ˚ (cid:15) (cid:15) ker c ˚ r´ s / / S p p π ˚ (cid:15) (cid:15) / / coker c / / i ˚ „ O O H ˚ p Cone p c qq pr ˚ / / I ˚ „ O O ker c r´ s / / S p p ι ˚ „ O O Assume that S is a homological splitting for Cone p c q which is product-compatible and define S “ P ˚ Sι ˚ . Then pr ˚ S “ pr ˚ P ˚ Sι ˚ “ π ˚ p pr ˚ S q ι ˚ “ π ˚ ι ˚ “ Cone p c q is homotopy equivalent to Cone p c q ,not just a homotopy retract.) Thus S is a homological splitting.The transported structure on Cone p c q is such that P ˚ , I ˚ , π ˚ , ι ˚ arering maps. Thus, if S is a ring map, so is S . (cid:3) Remark 5.7.
In view of Lemma 5.6 one can give a homotopy invariantgeneralization of Lemma 5.5: a product-compatible splitting exists assoon as the A -triple is homotopy equivalent to an A -triple which sat-isfies the assumptions of Lemma 5.5. It is unclear what is the most gen-eral sufficient condition for the existence of product-compatible split-tings.5.2. Splittings for cones of Floer continuation maps.
In this sec-tion W is a Weinstein domain with symplectic completion x W . Recallthe families of Hamiltonians t K λ u and t H λ,µ u from § Lemma 5.8.
Let W be a Weinstein domain of dimension n . For λ ă ă µ consider the map β : F C ˚ p K λ qr n s b F C ˚ p K λ qr n s Ñ F C ˚ p K µ sr n s of degree defined in Section 3. If n ě , then β vanishes identicallyfor any choice of defining data. Remark 5.9.
In the case n “ β can be made tovanish for a suitable choice of defining data. Proof of Lemma 5.8.
The map β decreases the action. Since the actionof 1-periodic orbits of K λ is ď
0, and ă K µ is ě
0, and ą β vanishes as soon as one of theinputs is a non-constant 1-periodic orbit. If the inputs are constantorbits, the output can only be a constant orbit as well. Thus we canassume without loss of generality that K λ “ K ´ ε and K µ “ K ε , with ε ą M C in the sequel. Morse homology groups will bedenoted
M H .Denote deg “ CZ ´ n the grading for Floer complexes shifted downby n . Under our assumption that W is a Weinstein domain, the chaincomplex M ˚ “ F C ˚` n p K ´ ε q “ M C ´˚ p K ´ ε q is supported in degrees t´ n, ´ n ` , . . . , ´ n u and the chain complex A ˚ “ F C ˚` n p K ε q “ M C ´˚ p K ε q is supported in degrees t´ n, ´ n ` , . . . , u . Let x, y P M ˚ be generators. In order for β p x, y q to have a nonzero coefficient in frontof a generator a P A we need deg a “ deg x ` deg y `
2, which via ´ n ď deg a “ deg x ` deg y ` ď ´ n ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 51 implies n ď
2. This shows that β ” n ě (cid:3) Lemma 5.10.
Let W n be a Weinstein domain satisfying the followingtwo conditions: ‚ W has a Weinstein decomposition with rk H n p W q n -handles; ‚ the map H n p W q Ñ SH p W q is injective.For λ ă ă µ consider the continuation map c “ c λ,µ : M ˚ “ F C ˚ p K λ qr n s Ñ A ˚ “ F C ˚ p K µ qr n s . Then, for a suitable choice ofhomotopy from K λ to K µ , the following hold:(i) the canonical map induced by inclusion is an isomorphism j ˚ “ j λ,µ ˚ : H ˚ p ker c q » Ñ ker c ˚ ; (ii) the canonical map induced by projection is an isomorphism pr ˚ “ pr λ,µ ˚ : coker c ˚ » Ñ H ˚ p coker c q . Proof.
We choose the homotopy from K λ to K µ as follows. As theslope varies in the negative range from λ to ´ ε with ε ą r λ , λ s with λ ď λ ă λ ď ´ ε the homotopy is con-stant in the region where the slopes of the Hamiltonians are largerthan λ . Similarly, as the slope varies in the positive range from ε to µ , on each interval r µ , µ s with ε ď µ ă µ ď µ the homo-topy is constant in the region where the slopes of the Hamiltoniansare smaller than µ . The continuation map c λ, ´ ε : F C ˚ p K λ qr n s Ñ F C ˚ p K ´ ε qr n s is then identified with the projection onto the quotientcomplex F C p´ ε, q˚ p K λ qr n s with kernel F C ă ˚ p K λ qr n s , whereas the con-tinuation map c ε,µ : F C ˚ p K ε qr n s Ñ F C ˚ p K µ qr n s is identified withthe inclusion of the subcomplex F C p ,ε q˚ p K µ qr n s . Denote c “ c ´ ε,ε : F C ˚ p K ´ ε qr n s Ñ F C ˚ p K ε qr n s . Then c factors as c “ c ε,µ c c λ, ´ ε and we have ker c “ F C ă ˚ p K λ qr n s ‘ ker c , coker c “ coker c ‘ F C ą ˚ p K µ qr n s . This condition implies that the canonical map H n p W q Ñ H n p W q _ is an iso-morphism. As such, it can be seen as a weak form of Poincar´e duality on theskeleton. Another point of view is to interpret it as an orientability condition: if W “ D ˚ M and M is non-orientable this condition forces R to be 2-torsion, so that M becomes R -orientable. Such a homotopy can only be realized C but not C in r , which is goodenough for our purposes. Alternatively, one can realize the homotopy smoothly byrequiring that, for any pair λ ă λ of slopes in a sequence of non-critical slopes, it isconstant in the region where the slopes of the Hamiltonians are slightly larger than λ . This is enough to guarantee that the continuation map c λ, ´ ε is identified withthe projection onto the quotient complex F C p´ ε, q˚ p K λ qr n s . A similar discussionholds for the homotopy in the range of positive slopes. From this point on the proofs of (i) and (ii) are dual to each other, andwe only give the details for (i).We first claim that it is enough to prove the statement for λ “ ´ ε .To see this, consider the commutative diagram with exact rows andcolumns0 / / F C ă ˚ p K λ qr n s / / ker c / / j ˚ (cid:15) (cid:15) ker c ´ ε,µ / / j ´ ε,µ (cid:15) (cid:15) . . . / / F C ă ˚ p K λ qr n s / / (cid:15) (cid:15) F C ˚ p K λ qr n s / / c (cid:15) (cid:15) F C ˚ p K ´ ε qr n s / / c ´ ε,µ (cid:15) (cid:15) . . . / / / / F C ˚ p K µ qr n s F C ˚ p K µ qr n s / / . . . / / F H ă ˚ p K λ qr n s / / H ˚ p ker c q / / j ˚ (cid:15) (cid:15) H ˚ p ker c ´ ε,µ q / / j ´ ε,µ ˚ (cid:15) (cid:15) . . .. . . / / F H ă ˚ p K λ qr n s / / (cid:15) (cid:15) F H ˚ p K λ qr n s / / c ˚ (cid:15) (cid:15) F H ˚ p K ´ ε qr n s / / c ´ ε,µ ˚ (cid:15) (cid:15) . . .. . . / / / / F H ˚ p K µ qr n s F H ˚ p K µ qr n s / / . . . Taking kernels of the lower vertical maps yields the following diagramfrom which by the 5-lemma proves the claim: . . . / / F H ă ˚ p K λ qr n s / / H ˚ p ker c q / / j ˚ (cid:15) (cid:15) H ˚ p ker c ´ ε,µ q / / j ´ ε,µ ˚ (cid:15) (cid:15) . . .. . . / / F H ă ˚ p K λ qr n s / / ker c ˚ / / ker c ´ ε,µ ˚ / / . . . Next we prove that j ˚ : H ˚ p ker c q » ÝÑ ker c ˚ is an isomorphism for j “ j ´ ε,ε . We can choose K ε “ K and K ´ ε “ ´ K for an exhaustingMorse function K : x W Ñ r , and view c as the continuation mapbetween the Morse cochain complexes c : M C ˚ p´ K q Ñ M C ˚ p K q .By hypothesis we can pick K without critical points of index ą n andwith exactly rk H n p W q critical points of index n . Then rk M C n p K q “ rk M H n p K q , so the differential M C n p K q Ñ M C n ´ p K q , or equivalentlythe differential M C n p´ K q Ñ M C n ` p´ K q , vanishes. Dualizing, thedifferential M C n ´ p K q Ñ M C n p K q also vanishes. Since M C ˚ p´ K q issupported in degrees n, . . . , n and M C ˚ p K q in degrees 0 , . . . , n , themap c can be nonzero only in degree n . Denoting c n : “ c | MC n p´ K q ,the map c thus has the form c “ ˆ c n
00 0 ˙ : M C n p´ K q ‘ M C ą n p´ K q Ñ M C n p K q ‘ M C ă n p K q , ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 53 where the differentials preserve the splittings and vanish on the firstsummands. It follows that ker c “ ker c n ‘ M C ą n p´ K q , the inducedmap on homology is c ˚ “ c n (in particular, it lives only in degree n ),and j ˚ is the composition of isomorphism H ˚ p ker c q – ker c n ‘ M H ą n p´ K q – ker c ˚ . After these preparations, we now prove that j λ,µ ˚ : H ˚ p ker c q Ñ ker c ˚ is an isomorphism. By the first claim, we may assume that λ “ ´ ε ,so that c “ c ε,µ c and c ˚ “ c ε,µ ˚ c ˚ . Since c ε,µ is injective, the firstequation implies ker c “ ker c . Since im c ˚ Ă H n p W q by the precedingdiscussion, and c ε,µ ˚ : H n p W q Ñ SH p W q is injective by hypothesis,the second equation implies ker c ˚ “ ker c ˚ . Hence j ´ ε,µ ˚ : H ˚ p ker c q Ñ ker c ˚ coincides with the map j ˚ : H ˚ p ker c q Ñ ker c ˚ , which wasshown above to be an isomorphism.As already mentioned, the proof of (ii) is dual to that of (i) and weomit it. (cid:3) In the next result we consider the symplectic homology group SH ˚ pt K λ uq .This sits in a long exact sequence(31) SH ´˚ p W qr´ n s c ˚ / / SH ˚ p W qr n s ι / / SH ˚ pt K λ uq π / / SH ´˚ p W qr´ n s induced by long exact sequences of cones for the maps c λ,µ , λ ă ă µ . Corollary 5.11.
Let W be a Weinstein domain of dimension n ě which satisfies the conditions: ‚ W has a Weinstein decomposition with rk H n p W q n -handles. ‚ the map H n p W q Ñ SH p W q is injective.Then the short exact sequences Ñ coker c ˚ Ñ SH ˚ pt K λ uq Ñ ker c ˚ r´ s Ñ arising from (31) admit a canonical splitting which is a ring map. (cid:3) Remark 5.12.
