Loop coproduct in Morse and Floer homology
aa r X i v : . [ m a t h . S G ] A ug LOOP COPRODUCT IN MORSE AND FLOERHOMOLOGY
KAI CIELIEBAK, NANCY HINGSTON, AND ALEXANDRU OANCEA
Abstract.
By a well-known theorem first proved by Viterbo, theFloer homology of the cotangent bundle of a closed manifold is iso-morphic to the homology of its loop space. We prove that, whenrestricted to positive
Floer homology resp. loop space homologyrelative to the constant loops, this isomorphism intertwines vari-ous constructions of secondary pair-of-pants coproducts with theloop homology coproduct. The proof uses compactified modulispaces of punctured annuli. We extend this result to reduced
Floerresp. loop homology (essentially homology relative to a point), andwe show that on reduced loop homology the loop product and co-product satisfy Sullivan’s relation. Along the way, we show that theAbbondandolo-Schwarz quasi-isomorphism going from the Floercomplex of quadratic Hamiltonians to the Morse complex of theenergy functional can be turned into a filtered chain isomorphismby using linear Hamiltonians and the square root of the energyfunctional.
Contents
1. Introduction 22. Loop coproduct 52.1. Topological description of the homology coproduct 62.2. Homology coproduct on the loop space of S Date : September 1, 2020. H ˚ Λ 71A.9. Isomorphism between symplectic homology and loop homology 72References 74 Introduction
For a closed manifold M there are canonical isomorphisms(1) H ˚ p Λ , Λ ; η q – F H ą ˚ p T ˚ M q – SH ą ˚ p D ˚ M q – SH ´˚ă p S ˚ M q . Here we use coefficients in any commutative ring R , twisted in the firstgroup by a suitable local system η which restricts to the orientation lo-cal system on the space Λ Ă Λ of constant loops (see Appendix A). Thegroups in the above chain of isomorphisms are as follows: H ˚ p Λ , Λ q denotes the homology of the free loop space Λ “ C p S , M q relative toΛ ; F H ą ˚ p T ˚ M q the positive action part of the Floer homology of a fi-brewise quadratic Hamiltonian on the cotangent bundle; SH ą ˚ p D ˚ M q the positive symplectic homology of the unit cotangent bundle D ˚ M ;and SH ´˚ă p S ˚ M q the negative symplectic cohomology of the trivialLiouville cobordism W “ r , s ˆ S ˚ M over the unit cotangent bun-dle S ˚ M . The first isomorphism is the result of work of many peoplestarting with Viterbo (see [35, 1, 5, 3, 31, 30, 14, 26, 6]); the secondone is obvious; and the third one is a restriction of the Poincar´e dualityisomorphism from [17]. OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 3
Restricting to field coefficients, all the groups in (1) carry natural co-products of degree 1 ´ n : ‚ the loop homology coproduct (in the sequel simply called loop coprod-uct ) λ on H ˚ p Λ , Λ ; η q defined by Sullivan [32] and further studiedby Goresky and the second author in [22], see also [24]; ‚ the (secondary) pair-or-pants coproduct λ F on F H ą ˚ p T ˚ M q definedby Abbondandolo and Schwarz [4]; ‚ the varying weights coproduct λ w on SH ą ˚ p D ˚ M q first describedby Seidel and further explored in [21]; ‚ the continuation coproduct λ cont on SH ą ˚ p D ˚ M q described in [16]; ‚ the Poincar´e duality coproduct σ _ P D on SH ´˚ă p S ˚ M q dual to thepair-of-pants product on SH ă ´˚ p S ˚ M q , described in [13].The first goal of this paper is to prove that all these coproducts areequivalent under the isomorphisms in (1). Remark 1.1.
There is a formal algebraic reason why we need to re-strict to field coefficients when speaking about homology coproducts.Given a chain complex C “ C ˚ and a chain map C Ñ C b C , we ob-tain a map H ˚ p C q Ñ H ˚ p C b C q . However, the latter factors through H ˚ p C q b H ˚ p C q only if the K¨unneth isomorphism H ˚ p C q b H ˚ p C q » Ñ H ˚ p C b C q holds, which is the case with field coefficients. All our co-products are defined at chain level with arbitrary coefficients, and wewould not need to restrict to field coefficients if we carried the discus-sion at chain level.More generally, let us define reduced loop homology H ˚ p Λ; η q “ H ˚ p C ˚ p Λ; η q{ χC ˚ p pt qq , where pt ã Ñ M is the inclusion of a basepoint viewed as a constantloop and χ equals the Euler characteristic of M . Remark 1.2.
A straightforward calculation shows that H ˚ p Λ; η q “ H ˚ p Λ; η q{ χ im p H ˚ p pt q Ñ H ˚ p Λ; η qq whenever the map χH p pt q Ñ H p Λ; η q is injective, and in particu-lar if one of the following conditions holds: (i) M is orientable; (ii) χ “
0; (iii) the coefficient ring is 2-torsion. (Note that, in these sit-uations, H ˚ p Λ; η q differs from H ˚ p Λ; η q only in degree zero. For M non-orientable and more general coefficients they may also differ indegree 1.)It was observed in [13, Corollary 1.5] that the loop coproduct has acanonical extension to H ˚ p Λ; η q under any of the above conditions.We prove in this paper that such a canonical extension always exists,whether or not these conditions hold. KAI CIELIEBAK, NANCY HINGSTON, AND ALEXANDRU OANCEA
It is proved in [13], recovering a result of Tamanoi [34], that the loopproduct descends to H ˚ p Λ; η q{ χ im p H ˚ p pt q Ñ H ˚ p Λ; η qq . In particu-lar, the loop product descends to H ˚ p Λ; η q under any of the aboveconditions. We conjecture that the loop product always descends to H ˚ p Λ; η q .Similarly, all the groups in (1) can be enlarged to suitable reducedhomologies with a chain of isomorphisms(2) H ˚ p Λ; η q – F H ˚ p T ˚ M q – SH ˚ p D ˚ M q – SH ´˚ď p S ˚ M q . Now we can formulate our main result.
Theorem 1.3.
All the coproducts on the groups in (1) have canonicalextensions to the corresponding reduced homologies, and the extendedcoproducts are equivalent under the isomorphisms in (2) . In [24] the authors have constructed (for M orientable) an extension“by zero” of the coproduct λ to the whole of H ˚ p Λ q . That extensiondiffers in general from the one in Theorem 1.3, even in the case where M has Euler characteristic zero and thus H ˚ p Λ q “ H ˚ p Λ q . Indeed,while the extension in [24] has no outputs involving constant loops,we show in § S .Another difference between the two extensions concerns the followingrelation between the loop product µ “ ‚ and the homology coproduct λ on H ˚ p Λ , Λ q conjectured for M orientable by Sullivan [32]:(3) λµ “ p b µ qp λ b q ` p µ b qp b λ q . This relation does not hold for the extension of λ to H ˚ p Λ q in [24]. Bycontrast, we prove in § M orientable or not, and underthe above conditions) Sullivan’s relation holds for the extension of λ to H ˚ p Λ; η q in Theorem 1.3. Structure of the paper.
The paper contains four sections and oneAppendix. These are more or less self-contained, although there areobvious interconnections.In § § § § S and seethat it has contributions from the constant loops. This shows that itdiffers from the extended coproduct in [24]. Finally, we prove in § § OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 5 reduced homologies. Our main contribution is the following: we showthat the Abbondandolo-Schwarz isomorphismΨ : SH ˚ p D ˚ M q » Ñ H ˚ p Λ; η q , which was originally constructed using asymptotically quadratic Hamil-tonians and as such did not preserve the natural filtrations (at thesource by the non-Hamiltonian action, and at the target by the squareroot of the energy), can be made to preserve these filtrations whenimplemented for the linear Hamiltonians used in the definition of sym-plectic homology. As such, Ψ becomes an isomorphism at chain level.This uses a length vs. action estimate inspired by [14].In § § § § T ˚ M and loop space ho-mology of M [26, 6, 5]. On the other hand, they can be useful inapplications [7]. Acknowledgements.
The second author is grateful for support overthe years from the Institute for Advanced Study, and in particularduring the academic year 2019-20. The third author was partiallyfunded by the Agence Nationale de la Recherche, France under thegrants MICROLOCAL ANR-15-CE40-0007 and ENUMGEOM ANR-18-CE40-0009. 2.
Loop coproduct
In this section we present direct topological and Morse theoretic defini-tions of the homology coproduct on H ˚ Λ, as well as a Morse theoreticproof of Sullivan’s relation. Moreover, we compute the coproduct forthe loop space of S where it has nontrivial contributions from the con-stant loops. For simplicity, we assume throughout this section that M is oriented and we use untwisted coefficients in a commutative ring R ; KAI CIELIEBAK, NANCY HINGSTON, AND ALEXANDRU OANCEA the necessary adjustments in the nonorientable case and with twistedcoefficients are explained in Appendix A. We denote S : “ R { Z and Λ : “ W , p S , M q . Topological description of the homology coproduct.