The assumptions of Corollary 5.11 hold true for thefollowing classes of Weinstein domains:(i) cotangent bundles of closed manifolds of dimension ě
3, witharbitrary coefficients if the manifold is orientable, and with Z { ě
3. Indeed, as shown in [6, Theorem 54],their boundaries admit contact forms whose closed Reeb orbitsare nondegenerate and have transverse Conley-Zehnder index ě (iv) Milnor fibers W of dimension 2 n ě ‰ ,
1. Indeed,the first assumption of Corollary 5.11 is satisfied by all Milnorfibers, and the condition on the indices ensures that the secondassumption is also satisfied because the non-constant orbits in-volved in the definition of symplectic homology cannot kill anygenerators of H n p W q “ SH p W q .Specific examples are the Milnor fillings of Brieskorn mani-folds Σ p ℓ, , . . . , q of dimension 2 n ´ ě
5. The indices havebeen computed explicitly by Ustilovsky [23] and van Koert [24],see also Uebele [22] and Fauck [8]. That all indices are ě Definition 5.13.
A Weinstein domain of dimension n ě is said tobe cotangent-like if it satisfies the conditions: ‚ W has a Weinstein decomposition with rk H n p W q n -handles. ‚ the map H n p W q Ñ SH p W q is injective. Corollary 5.11 is then rephrased as follows.
Corollary 5.11.bis.
Let W be a cotangent-like Weinstein domain ofdimension n ě . The short exact sequences Ñ coker c ˚ Ñ SH ˚ pt K λ uq Ñ ker c ˚ r´ s Ñ arising from (31) admit a canonical splitting which is a ring map.Recall from [4] that, given W “ D ˚ M the cotangent disc bundle ofa closed manifold M , the Rabinowitz Floer homology SH ˚ p S ˚ M q ofthe unit sphere bundle S ˚ M “ B D ˚ M was denoted q H ˚ Λ and called extended loop homology of the free loop space Λ “ Λ M . It was provedin [4, Theorem 1.4] that the short exact sequence Ñ coker c ˚ Ñ q H ˚ Λ Ñ ker c ˚ r´ s Ñ admits a canonical splitting compatible withproducts. Our next result is that this splitting coincides with the onedescribed previously in terms of cones. Proposition 5.14.
Let W “ D ˚ M be a cotangent disk bundle of amanifold of dimension n ě . Through the isomorphism SH ˚ pt K λ uq » SH ˚ p H _ q » q H ˚ Λ r n s from Theorem 4.2, the canonical splitting from Corollary 5.11 coincideswith the canonical splitting from [4, Theorem 1.4] . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 55
Proof.
Just like in the proof of Lemma 5.10, it is enough to prove theresult at energy zero. We thus choose ε ą H “ H ´ ε,ε , L “ H ´ ε, ´ ε , and ˘ K “ K ˘ ε . We work with the triples p M , c, A q “ p F C ˚ p´ K qr n s , c , F C ˚ p K qr n sq and p M , c , A q “ p F C ˚ p L qr n s , i H , F C ˚ p H qr n sq , which are homotopyequivalent. Here c , i H are the canonical continuation maps, and i H isidentified with an inclusion. The relevant part of diagram (6) becomes F C ˚ p´ K qr n s c (cid:15) (cid:15) π / / K & & ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ F C ˚ p L qr n s i H (cid:15) (cid:15) ι o o F C ˚ p K qr n s p / / F C ˚ p H qr n s i o o where the horizontal arrows are the canonical homotopy equivalencesinduced by monotone homotopies, and K : M r´ s “ F C ˚ p´ K qr n ´ s Ñ A “ F C ˚ p H qr n s is a homotopy between i H π and p c givingrise to the homotopy equivalence P “ ˆ p K π ˙ : Cone p c q „ ÝÑ Cone p i H q . See also the discussion and notation following Definition 2.6.By Corollary 5.11, the triple p M , c , A q admits a homological splittingwhich is a ring map S : ker p c q ˚ r´ s Ñ H ˚ p Cone p c qq , r ¯ x s ÞÑ rp , ¯ x qs defined for ¯ x P ker c r´ s and B ¯ x “
0. Following Lemma 5.6, weobtain an induced homological splitting S H “ P ˚ Sι ˚ for the triple p M , i H , A q . We identify F C ˚ p L q and F C ˚ p´ K q , so that π and ι become equal to the identity and the splitting S H is explicitly givenby the formula(32) S H : ker p i H q ˚ r´ s Ñ H ˚ p Cone p i H qq , r ¯ x s ÞÑ rp K p ¯ x q , ¯ x qs for ¯ x P ker c r´ s and B ¯ x “ § H induced by a Morse function f : M Ñ R , the critical points of H are of the type p F , p ˘ for each p P Crit p f q , with p F located on the zerosection, p ˘ located on S ˚ M , andind p p ` q “ ind p p q , ind p p ´ q “ ind p p q ` n ´ , ind p p F q “ ind p p q ` n. Denote B f the Morse coboundary operator for f . The Morse cobound-ary operator B for H was seen to satisfy B p ´ min “ p F min ` χ ¨ p ` max , B p ´ “ p F ` pB f p p qq ´ for all p ‰ p min . Also, the points t p F u generate a subcomplex which is identified with M C ˚ p f qr´ n s , and, for n ě
2, the points t p ` u generate a subcomplexwhich is identified with M C ˚ p f q .To prove these statements we started in [4, § h fiber on the n -disc, with three critical points p Ffiber of index n , p ´ fiber of index n ´ p ` fiber of index 0, and then grafted it in thecritical fibers of the lift to D ˚ M of a Morse function f : M Ñ R . SeeFigure 12. p Ffiberr p ˘ fiber p ` fiber | p Ffiber | “ n, | p ` fiber | “ , | p ´ fiber | “ n ´ p Ffiber p ´ fiber Figure 12.
Morse profile on the n -disc.This same procedure can be used in order to compute the homo-topy K on the complex generated by the points t p F u belonging toker c . To see this, consider first the fiber Morse cochain complex M C ˚ p h fiber q generated by p Ffiber , p ˘ fiber . Consider also the Morse pro-files ℓ fiber with only one critical point p Ffiber of index n and equal to h fiber on a smaller disc, and k fiber with only one critical point of in-dex 0, equal to a rescaling of ´ ℓ fiber . The inclusion of the subcomplexconsisting of p Ffiber , identified with
M C ˚ p ℓ fiber q , into the total com-plex M C ˚ p h fiber q , is homotopic to the zero map seen as a continuationmap from ℓ fiber to ´ k fiber to k fiber to h fiber . The homotopy, denoted K fiber : M C ˚ p ℓ fiber q Ñ M C ˚´ p h fiber q , is described as usual by count-ing index -1 connecting gradient trajectories in a 1-parameter homo-topy of homotopies. From the relation rB fiber , K fiber sp p Ffiber q “ p Ffiber
ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 57 we deduce the explicit formula K fiber p p Ffiber q “ p ´ fiber . By grafting thissituation fiberwise onto D ˚ M we obtain that the homotopy K satisfies K p p F q “ p ´ for all p P Crit p f q . Referring back to equation (32) which describes thesplitting S H , we obtain S H pr ¯ x F sq “ rp x ´ , ¯ x F qs for ¯ x F P ker c r´ s and B ¯ x F “ x F is a linear combinationof (shifted) generators p F , and x ´ denotes the corresponding linearcombination of generators p ´ ). Through the identification between Cone p i H q and C I ˚ p H qr n s given by projection, the splitting acts there-fore as S H pr ¯ x F sq “ r x ´ s for ¯ x F P ker c r´ s and B ¯ x F “
0. As such, the splitting coincides withthe canonical splitting from [4, § (cid:3) Mixed products and reduced homology
In this section we prove Theorem 1.2 from the Introduction. Let W be a cotangent-like Weinstein domain of dimension 2 n ě
6. We usethe notation µ for the primary product on SH ˚ p W q and λ for thesecondary coproduct on SH ą ˚ p W q dual to the continuation product σ c on SH ˚ą p W q , and we reserve the symbols λ and µ for action valueparameters.The statement of Theorem 1.2 is in effect a reinterpretation of thevarious components of the product m on SH ˚ pB W q in terms of thenatural duality pairing x¨ , ¨y : SH ´˚ p W qb SH ˚ p W q Ñ R , which induces x¨ , ¨y : SH ´˚ p W q b SH ˚ p W q Ñ R. We thus start with a discussion of the duality pairing, after which wedescribe the secondary pair-of-pants coproduct in terms of continuationmaps, and we close the section with the proof of the theorem.6.1.
The duality pairing.
Given a Liouville domain W there is acanonical evaluation map , or duality pairing x¨ , ¨y : SH ˚ p W q b SH ˚ p W q Ñ R defined as follows. Given any Hamiltonian H there is a canonical iden-tification F C ˚ p H q “ F C ˚ p H q _ , inducing also a canonical identification F C ˚p a,b q p H q “ F C p a,b q˚ p H q _ . This provides a canonical map F H ˚p a,b q p H q Ñ F H p a,b q˚ p H q _ which is compatible with the continuation maps and action truncationmaps which are involved in the definition of SH ˚ p W q and SH ˚ p W q .We obtain a canonical map(33) SH ˚ p W q Ñ SH ˚ p W q _ , or equivalently(34) x¨ , ¨y : SH ˚ p W q b SH ˚ p W q Ñ R. More precisely, recalling the family of Hamiltonians t K λ u from Exam-ple 3.5, we have SH ˚ p W q “ lim ÝÑ µ Ñ8 F H ˚ p K µ q and SH ˚ p W q “ lim ÐÝ µ Ñ8 F H ˚ p K µ q . The map SH ˚ p W q Ñ SH ˚ p W q _ is obtained by passing to the limit inthe directed system of maps F H ˚p´8 ,a q p K λ q / / F H p´8 ,a q˚ p K λ q _ F H ˚p´8 ,a q p K λ q / / O O F H p´8 ,a q˚ p K λ q _ O O defined for λ ď λ and a ď a .The same construction also provides a canonical duality pairing(35) x¨ , ¨y : SH ˚ pB W q b SH ˚ pB W q Ñ R. We now describe the duality pairing in terms of counts of pseudoholo-morphic curves. Let λ ă ă µ such that λ ` µ ď
0. Then K λ ` K µ ď π λ,µ : F C ´˚ p K λ q b F C ˚ p K µ q Ñ R defined by the count of genus 0 curves with 2 positive punctures andcylindrical ends at the positive punctures, solving a perturbed Cauchy-Riemann equation which reduces to the Floer equation for K λ , respec-tively K µ near the punctures. See Figure 13. π λ,µ K λ K µ Figure 13.
Moduli spaces for the duality pairing.The map π λ,µ can be equivalently phrased as a map π _ λ,µ : F C ´˚ p K λ q Ñ F C ˚ p K µ q _ . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 59
We have a system of squares which commute up to homotopy given bythe diagonal arrow
F C ´˚ p K λ q π _ λ,µ / / F C ˚ p K µ q _ F C ´˚ p K λ q π _ λ ,µ / / c λ ,λ O O π _ λ ,µ qqqqqqqqqqqqqqqqq F C ˚ p K µ q _ c _ µ,µ O O defined for λ ď λ ă ă µ ď µ such that λ ď ´ µ and λ ď ´ µ . Thesesquares reflect the possible splittings from Figure 14 below. Passing tothe inverse limit as λ Ñ ´8 and µ Ñ 8 we obtain π _ : SH ˚ p W q Ñ SH ˚ p W q _ . c µ,µ K λ K λ K µ π λ ,µ π λ,µ K µ „ K µ π λ ,µ K µ K λ „ K λ c λ ,λ Figure 14.
Splittings for duality pairing moduli spaces.
Proposition 6.1.