In thissubsection we define the homology coproduct λ on reduced singularhomology. It is induced by a densely defined operationΛ : C ˚ p Λ q Ñ C ˚` ´ n p Λ ˆ Λ q on singular chains defined as follows. We fix a small vector field v on M with nondegenerate zeroes such that the only periodic orbits of v withperiod ď v gradient-like near its critical points; then the periods ofnonconstant periodic orbits are uniformly bounded from below by aconstant c ą
0, so v { c has the desired property.) Denote by f t : M – ÝÑ M, t P R the flow of v , i.e. the solution of the ordinary differential equation ddt f t “ v ˝ f t . It follows that the only fixed points of f “ f are thezeroes of v , each zero p is nondegenerate as a fixed point, andsign 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 ` f p 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 orientation on Fix p f q “p f ˆ id q ´ p ∆ q endows q with the sign ind v p q q . Remark 2.1.
Alternatively, we could use the exponential map of someRiemannian metric to define a map M Ñ M by q ÞÑ exp q tv p q q . Al-though this map differs from f t above, for v sufficiently small it sharesits preceding properties and could be used in place of f t .For each q P M we denote the 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 λ P r , s we define the restrictions α | r ,λ s , α | r λ, s : r , s Ñ M by(4) α | r ,λ s p t q : “ α p λt q , α | r λ, s p t q : “ α ` λ ` p ´ λ q t ˘ . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 7
For paths α, β : r , s Ñ M with α p q “ β p q we define their concate-nation α β : r , s Ñ M by α β p t q : “ α p t q t ď { ,β p t ´ q t ě { . As in [11] we consider chains a : K a Ñ Λ, b : K b Ñ Λ defined oncompact oriented manifolds with corners, and define their product by a ˆ b : K a ˆ K b Ñ Λ ˆ Λ , p x, y q ÞÑ ` a p x q , b p y q ˘ . Consider now a chain a : K a Ñ Λ such that the evaluation mapev a : K a ˆ r , s Ñ M ˆ M, p x, s q ÞÑ ´ f ` a p x qp q ˘ , a p x qp s q ¯ is transverse to the diagonal ∆ Ă M ˆ M . Then K λ p a q : “ ev ´ a p ∆ q “ tp x, s q P K a ˆ r , s | a p x qp s q “ f ` a p x qp q ˘ u is a compact manifold with corners and we define λ p a q : K λ p a q Ñ Λ ˆ Λby λ p a qp x, s q : “ ´ a p x q| r ,s s π ´ a p x qp q , π a p x qp q a p x q| r s, s ¯ . See Figure 1 where α “ a p x q . At s “ s “ K λ p a q becomes a p x qp q “ q P Fix p f q , and denoting the constant loopat q by the same letter we find λ p a qp x, q “ ` a p x q q, q ˘ , λ p a qp x, q “ ` q, q a p x q ˘ . It follows that B λ p a q ´ λ pB a q “ ÿ q P Fix p f q ind v p q q ´ p a ‚ q q ˆ q ´ q ˆ p q ‚ a q ¯ , where q is viewed as a 0-chain and the loop products with the constantloop q are given by a ‚ q : K a ‚ q “ t x P K a | a p x qp q “ q u Ñ Λ , x ÞÑ a p x q q,q ‚ a : K q ‚ a “ t x P K a | q “ a p x qp qu Ñ Λ , x ÞÑ q a p x q . Here the signs ind v p q q arise from the discussion before Remark 2.1,noting that the restriction of ev a to s “ s “ K a Ñ M , x ÞÑ a p x qp q and the map M Ñ M ˆ M , q ÞÑ ` f p q q , q ˘ .Let us now fix a basepoint q P M and consider a such that the mapev a, : K a Ñ M, x ÞÑ a p x qp q is transverse to q . We choose all zeroes of v (i.e. fixed points of f )so close to q that ev a, is transverse to each q P Fix p f q . Then after KAI CIELIEBAK, NANCY HINGSTON, AND ALEXANDRU OANCEA identifying the domains K q ‚ a with K q ‚ a and transferring loops at q toloops at q we have(5) B λ p a q ´ λ pB a q “ χ ´ p a ‚ q q ˆ q ´ q ˆ p q ‚ a q ¯ , where χ “ ř q P Fix p f q ind v p q q is the Euler characteristic of M . Hence λ induces a coproduct on the reduced homology H ˚ Λ, i.e. the homologyof the quotient complex C ˚ p Λ q : “ C ˚ p Λ q{ R χq .2.2. Homology coproduct on the loop space of S . In this sub-section we use Z -coefficients. Recall from [18] that the degree shiftedhomology of the free loop space of S is the exterior algebra H ˚` p Λ S q – Λ p A, U q , | A | “ ´ , | U | “ , where the shifted degree | a | is related to the geometric degree by | a | “ deg a ´
3. Here A is the class of a point (of geometric degree 0) and U is represented by the descending manifold of the Bott family of simplegreat circles tangent at their basepoint to a given non-vanishing vectorfield on the sphere (of geometric degree 5). Since χ p S q “
0, thecoproduct λ is defined on H ˚` p Λ S q and has shifted degree ´ ´ Lemma 2.2.
The coproduct on H ˚ p Λ S q satisfies(a) λ p AU q “ A b A ,(b) λ p AU q “ A b AU ` AU b A ,(c) λ p U q “ A b ` b A .Proof. (a) We represent AU by the 2-chain a : K Ñ Λ S of all circleswith fixed initial point q and initial direction v . ( K is the 2-disc of all2-planes in R through q containing the vector v , whose boundary ismapped to q .) Then a p k qp q “ q for all k P K . Since the evaluationmap p k, t q ÞÑ a p k qp t q covers S once, there exists a unique p k, t q forwhich a p k qp t q “ f p q q . Therefore, λ p a q is homologous to the 0-cycle A b A .(b) We represent AU by the 4-chain a : K Ñ Λ S of all circles withfixed initial point q . ( K is a fibre bundle K Ñ K Ñ S , where S isthe 2-sphere of all initial directions at q and K is the 2-disc from (a)of all circles through q in a given initial direction.) Then a p k qp q “ q for all k P K . Recall that f p q q ‰ q is a point close to q . Let us fixsome initial direction v at q . For every sufficiently large circle (whosediameter is bigger than the distance from q to f p q q ) with initial point q and initial direction v there exist precisely two rotations of theinitial direction such that the rotated circles pass through f p q q . Oneof these rotated circles passes though f p q q near t “ t “
1. As the circle varies over the 2-chain K of all circles OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 9 with initial point q and initial direction v (and we let f p q q move to q ), these two families of rotated circles give rise to cycles representingthe classes A b AU and AU b A , respectively.(c) Fix two orthogonal non-vanishing vector fields v and w on the sphereand represent U by the 5-chain a : K Ñ Λ S of all circles tangent attheir basepoint q P S to w p q q . ( K is a fibre bundle K Ñ K Ñ S ,where S corresponds to the initial points and K is the 2-disc from(a).) Denote f “ f the time-one flow of v and recall that f p q q ‰ q is a point close to q for all q P S . Thus for every q P S there existsprecisely one circle a p x q q tangent to w p q q at its basepoint and passingthrough f p q q , and because all the circles constituting the chain a aresimple there is a unique s q such that a p x q qp s q q “ f p q q “ f p a p x q qp qq .Since v is orthogonal to w , each circle a p x q q is small and the resultingcycle λ p a q can be deformed in Λ ˆ Λ to the diagonal ∆ Ă Λ ˆ Λ . Inturn, this is represented in H ˚ p Λ q b H ˚ p Λ q by r q s b r Λ s ` r Λ s b r q s ,i.e. A b ` b A . (cid:3) Note that Lemma 2.2 is compatible with graded symmetry of λ andwith Sullivan’s relation (3), which in Sweedler’s notation reads λ p a ‚ b q “ a b p a ‚ b q ` p a ‚ b q b b . In fact, Sullivan’s relation together with the values λ p q “ λ p A q “ λ p U q “ A b ` b A inductively determines all values of λ to be(compare with [24, Proposition 3.15]) Corollary 2.3.
The coproduct on H ˚ p Λ S q satisfies for all k ě λ p U k q “ ÿ i,j ě , i ` j “ k ´ ` AU i b U j ` U i b AU j ˘ ,λ p AU k q “ ÿ i,j ě , i ` j “ k ´ AU i b AU j . (cid:3) Morse theoretic description of the homology coproduct.
In this subsection we describe the homology coproduct in terms ofMorse theory on the loop space. This description will be used to proveSullivan’s relation and to relate it to the secondary pair-of-pants co-product on symplectic homology. The analysis is identical to the onein [1, 19] and we refer to there for details.
The Morse complex.
Consider a smooth Lagrangian 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 a smoothaction 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 [2]. The latter con-dition means that for all a, b P Crit p S L q the unstable manifold W ´ p a q and the stable manifold W ` p b q with respect to ´ ∇ S L intersect trans-versely in a manifold of dimension ind p a q´ ind p b q , where ind p a q denotesthe Morse index with respect to S L .Let p M C ˚ , Bq be the Morse complex of S L with R -coefficients. It isgraded by the Morse index and the differential is given by B : M C ˚ Ñ M C ˚´ , a ÞÑ ÿ ind p b q“ ind p a q´ M p a ; b q b, where M p a ; b q denotes the signed count of points in the oriented0-dimensional manifold M p a ; b q : “ ` W ´ p a q X W ` p b q ˘ { R . Then
B ˝ B “
M H ˚ is isomorphic to the singularhomology 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 .We assume that L | M has a unique minimum q m and denote by χ “ χ p M q the Euler characteristic of M . Then Rχq m is a subcomplex of M C ˚ and we define the reduced Morse complex as the quotient complex M C ˚ : “ M C ˚ L Rχq m with reduced Morse homology M H ˚ . The loop product.
Recall from § α β ofpaths. For a, b, c P Crit p S L q set M 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 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 : p M C b M C q ˚ Ñ M C ˚´ n , a b b ÞÑ ÿ c M p a, b ; c q c. OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 11
If the dimension equals 1 it can be compactified to a compact 1-dimensional manifold with boundary B M p a, b ; c q “ ž ind p a q“ ind p a q´ M p a ; a q ˆ M p a , b ; c q> ž ind p b q“ ind p b q´ M p b ; b q ˆ M p a, b ; c q> ž ind p c q“ ind p c q` M p a, b ; c q ˆ M p c ; c q . corresponding to broken gradient lines. So we have(6) µ pB b id ` id b Bq ´ B µ “ µ induces a map on homology µ : p M H b M H q ˚ Ñ M H ˚´ n which agrees with the loop product under the canonical isomorphism M H ˚ – H ˚ Λ. By [13] (see also [34]), the loop product descends toreduced homology µ : p M H b M H q ˚ Ñ M H ˚´ n . The homology coproduct.
As in § v on M with transverse zeroes such that f “ f has nondegeneratefixed points, where f t : M Ñ M , t P R is the flow of v . Recall thedefinition of the path π q p t q : “ f t p q q from q P M to f p q q and its inversepath π ´ q , as well as the restriction and concatenation of paths. Nowfor a, b, c P Crit p S L q we set M p a ; b, c q : “ p τ, α, β, γ q P r , s ˆ W ´ p a q ˆ W ` p b q ˆ W ` p c q | β “ α τ , γ “ α τ ( with α τ p t q : “ α | r ,τ s π ´ α p q “ α p τ t q t ď { ,f ´ t ` α p q ˘ t ě { ,α τ p t q : “ π α p q α | r τ, s “ f t ` α p q ˘ t ď { ,α ` τ ´ ` p ´ τ q t ˘ t ě { . See Figure 1.Note that the matching conditions imply α p τ 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 p a ; b, c q is a transversely cut out manifoldof dimensiondim M p a ; b, c q “ ind p a q ´ ind p b q ´ ind p c q ` ´ n. ααπ ´ πγβ τ π β π ´ α p q α p τ q “ f p α p qq γ Figure 1.
Matching conditions for the definition of thehomology coproduct via Morse chains.If its dimension equals zero this manifold is compact and defines a map λ : M C ˚ Ñ p
M C b M C q ˚` ´ n , a ÞÑ ÿ b,c M 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 p a ; b, c q “ ž ind p a q“ ind p a q´ M p a ; a q ˆ M p a ; b, c q> ž ind p b q“ ind p b q` M p a ; b , c q ˆ M p b ; b q> ž ind p c q“ ind p c q` M p a ; b, c q ˆ M p c ; c q> M τ “ p a ; b, c q > M τ “ p a ; b, c q . Here the first three terms correspond to broken gradient lines and thelast two terms to the intersection of M p a ; b, c q with the sets t τ “ u and t τ “ u , respectively. So we have(7) pB b id ` id b Bq λ ` λ B “ λ ´ λ , where for i “ , λ i : M C ˚ Ñ p
M C b M C q ˚´ n , a ÞÑ ÿ b,c M τ “ i p a ; b, c q b b c. Let us look more closely at the map λ . For τ “ M 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 τ “ p a ; b, c q is empty if OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 13 c ‰ q m and M τ “ 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 that theterms on the right hand side corresponding to different q P Fix p f q are incanonical bijection to each other. By the discussion before Remark 2.1,the terms corresponding to a fixed point q come with the sign ind v p q q , sothe signed count M τ “ p a ; b, q m q is divisible by the Euler characteristic χ . A similar discussion applies to λ , so we have shown that λ p a q “ ÿ ind p b q“ ind p a q´ n M τ “ p a ; b, q m q b b q m P M C ˚´ n b χq m ,λ p a q “ ÿ ind p c q“ ind p a q´ n M τ “ p a ; q m , c q q m b c P χq m b M C ˚´ n . Equation (7) and the preceding discussion show that λ descends to achain map on the reduced Morse complex M C ˚ and thus induces acoproduct on reduced Morse homology λ : M H ˚ Ñ p
M H b M H q ˚` ´ n . It is clear from the construction that λ corresponds under the isomor-phism M H ˚ – H ˚ Λ to the homology coproduct on singular homologydefined in § Remark 2.4.
The above description of M τ “ p a ; b, q m q and M τ “ p a ; q m , c q implies that λ , λ : M C ˚ Ñ p
M C b M C q ˚´ n are chain maps. By equa-tion (7) they are chain homotopic, hence they induce the same primarycoproduct r λ s “ r λ s : M H ˚ Ñ p
M H b M H q ˚´ n and the preceding discussion recovers [4, Lemma 5.1]. Remark 2.5.
Alternatively, we could define the homology coproductusing the spaces Ă M p a ; b, c q : “ p τ, α, β, γ q P r , s ˆ W ´ p a q ˆ W ` p b q ˆ W ` p c q | β p t q “ f ´ t ˝ α p τ t q , γ p t q “ f ´ ´ t ˝ α p τ ` p ´ τ q t q ( . Again the matching conditions imply α p τ q “ f ˝ α p q , and Ă M p a ; b, c q isa transversely cut out manifold of dimension ind p a q ´ ind p b q ´ ind p c q ` ´ n whose rigid counts define a map r λ : M C ˚ Ñ p
M C b M C q ˚` ´ n , a ÞÑ ÿ b,c Ă M p a ; b, c q b b c. A discussion analogous to that for λ shows that r λ descends to a chainmap on the reduced Morse complex M C ˚ and thus induces a coproducton reduced Morse homology r λ : M H ˚ Ñ p
M H b M H q ˚` ´ n . The obvious homotopies between the loops α τ and α τ in the definitionof λ and the loops t ÞÑ f t ˝ α p τ t q and t ÞÑ f ´ ´ t ˝ α p τ ` p ´ τ q t q in the definition of r λ provides a chain homotopy between λ and r λ on the reduced Morse complex, so r λ agrees with λ on reduced Morsehomology. The proof of Sullivan’s relation in the following subsectionwill work more naturally for λ . We use the restriction of the map r λ toMorse chains modulo constants for the proof of Proposition 5.5. Themap r λ could be useful for establishing the isomorphism to the pair-of-pants coproduct on reduced Floer homology by extending the chain ofisomorphisms in § § Proof of Sullivan’s relation.
In this subsection we give a Morsetheoretic proof of Sullivan’s relation (3). (In [13] we gave a symplecticproof in dimension n ě Proposition 2.6 (Sullivan’s relation on chain level) . There exists alinear map
Γ : p M C b M C q ˚ Ñ p
M C b M C q ˚` ´ n whose commutatorwith the boundary operator satisfies the following relation on the reducedMorse complex M C ˚ : (8) rB , Γ s “ λµ ´ p b µ qp λ b q ´ p µ b qp b λ q . Proof.
For a, b, c, d P Crit p S L q we set M p a, b ; c, d q : “ p σ, τ, α, β, γ, δ q P R ˆ r , s ˆ W ´ p a q ˆ W ´ p b qˆ W ` p c q ˆ W ` p d q | p˚q ( with matching conditions p˚q that we will now describe for the variousranges of the parameters p σ, τ q . We denote the flow of ´ ∇ S L by φ s : Λ Ñ Λ , s ě . Region I: p σ, τ q P r ,
8q ˆ r , s . Here the matching conditions are γ “ ε τ , δ “ ε τ , ε “ φ σ ´ p α β q , which imply α p q “ β p q and ε p τ q “ f ` ε p q ˘ . See Figure 2. OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 15 βπ ´ πδγ τ ε φ σ ´ α Figure 2.
Matching conditions in Region I.Let us discuss the boundary terms. At σ “ τ ď { γ p t q “ α p τ t q t ď { ,f ´ t ` α p q ˘ t ě { ,δ p t q “ $’&’% f t ` α p q ˘ t ď { ,α ` τ p t q ˘ { ď t ď ν ,β ` η p t q ˘ t ě ν , with the canonical nondecreasing linear surjections τ : r { , ν s Ñ r τ, s , η : r ν , s Ñ r , s , ν “ ´ τ ´ τ . The expressions for τ ě { σ Ñ 8 the cylinder of length σ ´ e P Crit p S L q and we find ž ind p e q“ ind p a q` ind p b q´ n M p a, b ; e q ˆ M p e ; c, d q . At τ “ γ ” q P Fix p f q , so c “ q m . At τ “ δ ” q P Fix p f q , so d “ q m . Region II: p σ, τ q P p´8 , s ˆ r , { s . Here the matching conditionsare γ “ α τ , δ “ ε β, ε “ φ ´ σ p α τ q , which imply α p τ q “ f ` α p q ˘ and ε p q “ β p q . See Figure 3. απ ´ πγ τ φ ´ σ δ βε Figure 3.