The map π _ coincides with the duality pairing (33) .Proof. Fix µ ą
0. There is a canonical identification
F C ´˚ p K ´ µ q – F C ˚ p K µ q – F C ˚ p K µ q _ , and the key observation is that, via thisidentification, the degree 0 map π _´ µ,µ : F C ´˚ p K ´ µ q Ñ F C ˚ p K µ q _ is the identity for a suitable choice of defining data. Equivalently, π ´ µ,µ : F C ´˚ p K ´ µ q b F C ˚ p K µ q – F C ˚ p K µ q _ b F C ˚ p K µ q Ñ R isthe canonical duality pairing for a suitable choice of defining data.Indeed, consider on the cylinder R ˆ S Q p s, t q the Floer equation B s u ` J t p u qpB t u ´ X K µ p u qq “
0. Viewed as a continuation equationwhich does not depend on the parameter s , it induces the identity F C ˚ p K µ q Ñ F C ˚ p K µ q . On the other hand, we can turn the negativepuncture on the cylinder into a positive puncture and keep the equationunchanged, in which case we formally need to replace the Hamiltonian K µ at the negative puncture by the Hamiltonian K ´ µ “ ´ K µ . The re-sulting moduli spaces describe the map π ´ µ,µ . By construction, thereis a canonical bijection between the moduli spaces which describe thecontinuation map Id : F C ˚ p K µ q Ñ F C ˚ p K µ q and those which describethe map π ´ µ,µ . We infer that π ´ µ,µ is indeed the canonical dualitypairing, or equivalently π _´ µ,µ is the identity.The statement concerning the map π _ follows by noticing that thevarious continuation maps involved in the definitions can be suitablyfactored through maps π _´ µ,µ as above. (cid:3) Consider now the canonical map c ˚ : SH ´˚ p W q Ñ SH ˚ p W q involvedin the homology long exact sequence of the pair p W, B W q as the com-position SH ´˚ p W q ” SH ˚ p W, B W q Ñ SH ˚ p W q , where the first iso-morphism is Poincar´e duality. Denote SH ˚ p W q “ coker c ˚ , SH ´˚ p W q “ ker c ˚ . Proposition 6.2.
The duality pairing induces a reduced pairing π : SH ˚ p W q b SH ˚ p W q Ñ R, or equivalently a map π _ : SH ˚ p W q Ñ SH ˚ p W q _ . Proof.
Given λ ă ă µ consider the continuation map c λ,µ : F C ˚ p K λ q Ñ F C ˚ p K µ q and denote F H ˚ p K µ q “ coker p c λ,µ q ˚ , F H ˚ p K λ q “ ker p c λ,µ q ˚ . (The group F H ˚ p K µ q is independent on the choice of λ ă
0, and thegroup
F H ˚ p K λ q is independent on the choice of µ ą λ, λ ă ă µ such that λ, λ ď ´ µ , the pairing π λ,µ : F H ˚ p K λ q b F H ˚ p K µ q Ñ R vanishes onker p c λ,µ q ˚ b im p c λ ,µ q ˚ . In a different formulation, we must show that π λ,µ ˝ b p c λ ,µ q ˚ is zero if the first argument lies in ker p c λ,µ q ˚ . Thiscomposition is described by a count of genus 0 curves with 2 positivepunctures and cylindrical ends at the positive punctures, solving a per-turbed Cauchy-Riemann equation which reduces to the Floer equationfor K λ , respectively K λ near the punctures. See Figure 15. Turning thepositive puncture corresponding to K λ into a negative puncture andkeeping the equation fixed by formally replacing K λ by K ´ λ “ ´ K λ ,this count is equivalent to the count of curves which determine thecontinuation map c λ, ´ λ . For a suitable choice of defining data this lastmap factors as c µ, ´ λ ˝ c λ,µ , and therefore vanishes in homology if theinput corresponding to K λ lies in ker p c λ,µ q ˚ . (cid:3) c λ, ´ λ π λ,λ π λ,µ K λ „ “ K λ K µ c λ ,µ „ K µ K λ K ´ λ K ´ λ K λ K λ K λ c λ,µ c µ, ´ λ Figure 15.
Construction of the reduced pairing (Proposition 6.2).
ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 61
Reduced secondary pair-of-pants coproduct.
The purposeof this section is to describe a degree 1 secondary pair-of-pants coprod-uct λ : SH ˚ p W qr´ n s Ñ SH ˚ p W qr´ n s b on reduced symplectic homology in terms of continuation maps, un-der the assumption that W is a cotangent-like Weinstein domain ofdimension 2 n ě
6, see Definition 5.13.
Remark 6.3.
This recovers our other definitions from [4] in the case W “ D ˚ M , as a consequence of Proposition 4.10, Corollary 4.11, andProposition 5.14.Recall the family of Hamiltonians t K λ u from Example 3.5. Considerparameters λ , λ ă ă µ , µ , µ such that µ ď λ ` µ and µ ď µ ` λ . We define degree 0 chain maps (coproducts) m _ L : F C ˚ p K µ qr´ n s Ñ F C ˚ p K λ qr´ n s b F C ˚ p K µ qr´ n s and m _ R : F C ˚ p K µ qr´ n s Ñ F C ˚ p K µ qr´ n s b F C ˚ p K λ qr´ n s by counting index 0 pairs-of-pants with 1 positive puncture and 2 neg-ative punctures, asymptotic to 1-periodic orbits of K µ at the positivepuncture, and asymptotic to 1-periodic orbits of K λ and K µ at thenegative punctures in the case of m _ L , respectively asymptotic to 1-periodic orbits of K µ and K λ at the negative punctures in the case of m _ R . See Figure 16.We further define a degree 1 map ˜ λ “ ˜ λ µ, p µ ,µ q : F C ˚ p K µ qr´ n s Ñ F C ˚ p K µ qr´ n s b F C ˚ p K µ qr´ n s by counting index ´ K µ at the positive punctureand asymptotic to 1-periodic orbits of K µ and K µ at the negativepunctures, and subject to boundary conditions given by splitting off acontinuation cylinder from K λ to K µ at the first negative puncture,respectively splitting-off a continuation cylinder from K λ to K µ atthe second negative puncture. See Figure 16.The map ˜ λ satisfies the equation(37) rB , ˜ λ s “ p b c q m _ R ´ p c b q m _ L , where on the right hand side c stands for the continuation map c λ ,µ in the expression p c b q m _ L , and it stands for the continuation map c λ ,µ in the expression p b c q m _ R . K λ ˜ λ p b c q m _ R p c b q m _ L K µ K µ K λ K µ K µ K µ K µ K µ K µ K µ Figure 16.
Curves defining the map ˜ λ . Remark 6.4.
If one arranges the Hamiltonians K λ , λ ă t r ď u , the chain complexes coker c λ,µ are independentof λ ă µ ą F C ˚ p K µ q “ coker c λ,µ , λ ă , µ ą . Under our standing assumption that W is a Weinstein domain whichis cotangent-like, we have F H ˚ p K µ q “ H ˚ p F C ˚ p K µ qq . (This is thecontent of Lemma 5.10(ii).)Similarly, if we arrange the Hamiltonians K µ , µ ą t r ě u , the chain complexes ker c λ,µ are independent of µ ą λ ă F C ˚ p K λ q “ ker c λ,µ , λ ă , µ ą . Under our standing assumption that W is a Weinstein domain whichis cotangent-like, we have F H ˚ p K λ q “ H ˚ p F C ˚ p K λ qq . (This is thecontent of Lemma 5.10(i).)As a consequence of the relation (37), the map ˜ λ descends to a degree1 chain map F C ˚ p K µ qr´ n s Ñ F C ˚ p K µ qr´ n s b F C ˚ p K µ qr´ n s . Next we observe that the map ˜ λ vanishes on the image of c and there-fore factors through a chain map denoted(38) λ µ, p µ ,µ q : F C ˚ p K µ qr´ n s Ñ F C ˚ p K µ qr´ n s b F C ˚ p K µ qr´ n s . Indeed, the map ˜ λ c is dual to the map cσ from Lemma 5.8, so itsvanishing follows by the same argument as the vanishing of cσ in theproof of Lemma 5.8: for action reasons, ˜ λ c lives only on the constantorbits, where it vanishes for degree reasons. ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 63
The maps (38) fit in a direct system of commutative diagrams
F H ˚ p K µ qr´ n s λ µ, p µ ,µ q / / c (cid:15) (cid:15) F H ˚ p K µ qr´ n s b F H ˚ p K µ qr´ n s c b c (cid:15) (cid:15) F H ˚ p K µ qr´ n s λ µ , p µ ,µ q / / F H ˚ p K µ qr´ n s b F H ˚ p K µ qr´ n s defined for µ ď µ , µ ď µ , µ ď µ , subject to the conditions µ ď µ ` µ and µ ď µ ` µ . Using that SH ˚ p W qr´ n s “ lim ÝÑ µ Ñ8 F H ˚ p K µ qr´ n s , we define λ “ lim ÝÑ µ Ñ8 λ µ, p µ,µ q . Definition 6.5.
The secondary pair-of-pants coproduct on reducedsymplectic homology of a cotangent-like Weinstein domain is the de-gree coproduct λ : SH ˚ p W qr´ n s Ñ SH ˚ p W qr´ n s b . The map λ induces a map SH ˚ p W qr´ n s Ñ SH ą ˚ p W qr´ n s b by pro-jection in the target, and it is easy to see that this last map factorsthrough SH ą ˚ p W qr´ n s . We obtain in particular a secondary pair-of-pants coproduct on positive symplectic homology λ ą : SH ą ˚ p W qr´ n s Ñ SH ą ˚ p W qr´ n s b . As already mentioned, these coproducts coincide in the case W “ D ˚ M with our other equivalent definitions as a consequence of Proposi-tion 4.10, Corollary 4.11, and Proposition 5.14.6.3. Proof of Theorem 1.2.