Matching conditions in Region II.Let us discuss the boundary terms. At σ “ γ p t q “ α p τ t q t ď { ,f ´ t ` α p q ˘ t ě { ,δ p t q “ $’&’% f t ` α p q ˘ t ď { ,α ` τ p t q ˘ { ď t ď { ,β ` η p t q ˘ t ě { , with the canonical nondecreasing linear surjections τ : r { , { s Ñ r τ, s , η : r { , s Ñ r , s . As σ Ñ ´8 the cylinder of length ´ σ splits along some e P Crit p S L q and we find ž ind p e q“ ind p a q´ ind p c q` ´ n M p a ; c, e q ˆ M p e, b ; d q . At τ “ γ ” q P Fix p f q , so c “ q m .At τ “ { α p q “ q P Fix p f q and (denoting the constantloop at q by the same letter) the matching conditions become γ “ α q, δ “ φ ´ σ p q q β. Region III: p σ, τ q P p´8 , s ˆ r { , s . Here the matching conditionsare γ “ α ε, δ “ β τ ´ , ε “ φ ´ σ p β τ ´ q , which imply α p q “ ε p q and β p τ ´ q “ f ` β p q ˘ . Let us discuss theboundary terms. At σ “ OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 17 As σ Ñ ´8 the cylinder of length ´ σ splits along some e P Crit p S L q and we find ž ind p e q“ ind p b q´ ind p d q` ´ n M p b ; e, d q ˆ M p a, e ; c q . At τ “ γ ” q P Fix p f q , so c “ q m .At τ “ { β p q “ q P Fix p f q and the matching condi-tions become γ “ α φ ´ σ p q q , δ “ q β. Region IV: p σ, τ q P r , s ˆ r , { s . Here the matching conditionsare γ p t q “ α p τ t q t ď { ,f ´ t ` α p q ˘ t ě { ,δ p t q “ $’&’% f t { µ ` α p q ˘ t ď µ,α ` ξ p t q ˘ µ ď t ď ν,β ` η p t q ˘ t ě ν, which imply α p q “ β p q and α p τ q “ f ` α p q ˘ . See Figure 4. µ π ´ βανπδγ ττ Figure 4.
Matching conditions in Region IV.Here ξ : r µ, ν s Ñ r τ, s , η : r ν, s Ñ r , s are the canonical nondecreasing linear surjections, where ξ, η, µ, ν de-pend on the parameters p σ, τ q . We will either suppress the parameterdependence in the notation or indicate by ξ σ etc. the dependence on σ P r , s . With this notation, we set µ σ “ p ´ σ q ` σ , ν σ “ p ´ σ q ` σ ´ τ ´ τ . Let us discuss the boundary terms. At σ “ µ “ {
2, so thematching conditions agree with those for region I at σ “ τ P r , { s . π ´ πγ τ αδ β Figure 5.
Boundary term at σ “ σ “ µ “ { ν “ {
2, so the matching conditionsagree with those for region II at σ “
0. See Figure 5.At τ “ α p q “ q P Fix p f q , so γ “ q is the constantloop at q and c “ q m .At τ “ { α p q “ β p q “ q P Fix p f q , which implies that ν “ { ξ p t q ” γ “ α q, δ “ q β, q P Fix p f q . Region V: p σ, τ q P r , s ˆ r { , s . Here the discussion is entirelyanalogous to that of region IV, so we will not spell out all the formulasbut just discuss the boundary terms.At σ “ σ “ τ P r { , s , while at σ “ σ “ τ “ β p q “ q P Fix p f q , so δ “ q is the constantloop at q and d “ q m .At τ “ { τ “ { M p a, b ; c, d q . For generic choices it isa transversely cut out manifold of dimensiondim M p a, b ; c, d q “ ind p a q ` ind p b q ´ ind p c q ´ ind p d q ` ´ n. If the dimension equals zero its signed counts define a linear mapΓ : p M C b M C q ˚ Ñ p
M C b M C q ˚` ´ n , a b b ÞÑ ÿ c,d M p a, b ; c, d q c b d. OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 19
Claim. If M p a, b ; c, d q has dimension 1 it can be compactified to acompact 1-dimensional manifold with boundary B M p a, b ; c, d q “ ž ind p a q“ ind p a q´ M p a ; a q ˆ M p a , b ; c, d q> ž ind p b q“ ind p b q´ M p b ; b q ˆ M p a, b ; c, d q> ž ind p c q“ ind p c q` M p a, b ; c , d q ˆ M p c , c q> ž ind p d q“ ind p d q` M p a, b ; c, d q ˆ M p d , d q> ž ind p e q“ ind p a q` ind p b q´ n M p a, b ; e q ˆ M p e ; c, d q> ž ind p e q“ ind p a q´ ind p c q` ´ n M p a ; c, e q ˆ M p e, b ; d q> ž ind p e q“ ind p b q´ ind p d q` ´ n M p b ; e, d q ˆ M p a, e ; c q> M τ “ p a, b ; c, d q > M τ “ p a, b ; c, d q . Here the first four lines correspond to splitting off of Morse gradientlines, the 5th one to σ Ñ 8 in region I, the 6th one to σ Ñ ´8 inregion II, the 7th one to σ Ñ ´8 in region III, and the last one tothe intersection of M p a, b ; c, d q with the sets t τ “ u and t τ “ u ,respectively. It remains to show that all other boundary componentsof M p a, b ; c, d q cancel in pairs.By the discussion above region I matches regions IV and V along theset t σ “ u , and regions II and III match regions IV and V alongthe set t σ “ u , so the corresponding boundary terms are equal withopposite orientations and thus cancel out. Similarly, regions IV and Vmatch along the set t τ “ { u , so the corresponding boundary termscancel out.The intersections of regions II and III with the set t τ “ { u differ byswitching the roles of q and φ ´ σ p q q . To show that they cancel out, weconsider for a, c P Crit p S L q the set N p a ; c q : “ tp α, γ, q q P W ´ p a q ˆ W ` p c q ˆ M | γ “ α q u , where q denotes the constant loop at q and the matching conditionimplies α p q “ q . For generic choices this is a manifold of dimensiondim N p a ; c q “ ind p a q ´ ind p c q . We can choose the minimum q m P M of L | M to be a regular value ofthe projection P : N p a ; c q Ñ M, p α, γ, q q ÞÑ q. Then there exists an open neighbourhood U Ă M of q such that therestriction P : P ´ p U q Ñ U is a submersion, so all fibres N p a ; c ; q q : “ P ´ p q q “ tp α, γ q P W ´ p a q ˆ W ` p c q | γ “ α q u with q P U are diffeomorphic (canonically up to isotopy). In thisnotation the matching conditions for region II at t τ “ { u are p α, γ q P N p a ; c ; q q , p β, δ q P N ` b ; d ; φ ´ σ p q q ˘ for some q P Fix p f q , while those for region III at t τ “ { u are p α, γ q P N ` p a ; c ; φ ´ σ p q q ˘ , p β, δ q P N p b ; d ; q q . We can choose the set U to be invariant under the negative gradientflow of L | M . Moreover, we can choose the vector field v such that allits zeroes are contained in U , so Fix p f q Ă U . Then φ ´ σ p q q P U for q P Fix p f q and all σ ď
0, so by the preceding discussion N p a ; c ; q q is diffeomorphic to N ` p a ; c ; φ ´ σ p q q ˘ and N p b ; d ; q q is diffeomorphic to N ` b ; d ; φ ´ σ p q q ˘ (canonically up to isotopy). This shows that the inter-sections of regions II and III with the set t τ “ { u are diffeomorphicwith opposite orientations and thus cancel out. This concludes theproof of the claim.Now recall from the discussion above that for the spaces M τ “ p a, b ; c, d q and M τ “ p a, b ; c, d q in the last line, in each region at least one of theoutputs c, d equals the minimum q m and the signed counts of elementsin the relevant moduli spaces involve multiples of χq m . Therefore, theircontributions vanish on the reduced Morse complex. Hence the alge-braic counts of the boundary strata in B M p a, b ; c, d q yield equation (8)on M C ˚ applied to a b b and projected onto c b d , where the first fourlines give rB , Γ s and lines 5–7 give the three terms on the right handside. This concludes the proof of Proposition 2.6. (cid:3) Isomorphism between symplectic homology and loophomology
As before, M is a closed oriented manifold, T ˚ M its cotangent bundlewith its Liouville form λ “ p dq , and D ˚ M Ă T ˚ M its unit disc cotan-gent bundle viewed as a Liouville domain. The symplectic homology SH ˚ p D ˚ M q is isomorphic to the Floer homology F H ˚ p H q of a fibre-wise quadratic Hamiltonian H : S ˆ T ˚ M Ñ R . On the other hand, F H ˚ p H q is isomorphic to the loop homology H ˚ p Λ; σ q (Viterbo [35],Abbondandolo-Schwarz [1, 5], Salamon-Weber [31], Abouzaid [6]). Herewe use coefficients twisted by the local system σ defined by transgress-ing the second Stiefel-Whitney class, cf. Appendix A. We drop thelocal system σ from the notation in the rest of this section.The construction most relevant for our purposes is the chain mapΨ : F C ˚ p H q Ñ M C ˚ p S q OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 21 from the Floer complex of a Hamiltonian H : S ˆ T ˚ M Ñ R to theMorse complex of an action functional S : Λ Ñ R on the loop spacedefined in [4]. When applied to a fibrewise quadratic Hamiltonian H and the action functional S L associated to its Legendre transform L , itinduces an isomorphism on homologyΨ ˚ : SH ˚ p D ˚ M q – F H ˚ p H q Ñ M H ˚ p S L q – H ˚ Λintertwining the pair-of-pants product with the loop product [4]. Wewill prove in the next section that Ψ ˚ descends to reduced homologyand intertwines there the continuation-map coproduct with the loopcoproduct.One annoying feature of the map Ψ has been that, in contrast to itschain homotopy inverse Φ : M C ˚ p S L q Ñ F C ˚ p H q , it does not preservethe action filtrations. This would make it unsuitable for some of ourapplications in [13] such as those concerned with critical values. Usingan estimate inspired by [14] we show in this section that Ψ does preservesuitable action filtrations when applied to fibrewise linear Hamiltoniansrather than fibrewise quadratic ones.3.1. Floer homology.
Consider a smooth time-periodic Hamiltonian H : S ˆ T ˚ M Ñ R which outside a compact set is either fibrewisequadratic, or linear with slope not in the action spectrum. It inducesa smooth Hamiltonian action functional A H : C p S , T ˚ M q Ñ R , x ÞÑ ż ` x ˚ λ ´ H p t, x q dt ˘ . Its critical points are 1-periodic orbits x , which we can assume to benondegenerate with Conley–Zehnder index CZ p x q . Let J be a compati-ble almost complex structure on T ˚ M and denote the Cauchy–Riemannoperator with Hamiltonian perturbation on u : R ˆ S Ñ T ˚ M by B H u : “ B s u ` J p u q ` B t u ´ X H p t, u q ˘ . Let
F C ˚ p H q be the free R -module generated by Crit p A H q and gradedby the Conley–Zehnder index. The Floer differential is given by B F : F C ˚ p H q Ñ F C ˚´ p H q , x ÞÑ ÿ CZ p y q“ CZ p x q´ M p x ; y q y, where M p x ; y q denotes the signed count of points in the oriented0-dimensional manifold M p x ; y q : “ t u : R ˆ S Ñ T ˚ M | B H u “ , u p`8q “ x, u p´8q “ y u{ R . Then B F ˝ B F “ F H ˚ p H q is isomorphic to thesymplectic homology SH ˚ p T ˚ M q if H is quadratic. If H is linear, weobtain an isomorphism to SH ˚ p T ˚ M q in the direct limit as the slopegoes to infinity. The isomorphism Φ . Suppose now that H is fibrewise convexwith fibrewise Legendre transform L : S ˆ T M Ñ R . As in § p M C ˚ , Bq of the action functional S L : Λ Ñ R , S L p q q “ ż L p t, q, q q dt. Following [1], for a P Crit p S L q and x P Crit p A H q we consider the space M p a ; x q : “ t u : p´8 , s ˆ S Ñ T ˚ M |B H p u q “ , u p´8q “ x,π ˝ u p , ¨q P W ´ p a qu , where W ´ p a q denotes the stable manifold for the gradient flow of S L and π : T ˚ M Ñ M is the projection. (It is sometimes useful to view W ´ p a q as the unstable manifold for the negative gradient flow of S L .)For generic H this is a manifold of dimensiondim M p a ; x q “ ind p a q ´ CZ p x q . The signed count of 0-dimensional spaces M p a ; x q defines a chain map(9) Φ : M C ˚ p S L q Ñ F C ˚ p H q , a ÞÑ ÿ CZ p x q“ ind p a q M p a ; x q x. It was shown in [1] that the induced map on homology is an isomor-phism Φ ˚ : M H ˚ p S L q – ÝÑ F H ˚ p H q . For u P M p a ; x q consider the loop p q, p q “ u p , ¨q : S Ñ T ˚ M at s “ A H p q, p q “ ż ´ x p, q y ´ H p t, q, p q ¯ dt ď ż L p t, q, q q dt “ S L p q q . It follows that A H p x q ď A H ` u p , ¨q ˘ ď S L p q q ď S L p a q whenever M p a ; x q is nonempty, so Φ decreases action.3.3. The isomorphism Ψ . Consider once again a fibrewise quadraticHamiltonian H : S ˆ T ˚ M Ñ R as in § L . Following [4, 14], for x P Crit p A H q and a P Crit p S L q we define M p x q : “ t u : r ,
8q ˆ S Ñ T ˚ M | B H u “ , u p`8 , ¨q “ x, u p , ¨q Ă M u and M p x ; a q : “ t u P M p x q | u p , ¨q P W ` p a qu , where W ` p a q is the stable manifold of a for the negative gradient flowof S L , see Figure 6.For generic H these are manifolds of dimensionsdim M p x q “ CZ p x q , dim M p x ; a q “ CZ p x q ´ ind p a q . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 23 x Ma u
Figure 6.
Moduli spaces for the map Ψ.The signed count of 0-dimensional spaces M p x ; a q defines a chain mapΨ : F C ˚ p H q Ñ M C ˚ p S L q , x ÞÑ ÿ ind p a q“ CZ p x q M p x ; a q a. The induced map on homology is an isomorphismΨ ˚ “ Φ ´ ˚ : F H ˚ p H q – ÝÑ M H ˚ p S L q – H ˚ Λ , which is the inverse of Φ ˚ and intertwines the pair-of-pants prod-uct with the loop product. This was shown by Abbondandolo andSchwarz [4] with Z { H ˚ Λ by a suitable local system, see Appendix A.Moreover, Abouzaid proved that Ψ ˚ is an isomorphism of twisted BValgebras.Unfortunately, the map Ψ does not preserve the action filtrations. Thisalready happens for a classical Hamiltonian H p q, p q “ | p | ` V p q q : For u P M p x ; a q the loop q “ u p , ¨q : S Ñ M satisfies A H p x q ě A H ` u p , ¨q ˘ “ ´ ż V p q q dt ď ż ´ | q | ´ V p q q ¯ dt “ S L p q q ě S L p a q , so the middle inequality goes in the wrong direction (even if V “ An action estimate for Floer half-cylinders.
Now we willreplace the quadratic Hamiltonians from the previous subsections byHamiltonians of the shape used in the definition of symplectic homol-ogy. For Floer half-cylinders of such Hamiltonians, we will estimate thelength of their boundary loop on the zero section by the Hamiltonianaction at `8 .We equip M with a Riemannian metric and choose the following data.The Riemannian metric on M induces a canonical almost complexstructure J st on T ˚ M compatible with the symplectic form ω st “ dp ^ dq (Nagano [28], Tachibana-Okumura [33], see also [9, Ch. 9]). In geodesicnormal coordinates q i at a point q and dual coordinates p i it is givenby J st : BB q i ÞÑ ´ BB p i , BB p i ÞÑ BB q i . We pick a nondecreasing smooth function ρ : r ,
8q Ñ p , with ρ p r q ” r “ ρ p r q “ r for large r . Then J : BB q i ÞÑ ´ ρ p| p |q BB p i , ρ p| p |q BB p i ÞÑ BB q i (in geodesic normal coordinates) defines a compatible almost complexstructure on T ˚ M which agrees with J st near the zero section and iscylindrical outside the unit cotangent bundle.We view r p q, p q “ | p | as a function on T ˚ M . Then on T ˚ M z M we have λ “ rα, α : “ p dq | p | . Consider a Hamiltonian of the form H “ h ˝ r : T ˚ M Ñ R for asmooth function h : r ,
8q Ñ r , vanishing near r “
0. Then itsHamiltonian vector field equals X H “ h p r q R , where R is the Reebvector field of p S ˚ M, α q . The symplectic and Hamiltonian actions of anonconstant 1-periodic Hamiltonian orbit x : S Ñ T ˚ M are given by ż x α “ h p r q , A H p x q “ rh p r q ´ h p r q . Given a slope µ ą p S ˚ M, α q and any ε ą
0, we can pick h with the following properties: ‚ h p r q ” r ď h p r q ” µ for r ě ` δ , with some δ ą ‚ h p r q ą rh p r q ´ h p r q ´ ε ď h p r q ď rh p r q ´ h p r q for r Pp , ` δ q .Specifically, we choose 0 ă δ ď ε { µ , we consider a smooth function β : r ,
8q Ñ r , s such that β “ r , s , β “ r ` δ, and β is strictly increasing on p , ` δ q , and we define h : r ,
8q Ñ r , by h p r q “ µ ż r β p ρ q dρ. We have rh ´ h ´ h “ µ ` p r ´ q β ´ ş r β ˘ . This expression differ-entiates to µ p r ´ q β ě r , s , hence it is non-negative for r ě
0. On the other hand, we have an upper bound µ ` p r ´ q β ´ ş r β ˘ ď µδ for r P p , ` δ q , and indeed for r ě
0. Givenour choice δ ď ε { µ , this establishes the inequalities rh p r q ´ h p r q ´ ε ď h p r q ď rh p r q ´ h p r q for all r ě OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 25
These inequalities imply that for each nonconstant 1-periodic Hamil-tonian orbit x we have(10) A H p x q ´ ε ď ż x α ď A H p x q . With this choice of J and H , consider now as in the previous subsectiona map u : r ,
8q ˆ S Ñ T ˚ M satisfying B H u “ , u p`8 , ¨q “ x, u p , ¨q Ă M. Set q p t q : “ u p , t q and denote its length by ℓ p q q : “ ż | q | dt. The following proposition is a special case of [14, Lemma 7.2]. Sincethe proof was only sketched there, we give a detailed proof below.