For the proof we use our description of SH ˚ pB W qr n s as SH ˚ pt K λ uq , see Theorem 4.2. By Corollary 5.11 thesplitting SH ˚ pB W qr n s “ SH ˚ p W qr n s ‘ SH ´˚` p W qr n s arises from thesplitting of the short exact sequence0 Ñ coker c ˚ Ñ SH ˚ pt K λ uq Ñ ker c ˚ r´ s Ñ S : ker c ˚ r´ s Ñ H ˚ p Cone p c qq , r ¯ x s ÞÑ rp , ¯ x qs .Here ¯ x P ker c r´ s is a cycle, and c “ c λ,µ : F C ˚ p K λ qr n s Ñ F C ˚ p K µ qr n s is the continuation map for λ ă ă µ . We havecoker c ˚ “ H ˚ Λ r n s “ H ˚` n Λ , ker c ˚ r´ s “ H ´˚ Λ r n s “ H ´˚´ n Λ . The relations to be proved all follow from the description of the producton the cone given in Proposition 2.9. Let λ ´ ď λ ` and recall fromExample 3.5 the continuation maps induced by monotone homotopies c λ ´ ,λ ` : F C ˚ p K λ ´ qr n s Ñ F C ˚ p K λ ` qr n s . Given λ ď λ such that λ ´ ` λ ` ď λ and 2 λ ` ď λ , we haveconstructed operations µ , m L , m R , σ , τ L , τ R , β which assemble into abilinear map (product)(39) m “ m p λ ´ ,λ ` q , p λ ,λ q : Cone p c λ ´ ,λ ` q p a,b q b Cone p c λ ´ ,λ ` q p a,b q Ñ Cone p c λ ,λ q p a ` b, b q for all ´8 ă a ă b ă 8 , see (19). The product on SH ˚ pt K λ uq wasconstructed at homology level from the collection of these maps by alimiting procedure.Abbreviate c “ c λ ´ ,λ ` and c “ c λ ,λ . When passing to homology,each short exact sequence0 Ñ coker c ˚ Ñ H ˚ p Cone p c qq Ñ ker c ˚ r´ s Ñ H ˚ p Cone p c qq “ coker c ˚ ‘ ker c ˚ r´ s .We denote the map induced in homology by the products m from (39)by m (since it gives rise to the map m in Theorem 1.2 in the direct-inverse limit over µ, λ ). It has components m ``` , m `´` etc., where thesuperscripts refer to inputs, the subscripts refer to outputs, the symbol ` refers to an input or output in coker c ˚ , and the symbol ´ refers toan input or output in ker c ˚ r´ s . Each of these components is in turninduced by one of the maps µ , m L , m R , σ , τ L , τ R , β which enter intothe definition of m . Thus m ``` is induced by µ , m `´` is induced by τ R , m ´`` is induced by τ L , m `´´ is induced by m L , m ´`´ is induced by m R , m ´´´ is induced by σ , and m ´´` is induced by β . Also m ``´ “ m there is no map taking inputs in thetarget of c and output in the source of c . Recall also that the maps m L , m R , σ , τ L , τ R , β are shifts of the maps m L , m R , σ , τ L , τ R , β , asdescribed in the proof of Lemma 2.8. We use the following notation:given f P ker c ˚ , we denote ¯ f P ker c ˚ r´ s the same element with degreeshifted up by 1, i.e. | ¯ f | “ | f | ` m ``` “ µ follows from the fact that m ``` isinduced by µ .Proof of (2). The relation x m ´´´ p ¯ f , ¯ g q , a y “ p´ q | g | x f b g, λ p a qy followsfrom the fact that m ´´´ is induced by σ , in conjunction with equa-tions (13) and (48). Indeed, these equations give x σ p ¯ f , ¯ g q , a y “ p´ q | f | x σ p f, g q , a y“ p´ q | f | ¨ p´ q | f |`| g | x f b g, λ p a qy“ p´ q | g | x f b g, λ p a qy . Proof of (3). The relation m ``´ “ m ``´ “ m ´´` “ ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 65 m ´´` is induced by β which vanishes already at chain level for n ě m ´`` p ¯ f , a q “ ´p´ q | f | x f b , λ p a qy followsfrom the fact that m ´`` is induced by τ L , together with equations (16)and (49). These equations give τ L p ¯ f , a q “ τ L p f, a q “ ´p´ q | f | x f b , λ p a qy . (See also Remark 6.6 for an explanation of equation (49) in Floer the-oretic terms.)The relation m `´` p b, ¯ f q “ p´ q | b | x λ p b q , b f y is a consequence of thefact that m `´` is induced by τ R , together with equations (17) and (50).These equations give τ R p b, ¯ f q “ p´ q | b | τ R p b, f q “ p´ q | b | x λ p b q , b f y . Proof of (5). The relation x m ´`´ p ¯ f , a q , b y “ x f, µ p a, b qy is a consequenceof the fact that m ´`´ is induced by m R , together with equations (15)and (47). These equations give x m R p ¯ f , a q , b y “ x m R p f, a q , b y “ x f, µ p a, b qy . (We refer to Remark 6.7 for an explanation of equation (47) in Floertheoretic terms.)The relation x a, m `´´ p b, ¯ f qy “ p´ q | b | x µ p a, b q , f y is a consequence of thefact that m `´´ is induced by m L , together with equations (14) and (46).These give x a, m L p b, ¯ f qy “ p´ q | b | x a, m L p b, f qy “ p´ q | b | x µ p a, b q , f y . (cid:3) Remark 6.6.
We give in this remark an alternative, Floer theoreticpoint of view on equation (49). This equation was used in the proofof the formula describing m ´`` and it reads τ L p ¯ f , a q “ ´p´ q | f | x f b , λ p a qy . The equation is understood to hold at the level of symplectichomology groups, but since all operations are suitably compatible withdirect and inverse limits, the equation holds for suitable finite slopes ofthe Hamiltonians at the inputs and outputs. We view τ L acting as τ L : F C ˚ p K λ q b F C ˚ p K µ q Ñ F C ˚ p K µ q , the map x¨ , ¨y “ π λ, ´ λ : F C ˚ p K λ q b F C ˚ p K ´ λ q Ñ R is the pairing induced from (36), and λ : F C ˚ p K µ q Ñ F C ˚ p K ´ λ q b F C ˚ p K µ q is the secondary pair-of-pants coproduct from (38).We prove that the maps ´ τ L and p ev b qp b λ q : f b a ÞÑ p´ q | f | x f b , λ p a qy acting from F C ˚ p K λ q b F C ˚ p K µ q Ñ F C ˚ p K µ q coincide for a suitable choice of defining data. Then the same is true when passingto the reduced chain groups. Consider therefore chain representatives f P F C ˚ p K λ q , a P F C ˚ p K µ q and see these maps as providing chainlevel outputs in F C ˚ p K µ q .The map ´ τ L is described by a count of index ´ K λ and K µ and 1 negative punc-ture with asymptote K µ , parametrized by the 1-simplex, with bound-ary condition at the negative boundary of the simplex given by splittingoff a continuation cylinder from K λ to K µ at the first positive punc-ture, respectively boundary condition at the positive boundary of thesimplex given by splitting off a continuation cylinder from K λ ` µ to K µ at the negative puncture. See Figures 17 and 26. K λ K µ K µ K µ K µ K µ K λ K λ K µ K µ K λ ` µ Figure 17.
Curves defining the map ´ τ L in Floer theory.The composition p ev b qp b λ q : f b a ÞÑ p´ q | f | x f b , λ p a qy isdescribed, in view of equation (49), by attaching at the first negativepuncture of each curve in the count that defines λ an index 0 cylinderwith 2 positive punctures with asymptotes K λ and K ´ λ . The outcomeis a description of this composition as a count of index ´ K λ and K µ and 1negative puncture with asymptote K µ , parametrized by the 1-simplex,with boundary condition at the negative boundary of the simplex givenby splitting off at the first positive puncture a cylinder with 2 positivepunctures with asymptotes K λ and K ´ µ , and boundary condition at thepositive boundary of the simplex given by splitting off at the negativepuncture a continuation cylinder from K λ ` µ to K µ . See Figure 18,in which the symbol K ´ λ on the bended arcs means that the definingdata on the corresponding cylinder is given by the Floer equation for K ´ λ on some long neck that corresponds to gluing.We see that the two moduli spaces defining the map ´ τ L and the abovecomposition agree if in the left hand configuration in Figure 18 we rein-terpret the positive/negative puncture at K ´ µ as a negative/positivepuncture at K µ . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 67 K µ K λ ` µ K µ K µ K µ K µ K µ K ´ µ K ´ λ K λ K λ K ´ λ K ´ λ K λ Figure 18.
Curves defining the map f b a ÞÑ´p´ q | f | x f b , λ p a qy . Remark 6.7.
We describe in this remark a Floer theoretic point ofview on equation (47). Denote µ the product in order to distinguish itfrom slope values denoted µ . Working at fixed slope, the two sides ofthe equality x m R p f, a q , b y “ x f, µ p a, b qy with a, b P F H ˚ p K µ q , f P F H ˚ p K λ q , and λ ă ă µ such that λ ` µ ă
0, are both described at chain level by counts of index 0 genus 0 curveswith 3 positive punctures with asymptotes K λ , K µ and K µ . These twocounts appear at the ends of the moduli space parametrized by the1-simplex shown in Figure 19, so they agree on the level of homology. bK λ ` µ K µ „ m R x¨ , ¨y „ K λ µ K µ K µ K µ x¨ , ¨y K λ K µ aγ K µ K µ K λ γ a aγb b Figure 19.
Curves for interpreting m R in terms of theduality pairing.7. The pair-of-pants coproduct using varying weights
We restrict in this section to the homology negative action range y SH ă ˚ pB W q – SH ă ˚ p W, B W q – SH ´˚ą p W q . We introduce in § varying weights secondary product σ w on SH ă ˚ p W, B W q . We show in § duality product σ P D and with the continuation product σ c . The prod-uct σ w is not used elsewhere in the paper but, unlike the Poincar´eduality product σ P D and the continuation product σ c , it did appearpreviously in the literature. Its construction goes back to Seidel andwas further explored in [7], see also [2]. The purpose of this section isclarify its relationship to the constructions of the present paper.7.1. Definition of the varying weights secondary product.
LetΣ be the genus zero Riemann surface with three punctures, two ofthem labeled as positive z ` , z ` and the third one labeled as negative z ´ , endowed with cylindrical ends r ,
8q ˆ S at the positive puncturesand p´8 , s ˆ S at the negative puncture. Denote p s, t q , t P S theinduced cylindrical coordinates at each of the punctures. Consider asmooth family of 1-forms β ǫ P Ω p Σ q , ǫ P p , q satisfying the followingconditions: ‚ (nonnegative) dβ ǫ ě ‚ (weights) β ǫ “ dt near each of the punctures; ‚ (interpolation) we have β ǫ “ ǫdt on r , R p ǫ qs ˆ S in the cylin-drical end near z ` , and β ǫ “ p ´ ǫ q dt on r , R p ´ ǫ qs ˆ S in thecylindrical end near z ` , for some smooth function R : p , q Ñ R ą .In other words, the family t β ǫ u interpolates between a 1-form whichvaries a lot near z ` and very little near z ` , and a 1-form which variesa lot near z ` and very little near z ` ; ‚ (neck stretching) we have R p ǫ q Ñ `8 as ǫ Ñ ǫ close to 0 we have β ǫ “ f ǫ p s q dt in the cylindrical end at the positive puncture z ` , with f ǫ ě f ǫ “ `8 , and f ǫ “ ǫ on r , R p ǫ qs , and similarly for ǫ close to 1 on the end at z ` . dt R p ǫ q ǫdt dt dtdt p ´ ǫ q dt R p ´ ǫ q dtdt ǫ P p , q Figure 20.
Interpolating family of 1-forms with varying weights.
ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 69
Let H : x W Ñ R be a concave smoothing localized near B W of a Hamil-tonian which is zero on W and linear of negative slope on r ,
8q ˆ B W .The Hamiltonian H further includes a small time-dependent perturba-tion localized near B W , so that all 1-periodic orbits are nondegenerate.Assume the absolute value of the slope is not equal to the period of aclosed Reeb orbit. Denote P p H q the set of 1-periodic orbits of H . Theelements of P p H q are contained in a compact set close to W . Remark 7.1.