Proposition 3.1.
Let
H, J, u be as above with q “ u p , ¨q and a non-constant orbit x “ u p`8 , ¨q . Then ℓ p q q ď ż x α ď A H p x q . The first inequality is an equality if and only if u is contained in thehalf-cylinder over a closed geodesic q , in particular x is the lift of thegeodesic q . The idea of the proof is to show that0 ď ż p , S u ˚ dα “ ż x α ´ ℓ p q q . Since the image u ` p ,
8q ˆ S ˘ can hit the zero section M where α is undefined, the quantity ş p , S u ˚ dα has to be interpreted as animproper integral as follows. Given ε ą
0, let τ “ τ ε : r ,
8q Ñ r , be a smooth function with τ p r q ě r , τ p r q “ r “
0, and τ p r q “ r ě ε , and consider the globally defined 1-form on T ˚ M given by α ε : “ τ p| p |q p dq | p | . We now define (11) ż p , S u ˚ dα “ lim σ Œ lim ε Œ ż r σ, S u ˚ dα ε . The proof of Proposition 3.1 is based on the following lemma.
Lemma 3.2.
For any v P T p q,p q T ˚ Q , we have dα ε p v, J v q ě . At points where τ p| p |q ą , equality only holds for v “ , whereas atpoints where τ “ and τ ‰ equality holds if and only if v is a linearcombination of p B p and p B q .Proof. In geodesic normal coordinates we compute dα ε “ d ˜ÿ i τ p| p |q p i dq i | p | ¸ “ ÿ i τ p| p |q dp i ^ dq i | p | ` ÿ i,j p τ p| p |q| p | ´ τ p| p |qq p i p j dp i ^ dq j | p | . For a vector of the form v “ ř i a i ρ p| p |qB p i we obtain J ρ v “ ř i a i B q i and hence by the Cauchy-Schwarz inequality dα ε p v, J v q “ ÿ i τ p| p |q ρ p| p |q a i | p | ` ÿ i,j p τ p| p |q| p | ´ τ p| p |qq ρ p| p |q p i p j a i a j | p | “ τ p| p |q ρ p| p |q| p | p| a | | p | ´ x a, p y q ` τ p| p |q ρ p| p |q| p | x a, p y ě . At points where τ ą
0, equality only holds for a “
0, and at pointswhere τ “ τ ą a is a multiple of p . Similarly,for a general vector v “ ř i a i ρ p| p |qB p i ´ ř i b i B q i we get dα ε p v, J v q ě a “ b “ τ “ a and b aremultiples of p . (cid:3) Proof of Proposition 3.1.
The proof consists in 3 steps.
Step 1. We prove that ş p , S u ˚ dα ě . In view of Definition (11), it is enough to show that u ˚ dα ε ě p ,
8q ˆ S . To see this, recall that u satisfies the equation B s u ` J p u q ` B t u ´ X H p u q ˘ “
0, so that u ˚ dα ε “ dα ε pB s u, B t u q ds ^ dt “ dα ε ` B s u, J p u qB s u ` X H p u q ˘ ds ^ dt. Now at points in D ˚ M the Hamiltonian vector field X H vanishes. Atpoints outside D ˚ M we have X H “ h p r q R and α ε “ α (we can assumew.l.o.g. ε ď i X H dα ε “ h p r q i R dα “
0. In either case wehave u ˚ dα ε “ dα ε pB s u, J p u qB s u q ds ^ dt, which is nonnegative by Lemma 3.2. Step 2. Denote u σ “ u p σ, ¨q for σ ą . We have lim σ Œ lim ε Œ ż S u ˚ σ α ε “ ℓ p q q . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 27
To see this we consider the map˜ q : r ,
8q ˆ S Ñ T ˚ M, ˜ q p s, t q : “ ` q p t q , s q p t q ˘ , and denote as above ˜ q σ “ ˜ q p σ, ¨q for σ ą
0. Since J “ J st near thezero section, the maps u and ˜ q agree with their first derivatives alongthe boundary loop q at s “
0, hence u σ and ˜ q σ are C -close for σ close to 0. On the other hand α ε is C -bounded near the zero sectionuniformly with respect to ε Ñ
0. These two facts imply that theintegrals ş S u ˚ σ α ε and ş S ˜ q ˚ σ α ε are C -close for σ close to 0, uniformlywith respect to ε Ñ
0, and thereforelim σ Œ lim ε Œ ż S u ˚ σ α ε “ lim σ Œ lim ε Œ ż S ˜ q ˚ σ α ε . We now prove that(12) lim ε Œ ż S ˜ q ˚ σ α ε “ ℓ p q q for all σ ą
0, which implies the desired conclusion. Fix therefore σ ą
0. Let I ε “ t t P S : | σ q p t q| ď ε u , so that I ε Ă I ε for ε ď ε and ş ε ą I ε “ I “ t t : q p t q “ u . On the one hand we have ż S z I ε ˜ q ˚ σ α ε “ ż S z I ε ˜ q ˚ σ α “ ż S z I ε α p q p t q ,σ q p t qq ¨ ˜ q p t q“ ż S z I ε σ | q p t q| | σ q p t q| dt “ ż S z I ε | q p t q| dt “ ℓ p q | S z I ε q . We can therefore estimate ˇˇˇˇż S ˜ q ˚ σ α ε ´ ℓ p q | S z I ε q ˇˇˇˇ “ ˇˇˇˇ ż I ε ˜ q ˚ σ α ε ˇˇˇˇ “ ˇˇˇˇ ż I ε α ε p ˜ q σ p t qq ¨ ˜ q σ p t q dt ˇˇˇˇ ď C ¨ εσ ¨ m p I ε q Ñ ε Ñ . Here m p I ε q is the measure of I ε , uniformly bounded by the length ofthe circle, C ą C -bound on α ε near the 0-section, uniform withrespect to ε Ñ
0, and ε { σ is by definition the bound on | q p t q| on I ε .The estimate follows from ˜ q σ “ p q, σ : q q and the fact that the 1-form α ε only acts on the first component of the vector ˜ q σ .Because lim ε Œ ℓ p q | S z I ε q “ ℓ p q | S z I q “ ℓ p q q , equality (12) follows. Step 3. We prove ż p , S u ˚ dα “ ż x α ´ ℓ p q q . Indeed, for σ, ε ą ż r σ, S u ˚ dα ε “ ż x α ´ ż u σ α ε . (The 1-form α ε is equal to α near the orbit x .) The desired equalityfollows from the definition of ş p , S u ˚ dα and Step 2. Conclusion.
Combining Step 3 with Step 1 we obtain the first inequal-ity ℓ p q q ď ş x α in Proposition 3.1. Moreover, Lemma 3.2 (in the limit ε Ñ
0) shows that this inequality is an equality if and only if u iscontained in the half-cylinder over a closed geodesic.The second inequality ş x α ď A H p x q follows from (10). (cid:3) The isomorphism Ψ from symplectic to loop homology. Now we adjust the definition of Ψ to symplectic homology. For
J, H asin the previous subsection and x P Crit p A H q we define as before M p x q : “ t u : r ,
8q ˆ S Ñ T ˚ M | B H u “ , u p`8 , ¨q “ x, u p , ¨q Ă M u By Proposition 3.1 the loop q “ u p , ¨q satisfies ℓ p q q ď A H p x q . More-over, the loop q is smooth and in particular has Sobolev class H , hencefollowing Anosov [8] it has a unique H -reparametrization q : S Ñ M ,with | q | ” const and q p q “ q p q (we say that q is parametrized propor-tionally to arclength, or PPAL ). We have ℓ p q q “ ℓ p q q “ ż | q | dt “ ´ż | q | dt ¯ { “ E p q q { with the energy E : Λ Ñ R , E p q q : “ ż | q | dt. The energy defines a smooth Morse-Bott function on the loop spacewhose critical points are constant loops and geodesics parametrizedproportionally to arclength. We denote by W ˘ p a q the unstable/stablemanifolds of a P Crit p E q with respect to ∇ E . Now for x P Crit p A H q and a P Crit p E q we define M p x ; a q : “ t u P M p x q | u p , ¨q P W ` p a qu . An element u in this moduli space still looks as in Figure 6, wherenow the lop q “ u p , ¨q is reparametrized proportionally to arclengthand then flown into a using the flow of ´ ∇ E . By Proposition 3.1, for u P M p x ; a q we have the estimate(13) A H p x q ě ℓ p q q “ E p q q { ě E p a q { . To define the map Ψ, we now perturb H and E by small Morse func-tions near the constant loops on M and the closed geodesics, and wegenerically perturb the almost complex structure J from the previoussubsection. For generic such perturbations, each M p x ; a q is a manifoldof dimension dim M p x ; a q “ CZ p x q ´ ind p a q . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 29
The signed count of 0-dimensional spaces M p x ; a q defines a chain mapΨ : F C ˚ p H q Ñ M C ˚ p E { q , x ÞÑ ÿ ind p a q“ CZ p x q M p x ; a q a. Here
M C ˚ p E { q denotes the Morse chain complex of E : Λ Ñ R ,graded by the Morse indices of E , but filtered by the square root E { (which is decreasing under the negative gradient flow of E ). The ac-tion estimate (13) continues to hold for the perturbed data up to anarbitrarily small error, which we can make smaller than the smallestdifference between lengths of geodesics below a given length µ . ThusΨ preserves the filtrationsΨ : F C ă b ˚ p H q Ñ M C ă b ˚ p E { q . The induced maps on filtered Floer homologyΨ ˚ : F H p a,b q˚ p H q Ñ M H p a,b q˚ p E { q – H p a,b q˚ Λhave upper triangular form with respect to the filtrations with ˘ ˚ intertwines the pair-of-pants product with the loop product, as wellas the corresponding BV operators. Passing to the direct limit overHamiltonians H , we have thus proved Theorem 3.3.