The Hamiltonian H above has the standard shape usedin the definition of symplectic homology SH ˚ p W, B W q . However, theconstruction can accommodate more general Hamiltonians using meth-ods from [5, Lemmas 2.2 and 2.3].Let J “ p J zǫ q , z P Σ, ǫ P p , q be a generic family of compatiblealmost complex structures, independent of ǫ and s near the punctures,cylindrical and independent of ǫ and z in the symplectization r ,
8q ˆB W . For x , x , y P P p H q denote M p x , x ; y q : “ p ǫ, u q ˇˇ ǫ P p , q , u : Σ Ñ x W , p du ´ X H b β ǫ q , “ , lim s Ñ`8 z “p s,t qÑ z i ` u p z q “ x i p t q , i “ , , lim s Ñ´8 z “p s,t qÑ z ´ u p z q “ y p t q ( . In the symplectization r ,
8q ˆ B W we have H ď d p Hβ q ď
0, so that elements of the above moduli space are containedin a compact set. The dimension of the moduli space isdim M p x , x ; y q “ CZ p x q ` CZ p x q ´ CZ p y q ´ n ` . When it has dimension zero the moduli space M “ p x , x ; y q is com-pact. When it has dimension 1 the moduli space M “ p x , x ; y q admits a natural compactification into a manifold with boundary B M “ p x , x ; y q “ ž CZ p x q“ CZ p x q´ M p x ; x q ˆ M “ p x , x ; y q> ž CZ p x q“ CZ p x q´ M p x ; x q ˆ M “ p x , x ; y q> ž CZ p y q“ CZ p y q` M “ p x , x ; y q ˆ M p y ; y q> M ǫ “ p x , x ; y q > M ǫ “ p x , x ; y q . Here M ǫ “ p x , x ; y q and M ǫ “ p x , x ; y q denote the fibers of the firstprojection M “ p x , x ; y q Ñ p , q , p ǫ, u q ÞÑ ǫ near 1, respectivelynear 0. (By a standard gluing argument the projection is a trivialfibration with finite fiber near the endpoints of the interval p , q .) Consider the degree 1 operation σ w : F C ˚ p H qr n s b F C ˚ p H qr n s Ñ F C ˚ p H qr n s defined on generators by σ w p x b x q “ ÿ CZ p y q“ CZ p x q` CZ p x q´ n ` M “ p x , x ; y q y, where M “ p x , x ; y q denotes the count of elements in the 0-dimen-sional moduli space M “ p x , x ; y q with signs determined by a choiceof coherent orientations. Consider also the degree 0 operations σ iw : F C ˚ p H qr n s b F C ˚ p H qr n s Ñ F C ˚ p H qr n s , i “ , σ iw p x b x q “ ÿ CZ p y q“ CZ p x q` CZ p x q´ n M ǫ “ i p x , x ; y q y, where M ǫ “ i p x , x ; y q denotes the count of elements in the 0-dimen-sional moduli space M ǫ “ i p x , x ; y q with signs determined by a choiceof coherent orientations.The formula for B M “ p x , x ; y q translates into the algebraic relation(40) B F σ w ` σ w pB F b id ` id b B F q “ σ w ´ σ w . We now claim that(41) σ w | F C ă ˚ p H qr n sb F C ˚ p H qr n s “ , σ w | F C ˚ p H qr n sb F C ă ˚ p H qr n s “ . To prove the claim for σ w , note that this map can be expressed as a com-position µ ˝ p c b id q , where µ : F C ˚ p ǫH qr n s b F C ˚ p H qr n s Ñ F C ˚ p H qr n s is a pair-of-pants product, and c : F C ˚ p H qr n s Ñ F C ˚ p ǫH qr n s is a con-tinuation map. The action decreases along continuation maps, hence c p F C ă ˚ p H qr n sq Ă F C ă ˚ p ǫH qr n s . At the same time this last group van-ishes because ǫH has no nontrivial 1-periodic orbits of negative actionfor ǫ small enough. The argument for σ w is similar.It follows that σ w restricts to a degree 1 chain map(42) σ w : F C ă ˚ p H qr n s b F C ă ˚ p H qr n s Ñ F C ă ˚ p H qr n s . (This map lands in F C ă ˚ p H qr n s for action reasons.) Passing to thelimit we obtain a degree 1 product σ w : SH ă ˚ p W, B W qr n s b SH ă ˚ p W, B W qr n s Ñ SH ă ˚ p W, B W qr n s . Finally we apply a shift as in (13), namely σ w “ ´ σ w r´ , ´ ´ s , inorder to obtain a degree 0 product σ w : SH ă ˚ p W, B W qr n ´ s b Ñ SH ă ˚ p W, B W qr n ´ s . Explicitly σ w p ¯ x, ¯ x q “ ´p´ q | ¯ x | σ p x, x q , where x, x P SH ă ˚ p W, B W qr n s and ¯ x, ¯ x P SH ă ˚ p W, B W qr n ´ s denote their shifted images. We call ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 71 σ w the varying weights degree ´ n ` secondary product on SH ă ˚ p W, B W q .Equivalently, in view of the canonical isomorphism SH ă ˚ p W, B W q » SH ´˚ą p W q from [5], the above construction defines a degree n ´ secondary product on SH ˚ą p W q , denoted σ w . At the level of modulispaces this is described by exchanging the roles of the positive andnegative punctures, and reversing the sign of the Hamiltonian. Thusone considers curves with 2 negative punctures with varying weightstreated as cohomological inputs and 1 positive puncture treated as acohomological output. In this framework, the relevant Floer equationinvolves Hamiltonians on x W with positive slope on r ,
8q ˆ B W .In yet another equivalent formulation, the above construction defines a degree ´ n ` secondary coproduct on SH ą ˚ p W q , denoted λ w . Themoduli spaces are the same as for the secondary product on SH ˚ą p W q (2 negative punctures with varying weights and 1 positive puncture),except that the positive puncture is treated as a homological input andthe negative punctures are treated as homological outputs.7.2. Secondary products: varying weights and continuationmaps.
In view of Corollary 4.9 and Proposition 4.10, the negative ac-tion homology group SH ă ˚ p W, B W qr n ´ s – y SH ă ˚ pB W qr n ´ s carriestwo other products of degree 0 which coincide : ‚ the Poincar´e duality product σ P D , induced from the primaryproduct on Rabinowitz Floer homology via the Poincar´e dualityisomorphism from Theorem 1.1 restricted in negative action. ‚ the continuation product σ c “ σ constructed in § Lemma 7.2.
The continuation product σ c and the varying weightsproduct σ w coincide on SH ă ˚ p W, B W q .Proof. Going back to the definition of the unshifted varying weightsproduct σ w , we recall that the vanishing of the left boundary termin (41), i.e. σ w | F C ă ˚ p H qr n s b F C ˚ p H qr n s “ , was ensured by the fact that σ w could be expressed as a composition µ ˝p c b id q , where µ : F C ˚ p ǫH qr n sb F C ˚ p H qr n s Ñ F C ˚ p H qr n s is a pair-of-pants product, and c : F C ˚ p H qr n s Ñ F C ˚ p ǫH qr n s is a continuationmap. Since the action decreases along continuation maps, we have c p F C ă ˚ p H qr n sq Ă F C ă ˚ p ǫH qr n s , and the last group vanishes because ǫH has no nontrivial 1-periodic orbits of negative action for ǫ smallenough. Similarly, the boundary term σ w can be expressed as µ ˝p id b c q .In the case of the unshifted continuation product σ , the boundaryterms are expressed as µ ˝ p c b id q , respectively µ ˝ p id b c q , where c : F C ˚ p H qr n s Ñ F C ˚ p K ν qr n s is the continuation map towards aHamiltonian K ν which vanishes on W and has positive slope ν on r ,
8q ˆ B W . The continuation map c factors as F C ˚ p H qr n s c ÝÑ F C ˚ p ǫH qr n s Ñ F C ˚ p K ν qr n s and, when restricted to negative action,vanishes for all positive values of ν . As such, the 1-parameter familyof Floer problems with source a genus 0 Riemann surface with twopositive punctures and one negative puncture which defines σ can bechosen as follows: on a first interval we interpolate near the first posi-tive puncture from the continuation map c to the continuation map c .On a second interval we follow the 1-parameter family which defines σ w .And on a third interval we interpolate near the second positive punc-ture from the continuation map c to the continuation map c . Whenrestricted to negative action, the first and third parametrizing intervalsbring no contribution, so that σ c “ σ w for this choice of defining data.Finally, the continuation product σ c and the varying weights product σ w are defined by the same shift ´r´ , ´ ´ s from σ c and respectively σ w , so that they coincide as well. (cid:3) Appendix A. Multilinear shifts
We describe in this section our convention for shifts of multilinear maps.It is inspired by Lef`evre-Hasegawa’s discussion of shifts in the contextof A -algebras [12, § | ¨ | stands fordegree (of an element, resp. of a linear map).Given a chain complex p V, Bq , recall that V r k s is the chain complex inwhich we shift the degree down by k P Z and multiply the differentialby p´ q k , i.e. V r k s i “ V i ` k , Br k s “ p´ q k B . We denote s k : V Ñ V r k s the map induced by Id V . This is a chain isomorphism of degree ´ k ,and we denote its inverse, which is a chain isomorphism of degree k ,by ω k : V r k s Ñ V. Definition A.1.
Let V , . . . , V ℓ and V be chain complexes and let α : V b ¨ ¨ ¨ b V ℓ Ñ V be a multilinear map. Given k , . . . , k ℓ , k P Z we denote k “ p k , . . . , k ℓ ; k q and define the k -shift of α to be α r k s : V r k s b ¨ ¨ ¨ b V ℓ r k ℓ s Ñ V r k s ,α r k s “ s k αω k b ¨ ¨ ¨ b ω k ℓ . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 73
In an equivalent formulation, the shift α r k s is defined from the com-mutative diagram  i V i r k i s α r k s / / b i ω ki (cid:15) (cid:15) V r k s  i V i α / / V s k O O Yet equivalently, given elements x i P V i and x P V , denote x i “ s k i x i P V r k i s and x “ s k x P V r k s . We then have α r k sp ¯ x , . . . , ¯ x ℓ q “ p´ q | x |p k `¨¨¨` k ℓ q`¨¨¨`| x ℓ ´ | k ℓ α p x , . . . , x ℓ q . The degrees of α and α r k s are related by | α r k s| “ | α | ` k ` ¨ ¨ ¨ ` k ℓ ´ k, and we have “ B , α r k s ‰ “ p´ q k rB , α sr k s . In particular, if α is a chain map then so is α r k s . Example A.2.
Let t µ d A u , d ě be an A -algebra structure on A , with µ d A : A r´ s b d Ñ A r´ s of degree ´ and r µ, µ s “ . Denoting B “ µ A r s “ ´ µ A the shifted differential on A “ A r´ sr s , the shifted product µ “ µ A r ,
1; 1 s : A b A Ñ A is a chain map with respect to B and is defined on elements as µ p a, b q “ p´ q | a | µ A p a, b q . This is exactly our convention (4) . Moreover, formula (5) shows thatthe associator for µ is precisely given by f “ µ A r , ,
1; 1 s , i.e. µ p µ p a , a q , a q ´ µ p a , µ p a , a qq “ rB , f sp a , a , a q . Example A.3. If f : V Ñ V is a graded map, then f r k s i “ f i ` k . Inparticular f r k sr´ k s “ f (the shift is “involutive” on linear maps). Remark A.4 (Non-involutivity) . Given k “ p k , . . . , k ℓ ; k q , denote ´ k “ p´ k , . . . , ´ k ℓ ; ´ k q . Then α r k sr´ k s “ p´ q k p k `¨¨¨` k ℓ q`¨¨¨` k ℓ ´ k ℓ α. Therefore the shift on multilinear maps is only involutive up to sign.
Remark A.5 (Non-commutativity) . Given k “ p k , . . . , k ℓ ; k q and t “p t , . . . , t ℓ ; t q , define k ` t “ p k ` t , . . . , k ℓ ` t ℓ ; k ` ℓ q . Then p´ q t p k `¨¨¨` k ℓ q`¨¨¨` t ℓ ´ k ℓ α r k sr t s“ α r k ` t s “ p´ q k p t `¨¨¨` t ℓ q`¨¨¨` k ℓ ´ t ℓ α r t sr k s , and thus α r k sr t s differs in general from α r t sr k s , and from α r k ` t s , bya sign. (This is also the reason for non-involutivity.) Remark A.6 (Associativity) . Let us view the shift r k s as acting fromthe right on the space of multilinear maps L p V b ¨ ¨ ¨ b V ℓ , V q . Then pr k sr t sqr s s “ r k spr t sr s sq . Indeed, a straightforward computation shows that both terms are equalto p´ q s p t `¨¨¨` t ℓ q` ...s ℓ ´ t ℓ p´ q s p k `¨¨¨` k ℓ q` ...s ℓ ´ k ℓ ˆ p´ q t p k `¨¨¨` k ℓ q`¨¨¨` t ℓ ´ k ℓ r k ` t ` s s . Appendix B. A ` -structures It often comes as a surprise that signs in differential graded (dg) linearalgebra miraculously fit together using the Koszul sign rule. We givein this section our point of view on this matter. We deduce all signconventions from the single rule for the differential on a tensor product,and discuss a useful pictorial representation. As an application, wedefine an A ` -structure on a chain complex A and show how it inducesan A -triple p A _ , c, A q as in Example 2.14.Let R be a commutative ring. The notions of left R -module, right R -module and symmetric R -bimodule coincide, so that we use the sim-plified terminology “ R -module”. We denote generically the differentialof a dg R -module by B , and will further specify the notation only ifnecessary. We work in homological grading convention: the differential B has degree ´
1. The degree of elements or maps is denoted | ¨ | .B.1.
Fundamental conventions.
Let A , B , C , D be dg R -modules.The fundamental conventions of dg linear algebra are the following.
1. Differential on the tensor product.
Let A b B be the R -module whose degree k part is ‘ i ` j “ k A i b B j . One defines a differentialby the formula (43) Bp a b b q “ B a b b ` p´ q | a | a b B b. In view of the Koszul sign rule below, this can also be rewritten B A b B “ B A b Id B ` Id A bB B .
2. Differential on the
Hom module.
We denote Hom p A , B q the R -module whose degree r part consists of R -module maps f : A ˚ Ñ This formula has geometric roots. From a purely algebraic perspective, it is theessentially unique possible choice in order to achieve the relation B “
0. The otherpossibility would be Bp a b b q “ p´ q | b | B a b b ` a b B b , which is equivalent to (43)under the twist isomorphism A b B » ÝÑ B b A , a b b ÞÑ p´ q | a |¨| b | b b a . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 75 B ˚` r . The differential is defined to be B f “ B ˝ f ´ p´ q | f | f ˝ B . We also write B f “ rB , f s . A particular case ofthe previous construction is that of the dual R -module A _ “ Hom p A , R q . Here the base ring R is understood to be supported in degree 0. Asa consequence A _ is graded as A _˚ “ Hom p A ´˚ , R q (note the minussign!), and is endowed with the differential B f “ ´p´ q | f | f ˝ B .
3. The evaluation map is a chain map.
The evaluation map isev : Hom p A , B q b A Ñ B , f b a ÞÑ f p a q . We also write f p a q “ x f, a y . This has degree 0 and is a chain mapunder conventions 1 and 2. Lemma B.1.
Assuming any of the conventions 1, 2 or 3, the two otherconventions are equivalent. (For the implication ` ñ we assumethat B A b B has the form a b b ÞÑ ˘B a b b ˘ a b B b .) (cid:3) The evaluationpairing is ev : A _ b A Ñ R, f b a ÞÑ f p a q “ : x f, a y . This has degree 0 and is a chain map under conventions 1 and 2bis.
Lemma B.2.
Assuming convention 1, conventions 2bis and 3bis areequivalent. (cid:3)
4. The twist isomorphism is a chain map.
The twist isomorphism is the dg R -module isomorphism(44) τ : A b B » ÝÑ B b A , a b b ÞÑ p´ q | a |¨| b | b b a. Under convention 1, a chain isomorphism A b B » ÝÑ B b A of theform a b b ÞÑ ε p| a | , | b |q b b a with ε p| a | , | b |q P t˘ u is necessarily of theform (44) up to global sign change. As an example, while a b b ÞÑ b b a is an R -module isomorphism, it is not in general a chain map.
5. The Koszul sign rule.
There is a chain map T : Hom p A , B q b Hom p C , D q Ñ Hom p A b C , B b D q defined by the commutative diagram(45) Hom p A , B q b Hom p C , D q b A b C b τ b / / T b b (cid:15) (cid:15) Hom p A , B q b A b Hom p C , D q b C ev b ev (cid:15) (cid:15) Hom p A b C , B b D q b A b C ev / / B b D . Here τ is the twist isomorphism applied on the 2nd and 3rd factors ofthe tensor product. With a slight abuse of notation we denote T p f b g q by f b g . Then the previous commutative diagram amounts to thefamiliar Koszul sign rule x f b g, a b c y “ p´ q | g |¨| a | f p a q b g p c q . We thus see that the Koszul sign rule is a consequence of convention 4for the twist map, which in turn is a consequence of convention 1 forthe differential on the tensor product.The bottom line of the discussion is the following: ‚ conventions 4 and 5 are consequences of convention 1. ‚ assuming any one of the conventions 1, 2 or 3, the two other onesare equivalent.At this point it is instructive to quote Loday-Vallette [13, 1.5.3]: “[TheKoszul convention] permits us to avoid complicated signs in the formu-las provided one works with the maps (or functions) without evaluatingthem on the elements. When all the involved operations are of degree , the formulas in the nongraded case apply mutatis mutandis to thegraded case.” The point of the previous discussion was to deduce theconvention for the twist isomorphism and for the Koszul sign rule fromconvention 1, which is ultimately of geometric nature.B.2.
Consequences of the fundamental conventions.
From thispoint on we will be able to avoid all signs by working exclusively withfunctional equalities, or commutative diagrams.1. We have a canonical chain map T : B b C _ ÝÑ Hom p C , B q obtained by specializing (45) with A “ D “ R . This is equivalentlydefined by the commutative diagram B b C _ b C b ev / / T b (cid:15) (cid:15) B Hom p C , B q b C ev / / B . If C is free and finite dimensional over R in each degree the map T isa chain isomorphism. We have T p b b f qp c q “ b ¨ f p c q . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 77
There is of course also a canonical chain map r T : C _ b B ÝÑ Hom p C , B q obtained by suitably specializing (45). This is alternatively defined bythe commutative diagram C _ b B b C τ b / / r T b (cid:15) (cid:15) B b C _ b C b ev / / B Hom p C , B q b C ev / / B . Thus r T “ T ˝ τ and r T p f b b qp c q “ p´ q | b |¨| f | b ¨ f p c q . As expected,there is no sign involved when expressing the operation in terms of acommutative diagram, i.e. in functional terms. The sign appears onlywhen explicitly evaluating on elements.2. We have canonical adjunction isomorphisms F : Hom p A , Hom p B , C qq » ÐÑ Hom p A b B , C q : G. The map F acts by F p f qp a b b q “ f p a qp b q . Its inverse G “ F ´ actsby G p g qp a qp b q “ g p a b b q . The fact that the adjunction isomorphismsare chain maps ties together conventions 1 and 2 in the same manneras does convention 3.3. There is a canonical map A _ b B _ Ñ p A b B q _ realized by the composition F r T : A _ b B _ Ñ Hom p A , B _ q » Ñ p A b B q _ .Also, there is a canonical map B _ b A _ Ñ p A b B q _ obtained from the previous one by pre-composing with τ , and thusrealized by F T : B _ b A _ Ñ Hom p A , B _ q » Ñ p A b B q _ . These mapsare isomorphisms if either A or B is free and finite dimensional over R .4. For sequel use, it is useful to consider the coevaluation map ev _ : R Ñ p A _ b A q _ , dual to the evaluation map, and also the canonical map ι : A Ñ A __ defined by the commutative diagram A b A _ τ / / ι b (cid:15) (cid:15) A _ b A ev / / R A __ b A _ ev / / R. Explicitly, the map ι acts as a ÞÑ ` f ÞÑ p´ q | a |¨| f | x f, a y ˘ . (Formally wehave x a, f y “ p´ q | a |¨| f | x f, a y .) If A is free and finite dimensional over R then the map ι : A Ñ A __ is an isomorphism, the modules A _ and A __ are also free and finitedimensional over R , and ev _ can be interpreted as a chain map ev _ : R Ñ A b A _ . We then have p ev b qp b ev _ q “ Id A _ and p b ev qp ev _ b q “ Id A .B.3. The language of trees.
We depict operations involving multipleinputs and outputs in A and A _ by trees whose half-edges carry twodifferent labels: 1. input or output , signified by an ingoing or outgoingarrow, and 2. A or A _ . In our graphical representation, the inputedges labeled A or A _ are directed “upwards”, resp. “downwards”,and the output edges labeled A or A _ are directed “downwards”, resp.“upwards”. The composition of operations is represented by stackingtrees one on top of the other.For example, the graphical representation of the maps ev and ev _ inthe case where ι is an isomorphism is A _ A ev AA _ ev _ The relations p ev b qp b ev _ q “ Id A _ and p b ev qp ev _ b q “ Id A become graphically A _ A ev A _ ev _ “ A _ A _ Id A _ AA _ ev _ A ev “ AA Id A The graphical representation is more suggestive than the formulas.Using the evaluation map, an input (output) in A can be converted toan input (output) in A _ : add a tensor factor A for each new outputin A _ , and apply the evaluation map as dictated by the tree. For In this situation, given a basis e , . . . , e ℓ of A and denoting e ˚ , . . . , e ˚ ℓ the dualbasis of A _ , we have ev _ p q “ ř i e i b e ˚ i . This is the so-called Casimir element . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 79
Figure 21.
Definition of m L and m R example, suppose we are given a bilinear product µ on A (which is notassumed to be commutative or associative) depicted by the trivalenttree with one vertex: µ A AA
Here the labels 1 and 2 on the incoming edges express the order of thearguments for the product µ : A b A Ñ A . The same tree then definesleft and right multiplications m L : A b A _ Ñ A _ and m R : A _ b A Ñ A _ by converting inputs and outputs as in Figure 21: According to ourrules, the operation m L is defined by the commuting diagram A b A _ b A τ τ (cid:15) (cid:15) m L b / / A _ b A ev $ $ ❍❍❍❍❍❍❍❍❍❍ A _ b A b A b µ / / A _ b A ev / / R, where τ τ is the permutation moving the first tensor factor to thethird position. This writes out asev p m L b q “ ev p b µ q τ τ , which means x m L p a, f q , b y “ p´ q | a |¨p| f |`| b |q x f, µ p b, a qy in terms of ele-ments. In accordance with the discussion in § B.1, the defining formuladoes not involve signs when written in functional terms: the signsappear only when the formula is evaluated on elements. Using the no-tation x a, f y “ p´ q | a |¨| f | x f, a y from above, the last formula can also bewritten in equivalent form without signs as(46) x b, m L p a, f qy “ x µ p b, a q , f y . Similarly, the right multiplication readsev p m R b q “ ev p b µ q , which in terms of elements means(47) x m R p f, a q , b y “ x f, µ p a, b qy . Remark B.3.