The map Ψ induces isomorphisms on filtered symplectichomology Ψ ˚ : SH p a,b q˚ p D ˚ M q – ÝÑ M H p a,b q˚ p E { q – H p a,b q˚ Λ , where the left hand side is filtered by non-Hamiltonian action and theright hand side by the square root of the energy. These isomorphismsintertwine the pair-of-pants product with the Chas-Sullivan loop prod-uct, as well as the corresponding BV operators. (cid:3) The isomorphism Ψ on reduced homology. Consider again aHamiltonian H : T ˚ M Ñ R as in § M is connectedand orientable. We perturb H so that all its critical points lie onthe zero section, and near the zero section H p q, p q “ ε | p | ` V p q q for asmall ε ą V : M Ñ R such that all nonconstantcritical points of A H have action larger than min V . Then the constantcritical points of A H define a subcomplex F C “ ˚ p H q of F C ˚ p H q whichagrees with the Morse cochain complex of V on M , with degrees of q P Crit p V q related by CZ p q q “ n ´ ind V p q q .Next we choose V to have a unique local maximum q P M . Then wehave subcomplexes R ¨ χq Ă F C “ ˚ p H q Ă F C ˚ p H q , where χ denotes the Euler characteristic of M and R is the coefficientring. Note that CZ p q q “
0. We obtain quotient complexes
F C ˚ p H q : “ F C ˚ p H q{ R ¨ χq and F C ą ˚ p H q : “ F C ˚ p H q{ F C “ ˚ p H q with projections F C ˚ p H q Ñ F C ˚ p H q Ñ F C ą ˚ p H q . The resulting homologies
F H ˚ p H q and F H ą ˚ p H q are called the reduced resp. positive Floer homology . The same constructions on the loopspace side yield the reduced resp. positive loop homology M H ˚ p E { q – H ˚ Λ and
M H ą ˚ p E { q – H ˚ p Λ , Λ q , where Λ Ă Λ is the subset of constant loops. By construction, themap Ψ from the previous subsection preserves the reduced and positivesubcomplexes, so we have
Corollary 3.4.
The isomorphism Ψ ˚ from Theorem 3.3 descends toisomorphisms Ψ ˚ : SH ˚ p D ˚ M q – ÝÑ H ˚ Λ and Ψ ˚ : SH ą ˚ p D ˚ M q – ÝÑ H ˚ p Λ , Λ q on reduced and positive homology. These isomorphisms preserve thefiltrations by the non-Hamiltonian action on the symplectic homologyside, respectively by the square root of the energy on the Morse side. (cid:3) Continuation coproduct and loop coproduct
We keep the setup from the previous section, so M is a closed orientedRiemannian manifold and D ˚ M Ă T ˚ M its unit disc cotangent bundle.In this section we prove Theorem 4.1.
The isomorphism Ψ ˚ : SH ˚ p D ˚ M q – ÝÑ H ˚ Λ fromCorollary 3.4 intertwines the continuation coproduct λ cont from [16] with the loop coproduct λ . Continuation coproduct.
In [16, § λ isdefined in terms of continuation maps on the reduced symplectic homol-ogy of any “cotangent-like Weinstein domain” W . In this subsectionwe recall its definition for W “ D ˚ M ; we will call it the continuationproduct and denote it by λ cont .The definition in [16] is described in terms of real parameters λ , λ ă ă µ , µ , µ satisfying µ ď min p λ ` µ , µ ` λ q . For simplicity, wechoose the parameters as λ “ λ “ ´ µ and µ “ µ “ µ for some µ ą
0. We assume that µ and 2 µ do not belong to the action spectrumof S ˚ M .As before, we denote by r “ | p | the radial coordinate on T ˚ M . Let K “ K µ be a convex smoothing of the Hamiltonian which is zeroon D ˚ M and equals r ÞÑ µr outside D ˚ M . Then 2 K “ K µ and OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 31 ´ K “ K ´ µ are the corresponding Hamiltonians of slopes 2 µ and ´ µ ,respectively.Let Σ be the 3-punctured Riemann sphere, where we view one punctureas positive (input) and the other two as negative (outputs). We fixcylindrical coordinates p s, t q P r ,
8q ˆ S near the positive punctureand p s, t q P p´8 , s ˆ S near the negative punctures. Consider a 1-form β on Σ which equals B dt near the positive puncture and A i dt near the i -th negative puncture p i “ ,
2) for some A i , B P R . We saythat β has weights B, A , A . We moreover require dβ ď
0, which ispossible iff A ` A ě B. We consider maps u : Σ Ñ T ˚ M satisfying the perturbed Cauchy-Riemann equation p du ´ X K b β q , “ . Near the punctures this becomes the Floer equation for the Hamiltoni-ans BK and A i K , respectively, and the algebraic count of such mapsdefines a (primary) coproduct F H ˚ p BK q Ñ F C ˚ p A K q b F C ˚ p A K q which has degree ´ n and decreases the Hamiltonian action.To define the secondary coproduct λ cont , we choose a 1-parameter fam-ily of 1-forms β τ , τ P p , q , with the following properties (see Figure 7): ‚ dβ τ ď τ ; ‚ β τ equals dt near the positive puncture and ` dt near each negativepuncture, i.e., β τ has weights 1 , , ‚ as τ Ñ β τ equals ´ dt on cylinders near the first negative punc-ture whose length tends to , so that β consists of a 1-form on Σwith weights 1 , ´ , ´ , ‚ as τ Ñ β τ equals ´ dt on cylinders near the second negativepuncture whose length tends to , so that β consists of a 1-formon Σ with weights 1 , , ´ ´ , r λ cont : F C ˚ p K q Ñ F C ˚ p K q b F C ˚ p K q which has degree 1 ´ n and decreases the Hamiltonian action.Let us analyze the contributions from τ “ ,
1. The algebraic count ofcylinders with weights ´ , continuation map (of degree 0) c “ c ´ K, K : F C ˚ p´ K q Ñ F C ˚ p K q . It is shown in [16] that the image of c is R ¨ χq , where χ is the Eulercharacteristic of M and q P M the basepoint (as before, we perturb Figure 7.
The continuation coproduct λ cont K by a function on M which attains its maximum at q ), so F C ˚ p K q{ im c “ F C ˚ p K q{ R ¨ χq “ F C ˚ p K q is the reduced Floer complex. Similarly, the algebraic count of cylinderswith weights ´ , c “ c ´ K,K : F C ˚ p´ K q Ñ F C ˚ p K q with image R ¨ χq , so that F C ˚ p K q{ im c “ F C ˚ p K q{ R ¨ χq “ F C ˚ p K q . It is shown in [16] that λ cont descends to a map λ cont : F C ˚ p K q Ñ F C ˚ p K q b F C ˚ p K q , which is a chain map and thus induces on homology the continuationcoproduct λ cont : F H ˚ p K q Ñ F H ˚ p K q b F H ˚ p K q . Letting the slope of K go to infinity we obtain in the limit the map λ cont : SH ˚ p D ˚ M q Ñ SH ˚ p D ˚ M q b SH ˚ p D ˚ M q , which we call continuation coproduct in symplectic homology . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 33
Conformal annuli.