Let us emphasize that in the preceding discussion aswell as in the sequel we use only the evaluation map and not its dual,so everything works in the infinite dimensional case. If A is free andfinite dimensional, then we can view the conversion of inputs/outputsin A to outputs/inputs in A _ as adding the pictures for ev resp. ev _ at the corresponding edge. Remark B.4.
One remarkable aspect about the language of trees isthat the definitions are forced upon us by TQFT-type relations: twocomposite expressions which give rise to the same tree by formal gluingare equal. In the situation at hand, this requirement leaves as anavailable choice the order on the input half-edges of µ , but this is fixedby the requirement that m L defines a left module structure, and m R defines a right module structure, in the case that the product µ isassociative (so in this case A _ becomes a bimodule over A ).B.4. A ` -structures. In this subsection we discuss in detail Exam-ple 2.14. Let us emphasize that the R -module A need not be free orof finite rank. (Of particular interest will be the case where A is freeover R , but possibly of infinite rank, see § C.)
Definition B.5.
Let p A , Bq be a dg R -module. An A ` -structure on p A , Bq consists of the following maps: ‚ the continuation quadratic vector c : R Ñ A b A , of degree ; ‚ the product µ : A b A Ñ A , of degree ; ‚ the secondary coproduct λ : A Ñ A b A , of degree ; ‚ the cubic vector B : R Ñ A b A b A , of degree . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 81
Figure 22.
The symmetry conditions on c and µ Figure 23.
The secondary coproduct λ The continuation quadratic vector c gives rise to the continuation map c : “ p ev b qp b c q “ p ev b qp b τ c q : A _ Ñ A . These maps are subject to the following conditions:(1) the continuation quadratic vector is symmetric, i.e. τ c “ c ;(2) the product µ commutes with B and is associative up to chainhomotopy;(3) the image of c lies in the center of µ , i.e. µ p c b q “ µ p b c q τ ;(4) rB , λ s “ p µ b qp b c q ´ p b µ qp c b q .(5) B B “ p b λ q c ` p λ b q c ´ τ p λ b q c . See Figures 22, 23 and 24 for pictorial representations of conditions (1),(3), (4) and (5).
Figure 24.
The cubic vector B Remark B.6.
Condition (1) is equivalent to ιc “ c _ in terms of thecanonical map ι : A Ñ A __ . If ι is an isomorphism, then c can berecovered from c by the formula c “ p b c q ev _ and condition (1) isequivalent to c “ c _ .The main result of this appendix is Proposition B.7. An A ` -structure on p A , Bq canonically gives rise toan A -triple p A _ , c, A q .Proof. The maps m L , m R , τ L , τ R , σ, β defining the A -triple are givenby explicit formulas in terms of µ , λ , B .
1. The degree maps m L and m R which define an A -bimodulestructure on A _ are defined as in § B.3. They are determined by µ ,
2. The degree map σ : A _ b A _ Ñ A _ is determined by λ and isdefined by Figure 25, which amounts to the equalityev p σ b q “ p ev b ev q τ p b b λ q ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 83
Figure 25.
The map σ between maps A _ b A _ b A Ñ R . In terms of elements this means(48) x σ p f, g q , a y “ p´ q | f |`| g | x f b g, λ p a qy . Said differently, σ equals the composition A _ b A _ Ñ p A b A q _ λ _ ÝÑ A _ , see § B.2.3 for the first canonical map. Thus “ σ is the dual of λ ”.The relation rB , σ s “ m R p b c q ´ m L p c b q is equivalent to rB , λ s “ p µ b q τ p b c q ´ p b µ q τ p c b q . In view of µ p c b q “ µτ p c b q , it follows that the latter is equivalent tothe defining relation for λ , i.e. rB , λ s “ p µ b qp b c q ´ p b µ qp c b q .
3. The degree map τ L : A _ b A Ñ A is defined by Figure 26,which amounts to the equality τ L “ ´p ev b qp b λ q . This means(49) τ L p f, a q “ ´p´ q | f | x f b , λ p a qy in terms of elements.At this point the reason why we have imposed that the image of c liesin the center of µ becomes clear: at the left endpoint of the interval ofparametrization of ´ τ L we see the operation µ p c b q , whereas at theleft endpoint of the interval of parametrization of p ev b qp b λ q we seethe operation µτ p c b q . Should we wish to express the map τ L in termsof λ , it is necessary to assume µ p c b q “ µτ p c b q . A similar discussionholds for the right endpoint of the interval of parametrization of τ L ,where the same condition becomes necessary. Figure 26.
The map τ L Figure 27.
The map τ R
4. The degree map τ R : A b A _ Ñ A is defined by Figure 27,which amounts to the equality τ R “ p b ev q τ p λ b q . This means(50) τ R p a, f q “ x λ p a q , b f y in terms of elements.A similar discussion as the one for τ L holds for τ R , and the condition µ p c b q “ µτ p c b q is necessary in order to express τ R in terms of λ . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 85
Figure 28.
The map β
5. The degree map β : A _ b A _ Ñ A is defined by Figure 28,which amounts to β “ p ev b ev b q τ τ p b b B q . This means β p f, g q “ x f b g b , B y in terms of elements.Comparison of Figures 28 and 24 shows that the boundary relation for β is equivalent to the boundaary relation for B . Note that on the sideof the triangle which is labeled by cσ there is a discrepancy, which isabsorbed again by the condition µ p c b q “ µτ p c b q .This concludes the proof of Propostion B.7. (cid:3) Remark B.8.
The terminology A ` -structure was chosen because sucha structure comprises a product µ , plus additional operations needed toinduce a product on the cone. Note that the operations are parametrizedby simplices of dimension 0 (the product µ with 1 output), of dimen-sion 1 (the secondary coproduct λ with 2 outputs), and of dimension2 (the cubic vector B with 3 outputs). The construction of the full A structure on the cone makes use of the full enrichment by simplices of arbitrary dimension. More generally, one can consider (noncom-pact) TQFT-type structures with operations parametrized by topo-logical types of 2-dimensional surfaces with boundary and enriched insimplices: whenever a given surface has k outputs, we attach to it a p k ´ q -dimensional simplex. The paper [7] is relevant for this line ofthought; we will take up this discussion elsewhere. Appendix C. Extended loop homology product via Morsetheory
We give in this section a direct description of the product on extendedloop homology in terms of Morse theory via cones. This is based onthe Morse theoretic description of the loop cohomology product on H ˚ p Λ , M q and of its extension to reduced loop cohomology H ˚ Λ dis-cussed in [3]. We assume in this section that the manifold M is ori-ented and spin. The construction goes through verbatim with suitabletwisted coefficients in the general case, as explained in [3, AppendixA].We adopt here the setup of [3], see also [1]. Consider a smooth La-grangian L : S ˆ T M Ñ R which outside a compact set has the form L p t, q, v q “ | v | ´ V p t, q q for a smooth potential V : S ˆ M Ñ R .It induces an action functional S L : Λ Ñ R , q ÞÑ ż L p t, q, q q dt, which we can assume to be a Morse function whose negative flow withrespect to the W , -gradient ∇ S L is Morse–Smale. We denote M C ˚ , respectively M C ˚ , the Morse chain complex, resp. the Morse cochain complex of S L ,with R -coefficients, graded by the Morse index. Their homologies M H ˚ and M H ˚ are isomorphic to the singular (co)homology groups H ˚ Λ, resp. H ˚ Λ. We will assume in addition that near the zero sec-tion L p t, q, v q “ | v | ´ V p q q for a time-independent Morse function V : M Ñ R such that all nonconstant critical points of S L have actionlarger than ´ min V . Then the constant critical points define a subcom-plex M C “ ˚ of M C ˚ which agrees with the Morse cochain complex of V on M , with degrees of q P Crit p V q related by ind S L p q q “ n ´ ind V p q q .Similarly, the constant critical points define a quotient complex M C ˚“ of M C ˚ which agrees with the Morse chain complex of V on M .We assume that L | M has a unique minimum q m . Then Rq m is a quotientcomplex of M C ˚ , and Rq m is a subcomplex of M C ˚ . Denote by χ “ χ p M q the Euler characteristic of M . The extended loop homology Morsecomplex is } M C ˚ “ Cone p c q , ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 87 where c : M C ´˚ Ñ M C ˚ is the map M C ´˚ c / / (cid:15) (cid:15) M C ˚ Rq m ¨ χ / / Rq m O O Our goal in this section is to describe the product structure on extendedloop homology ~ M H ˚ . To this end, we describe the structure of an A -triple on p M ˚ , c, A ˚ q “ p M C ´˚´ n , c r n s , M C ˚` n q . We are in the precise context of § B.4, see also Example 2.14. There wesaw that, in the case M ˚ “ A _˚ , one can construct the structure of an A -triple by specifying an A ` -structure on A , which amounts to thedata of chain level operations as follows: ‚ the continuation quadratic vector c : R Ñ M C b ˚` n , of degree 0, ‚ the loop product µ : M C ˚` n b M C ˚` n Ñ M C ˚` n , of degree 0, ‚ the secondary loop coproduct λ : M C ˚` n Ñ M C b ˚` n , of degree 1, ‚ the cubic vector B : R Ñ M C b ˚` n , of degree 2.These have to satisfy specific compatibility conditions. We define c “ χq m b q m , so that p ev b qp b c q “ c and τ c “ c . We will also see below thatit is possible to take B “ , so that we are left to specify the operations µ and λ . These are theMorse theoretic descriptions of the loop product and loop coproductdiscussed in [3], which we now recall. The loop product µ . Given paths α, β : r , s Ñ M such that α p q “ β p q , their concatenation α β is defined by α β p t q “ " α p t q , ď t ď { ,β p t ´ q , { ď t ď . For a, b, c P Crit p S L q set M µ p a, b ; c q : “ p α, β, γ q P W ´ p a q ˆ W ´ p b q ˆ W ` p c q | γ “ α β ( , which is a transversely cut out manifold of dimensiondim M µ p a, b ; c q “ ind p a q ` ind p b q ´ ind p c q ´ n. If its dimension equals zero this manifold is compact and defines a map µ : M C ˚` n b M C ˚` n Ñ M C ˚` n , a b b ÞÑ ÿ c M µ p a, b ; c q c. If the dimension equals 1 it can be compactified to a compact 1-dimensional manifold with boundary B M µ p a, b ; c q “ ž ind p a q“ ind p a q´ M p a ; a q ˆ M µ p a , b ; c q> ž ind p b q“ ind p b q´ M p b ; b q ˆ M µ p a, b ; c q> ž ind p c q“ ind p c q` M µ p a, b ; c q ˆ M p c ; c q . corresponding to broken gradient lines. So we have(51) rB , µ s “ . The secondary loop coproduct λ . We fix a small vector field v on M with nondegenerate zeroes such that the only periodic orbits of v with period ď f t : M – ÝÑ M the flow of v . It follows that the only fixed points of f “ f are thezeroes of v , and they are nondegenerate. For each q P M we denotethe induced path from q to f p q q by π q : r , s Ñ M, π q p t q : “ f t p q q and the inverse path by π ´ q : r , s Ñ M, π ´ q p t q : “ f ´ t p q q . For a path α : r , s Ñ M and s P r , s we define the restrictions α | r ,s s , α | r s, s : r , s Ñ M by(52) α | r ,s s p t q : “ α p st q , α | r s, s p t q : “ α ` s ` p ´ s q t ˘ . For a, b, c P Crit p S L q we set M λ p a ; b, c q : “ p s, α, β, γ q P r , s ˆ W ´ p a q ˆ W ` p b q ˆ W ` p c q | β “ α s , γ “ α s ( We have sign det p T p f ´ Id q “ ind v p p q , where ind v p p q is the index of p as a zero of v . The map f ˆ Id : M Ñ M ˆ M , q ÞÑ p f p q q , q q is transverse to the diagonal ∆ Ă M ˆ M and p f ˆ Id q ´ p ∆ q “t q P M | f p q q “ q u “ Fix p f q . Since for q P Fix p f q the map T q M Ñ T q M ˆ T q M , w ÞÑ ` p T q f ´ Id q w, ˘ fills up the complement to T p q,q q ∆, the induced orientationon Fix p f q “ p f ˆ Id q ´ p ∆ q endows q with the sign ind v p q q . ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 89 with α s p t q : “ α | r ,s s π ´ α p q “ α p st q t ď { ,f ´ t ` α p q ˘ t ě { ,α s p t q : “ π α p q α | r s, s “ f t ` α p q ˘ t ď { ,α ` s ´ ` p ´ s q t ˘ t ě { . Note that the matching conditions imply α p s q “ f ˝ α p q . Since f ˆ Id : M Ñ M ˆ M is transverse to the diagonal, this is a codi-mension n condition and M λ p a ; b, c q is a transversely cut out manifoldof dimensiondim M λ p a ; b, c q “ ind p a q ´ ind p b q ´ ind p c q ` ´ n. If its dimension equals zero this manifold is compact and defines adegree 1 map λ : M C ˚` n Ñ M C ˚` n b M C ˚` n , a ÞÑ ÿ b,c M λ p a ; b, c q b b c. If the dimension equals 1 it can be compactified to a compact 1-dimensional manifold with boundary B M λ p a ; b, c q “ ž ind p a q“ ind p a q´ M p a ; a q ˆ M λ p a ; b, c q> ž ind p b q“ ind p b q` M λ p a ; b , c q ˆ M p b ; b q> ž ind p c q“ ind p c q` M λ p a ; b, c q ˆ M p c ; c q> M λ,s “ p a ; b, c q > M λ,s “ p a ; b, c q . Here the first three terms correspond to broken gradient lines and thelast two terms to the intersection of M λ p a ; b, c q with the sets t s “ u and t s “ u , respectively. So we have(53) rB , λ s “ λ ´ λ , where the degree 0 maps λ i for i “ , λ i : M C ˚` n Ñ M C ˚` n b M C ˚` n , a ÞÑ ÿ b,c M λ,s “ i p a ; b, c q b b c. Let us look more closely at the map λ . For s “ M λ p a ; b, c q imply that α p q “ q is a fixed point of f and γ “ q is the constant loop at q . Assuming that L | M has a uniqueminimum q m and the fixed points of f are in general position withrespect to the stable and unstable manifolds of L | M , the condition q P W ` p c q is only satisfied for c “ q m . Thus M λ,s “ p a ; b, c q is empty if c ‰ q m and M λ,s “ p a ; b, q m q – ž q P Fix p f q p α, β q P W ´ p a q ˆ W ` p b q | β “ α q ( . Choosing all fixed points of f closely together, we can achieve thatthe terms on the right hand side corresponding to different q P Fix p f q are in canonical bijection to each other. By the previous discussionthe terms corresponding to a fixed point q come with the sign ind v p q q .Since ř q ind v p q q “ χ we obtain λ p a q “ χµ p a b q m q b q m , or equivalently λ “ p µ b qp b c q . Similarly we have λ p a q “ χq m b µ p q m b a q , or equivalently λ “ p b µ qp c b q . In conclusion we obtain rB , λ s “ p µ b qp b c q ´ p b µ qp c b q . The cubic vector B . The cubic vector B can be chosen to be zero.To see this we inspect the boundary rB , B s “ p b λ q c `p λ b q c ´p b λ q c and see that each of the three terms on the right hand sideof this equation must necessarily vanish. Indeed, they are all definedusing moduli spaces of the form M λ p q m ; b, c q , and these are emptybecause we chose our data such that q m R Fix p f q . As a conclusion,regardless of how one defines the cubic vector B , that choice will behomotopic to B “
0. We therefore choose B “ Extended loop homology.
The loop product µ defines maps m L and m R , and the secondary loop product λ defines maps σ , τ L and τ R ,see § B.4 and Example 2.14. All these operations involve only the twokinds of moduli spaces described above, i.e. M µ p a, b ; c q and M λ p a ; b, c q .Explicitly, with A ˚ “ M C ˚` n and A _˚ “ M C ´˚´ n : m L : A b A _ Ñ A _ , x m L p a, f q , b y “ p´ q | a |¨p| f |`| b |q x f, µ p b, a qy ,m R : A _ b A Ñ A _ , x m R p f, a q , b y “ x f, µ p a, b qy ,σ : A _ b A _ Ñ A _ , x σ p f, g q , a y “ p´ q | f |`| g | x f b g, λ p a qy ,τ L : A _ b A Ñ A , τ L p f, a q “ ´p´ q | f | x f b , λ p a qy ,τ R : A b A _ Ñ A , τ R p a, f q “ x λ p a q , b f y . The product structure on the shifted extended loop homology q H ˚` n Λ “ H ˚ p Cone p c qq is defined by the formula given in Proposition 2.9, namely ULTIPLICATIVE STRUCTURES ON CONES AND DUALITY 91 (with β “ m : Cone p c q b Cone p c q Ñ Cone p c q ,m ` p a, ¯ f q , p b, ¯ g q ˘ “ ` µ p a, b q ` p´ q | a | τ R p a, g q ` τ L p f, b q , p´ q | a | m L p a, g q ` m R p f, b q ´ p´ q | ¯ f | σ p f, g q ˘ . We recover in particular the formulas from Theorem 1.2. As a corollary,the construction of extended loop homology given in this section isequivalent to the symplectic one from [4] for n ě
3, and conjecturallyalso for n “ Reduced homology and canonical splitting.
We define the reduced Morse complex as the quotient complex, resp. subcomplex
M C ˚ : “ M C ˚ L Rχq m , M C ˚ : “ ker p M C ˚ Ñ Rq m q , with reduced Morse (co)homology M H ˚ , resp. M H ˚ . We then have acanonical splitting ~ M H ˚ “ M H ´˚` ‘ M H ˚ which is compatible with the ring structure by Proposition 5.5 : themap induced by inclusion j ˚ : H ˚ p ker c q » ÝÑ ker c ˚ is an isomorphism,and the map β defined by the cubic vector B trivially satisfies im β Ă im c because it is zero. We recover in this way the canonical splittingfrom [4]. References [1] A. Abbondandolo and M. Schwarz. Floer homology of cotangent bundles andthe loop product.
Geom. Topol. , 14(3):1569–1722, 2010.[2] A. Abbondandolo and M. Schwarz. On product structures in Floer homologyof cotangent bundles. In
Global differential geometry , volume 17 of
SpringerProc. Math. , pages 491–521. Springer, Heidelberg, 2012.[3] K. Cieliebak, N. Hingston, and A. Oancea. Loop coproduct in Morse and Floerhomology. arXiv preprint, 2020.[4] K. Cieliebak, N. Hingston, and A. Oancea. Poincar´e duality for loop spaces.arXiv preprint, 2020.[5] K. Cieliebak and A. Oancea. Symplectic homology and the Eilenberg–Steenrodaxioms.
Algebr. Geom. Topol. , 18(4):1953–2130, 2018.[6] T. Ekholm and Y. Lekili. Duality between Lagrangian and Legendrian invari-ants. arXiv:1701.01284, 2017.[7] T. Ekholm and A. Oancea. Symplectic and contact differential graded algebras.
Geom. Topol. , 21(4):2161–2230, 2017.[8] A. Fauck.
Rabinowitz-Floer homology on Brieskorn manifolds . PhD thesis,Humboldt-Universit¨at zu Berlin, Apr 2016.[9] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono.
Lagrangian intersection Floertheory: anomaly and obstruction. Part I , volume 46 of
AMS/IP Studies inAdvanced Mathematics . American Mathematical Society, Providence, RI; In-ternational Press, Somerville, MA, 2009.[10] J. Herzog and Y. Takayama. Resolutions by mapping cones.
Homology Homo-topy Appl. , 4(2, part 2):277–294, 2002. The Roos Festschrift volume, 2. [11] M. Kontsevich and Y. Soibelman. Homological mirror symmetry and torusfibrations. In
Symplectic geometry and mirror symmetry (Seoul, 2000) , pages203–263. World Sci. Publ., River Edge, NJ, 2001.[12] K. Lef`evre-Hasegawa.
Sur les A -cat´egories . Thesis, Universit´e Paris-Diderot- Paris VII, Nov. 2003.[13] J.-L. Loday and B. Vallette. Algebraic operads , volume 346 of
Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of MathematicalSciences] . Springer, Heidelberg, 2012.[14] M. Markl. Transferring A (strongly homotopy associative) structures. Rend.Circ. Mat. Palermo (2) Suppl. , (79):139–151, 2006.[15] M. Nagata.
Local rings . Interscience Tracts in Pure and Applied Mathematics,No. 13. Interscience Publishers a division of John Wiley & Sons New York-London, 1962.[16] A. Polishchuk. Field guide to A sign conventions. https://pages.uoregon.edu/apolish/ainf-signs.pdf .[17] M. Rivera and Z. Wang. Singular Hochschild cohomology and algebraic stringoperations. J. Noncommut. Geom. , 13(1):297–361, 2019.[18] M. Schwarz. On the action spectrum for closed symplectically aspherical man-ifolds.
Pacific J. Math. , 193(2):419–461, 2000.[19] P. Seidel. A -subalgebras and natural transformations. Homology HomotopyAppl. , 10(2):83–114, 2008.[20] P. Seidel.
Fukaya categories and Picard-Lefschetz theory . Zurich Lectures inAdvanced Mathematics. European Mathematical Society (EMS), Z¨urich, 2008.[21] J. D. Stasheff. Homotopy associativity of H -spaces. II. Trans. Amer. Math.Soc. , 108:293–312, 1963.[22] P. Uebele. Symplectic homology of some Brieskorn manifolds.
Math. Z. , 283(1-2):243–274, 2016.[23] I. Ustilovsky. Infinitely many contact structures on S m ` . Internat. Math.Res. Notices , (14):781–791, 1999.[24] O. van Koert. Contact homology of Brieskorn manifolds.
Forum Math. ,20(2):317–339, 2008.[25] S. Venkatesh. Rabinowitz Floer homology and mirror symmetry.
J. Topol. ,11(1):144–179, 2018.
Universit¨at AugsburgUniversit¨atsstrasse 14, D-86159 Augsburg, Germany
E-mail address : [email protected] Sorbonne Universit´e, Universit´e Paris Diderot, CNRSInstitut de Math´ematiques de Jussieu-Paris Rive Gauche, IMJ-PRGParis, France
E-mail address ::