To prove Theorem 4.1 we will produce, foreach Hamiltonian K “ K µ as in the previous subsection, a chain ho-motopy Θ : F C ˚ p K µ q Ñ M C ˚ p E { q b M C ˚ p E { q satisfying B Θ ` Θ B F “ p Ψ b Ψ q λ cont ´ λ Ψ . The map Θ will be defined by a count of Floer maps to T ˚ M definedover a 2-parametric family of punctured annuli. In this subsection wedescribe the underlying moduli space of conformal annuli.A (conformal) annulus is a compact genus zero Riemann surface withtwo boundary components. By the uniformization theorem (see forexample [10]), each annulus is biholomorphic to r , R s ˆ R { Z with itsstandard complex structure for a unique R ą (conformal)modulus . The exponential map s ` it ÞÑ e π p s ` it q sends the standardannulus onto the annulus A R “ t z P C | ď | z | ď e πR u Ă C . It will be useful to consider slightly more general annuli in the Riemannsphere S “ C Y t8u . A circle in S is the transverse intersection of S Ă R with a plane. We will call a disc in S an open domain D Ă S bounded by a circle, and an annulus in S a set D z D for two discs D, D Ă S satisfying D Ă D (with the induced complex structure). Lemma 4.2.
Every annulus A in S of conformal modulus R can bemapped by a M¨obius transformation onto the standard annulus A R Ă C Ă S above.Proof. Write A “ D z D for discs D, D Ă S . After applying a M¨obiustransformation, we may assume that D is the disc t z P C | | z | ă e πR u .Let D Ă D be the unit disc. There exists a M¨obius transformation φ of D sending a point z P B D to a point z P B D and the positive tangentdirection to B D at z to the positive tangent direction to B D at z .Thus φ sends B D to a circle tangent to B D at z , and since the annuli D z D and D z D both have modulus R we must have φ pB D q “ B D ,hence φ p D q “ D . (cid:3) For each R , the standard annulus r , R s ˆ R { Z carries two canonicalfoliations: one by the line segments r , R s ˆ pt and one by the circlespt ˆ R { Z . Moreover, these two foliations are invariant under the au-tomorphism group of the annulus. Hence by Lemma 4.2 each annulusin S also carries two canonical foliations, one by circle segments con-necting the two boundary components and one by circles, such that thefoliations are orthogonal and the second one contains the two bound-ary loops. These two foliations can be intrinsically described as follows:the automorphism group of an annulus A is Aut p A q » S . The firstfoliation consists of the orbits of the S -action. The second foliation Figure 8.
Conformal annuli and their canonical foliationsis the unique foliation orthogonal to the first one. Its leaves connectthe two boundary components because this is the case for a standardannulus.Figure 8 shows a 1-parametric family of annuli in C whose conformalmoduli tend to 0 together with their canonical foliations. The domainat modulus 0 is the difference of two discs touching at one point, thenode. Putting the node at the origin, the inversion z ÞÑ { z mapsthis domain onto a horizontal strip in C (with the node at ) withits standard foliations by straight line segments and lines. Openingup the node, we can conformally map it onto the standard disc withtwo boundary points corresponding to the node (since the map is nota M¨obius transformation, the two foliations will not be by circle seg-ments). Annuli with aligned marked points.
The relevant domains forour purposes are annuli with 3 marked points, one interior and one oneach boundary component. We require that the 3 points are aligned , bywhich we mean that they lie on the same leaf of the canonical foliationconnecting the two boundary components. (In the next subsectionthe interior marked point will correspond to the input from the Floercomplex and the boundary marked points will be the initial points ofthe boundary loops on the zero section.)Figure 9 shows the moduli space of such annuli with fixed finite con-formal modulus. It is an interval over which the interior marked point
OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 35
Figure 9.
Annuli with aligned marked points and fixed modulusmoves from one boundary component to the other. Each end of the in-terval corresponds to a rigid nodal curve consisting of an annulus withone boundary marked point and a disc with an interior and a boundarymarked point, where the marked point and the node are aligned in theannulus, and the two marked points and the node are aligned in thedisc (i.e., they lie on a circle segment perpendicular to the boundary).Figure 10 shows the moduli space of such annuli with varying confor-mal modulus. It is a pentagon in which we will view the two lowersides as being “horizontal” (although they meet at an actual corner).Then in the vertical direction the conformal modulus increases from0 (on the top side) to (on the two lower sides), while in the hori-zontal direction the interior marked point moves from one boundarycomponent to the other. In all configurations the marked points andnodes are aligned. The interior nodes occuring along the bottom sidescarry asymptotic markers (depicted as arrows) that are aligned withthe boundary marked points. In particular, each interior node comeswith an orientation reversing isomorphism between the tangent circlesmatching the asymptotic markers (this is the “decorated compactifica-tion”).4.3. Floer annuli.
Now we define a moduli space of Floer maps into T ˚ M over the moduli space P of annuli in Figure 10. For this, we Figure 10.
Annuli with aligned marked points andvarying moduluschoose a family of 1-forms β τ , τ P P , with the following properties (seeFigure 11): ‚ dβ τ ď τ ; ‚ β τ equals dt near the (positive) interior puncture, and 2 dt in coordi-nates p s, t q P r , ε qˆ R { Z near each (negative) boundary component,i.e., it has weights 1 , , ‚ on annuli of infinite modulus, β τ has weights at the positive/negativepunctures as shown in the figure.In the figure the black circles are boundary components, blue circlesare interior punctures (viewed as positive/negative when going up-wards/downwards), and red numbers denote the weights. Such a family β τ exists because on each component of each broken curve the sum ofnegative weights is greater or equal to the sum of positive weights.The annuli carry two marked points on their boundary circles (depictedas black dashes) which are aligned with the interior puncture. Again,all interior punctures carry asymptotic markers (not drawn) that arealigned with the boundary marked points, also over broken curves andare matching across each pair of positive/negative punctures.Note that the bottom corner of the pentagon in Figure 10 has beenreplaced by a new side over which the underlying stable domain is fixed,but the weights at the positive/negative puncture vary as depicted with OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 37
Figure 11.
The hexagon of Floer annuli a P r´ , s . Thus the conformal modulus is 0 along the top side, and along the three bottom sides.We fix a nonnegative Hamiltonian K : T ˚ M Ñ R as in § τ P P we denote by Σ τ the corresponding (possibly broken) annulus withone positive interior puncture z ` and two numbered boundary markedpoints z , z on the boundary components C , C , equipped with the1-form β τ . Given x P F C ˚ p K q we define the moduli space P p x q : “ tp τ, u q | τ P P , u : Σ τ Ñ T ˚ M, p du ´ X K b β τ q , “ ,u p z ` q “ x, u p C i q Ă M for i “ , u , where the condition u p z ` q “ x is understood as being C -convergence u p s, ¨q Ñ x as s Ñ 8 in cylindrical coordinates p s, t q P r ,
8q ˆ S near the positive puncture z ` . By Anosov [8], the restriction u | C i canbe uniquely parametrized over r , s as an H -curve proportionally toarclength such that time 0 corresponds to the marked point z i , i “ ,
2. Viewing u | C i with these parametrizations thus yields a boundaryevaluation mapev B : P p x q Ñ Λ ˆ Λ , p τ, u q ÞÑ p u | C , u | C q . Note that this map is also canonically defined over the boundary of P .Indeed, this is clear everywhere except possibly over the two verticalsides where one boundary loop is split into two. There one componentof Σ τ is an annulus without interior puncture, on which the map u is therefore constant (see the next subsection). Hence in the split bound-ary loop one component is constant, and we map it simply to the othercomponent parametrized proportionally to arclength.The expected dimension of P p x q isdim P p x q “ nχ p Σ τ q ` CZ p x q ` dim P “ CZ p x q ` ´ n, where χ p Σ τ q “ ´ P p x q is not transversely cut out over thevertical sides of P . Indeed, the moduli space of non-punctured annuliappearing there has Fredholm index nχ p A q ` “
1, where χ p A q “ A and the ` n `
1, where n is the dimension of the space of constant maps A Ñ M .In the following subsections we explain how to achieve transversality byperturbing the Floer equation by a section in the obstruction bundle.4.4. Moduli problems and obstruction bundles.
To facilitate thediscussion in the next subsection, we introduce in this subsection ageneral setup for moduli problems and obstruction bundles. Our notionof a moduli problem will be a slight generalization of that of a G -moduliproblem in [15] for the case of the trivial group G , which allows us towork with integer rather than rational coefficients.A moduli problem is a quadruple p B , F , S , Z q with the following prop-erties: ‚ p : F Ñ B is a Banach fibre bundle over a Banach manifold; ‚ Z Ă F is a Banach submanifold transverse to the fibres; ‚ S : B Ñ F is a smooth section such that the solution set M : “ S ´ p Z q Ă B is compact and for each b P M the composed operator D b S : T b B T b S ÝÑ T S p b q F ÝÑ T S p b q F { T S p b q Z is Fredholm with constant index ind p S q “ ind p D b S q , and its deter-minant bundledet p S q “ Λ top ker p D S q b Λ top coker p D S q ˚ Ñ M is oriented.A morphism between moduli problems p B , F , S , Z q and p B , F , S , Z q is a pair p ψ, Ψ q with the following properties: ‚ ψ : B ã Ñ B is a smooth embedding; In particular, T z Z Ă T z F is a closed subspace which has a closed complementfor all z P Z . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 39 ‚ Ψ : F Ñ F is a smooth injective bundle map covering ψ such that S ˝ ψ “ Ψ ˝ S , M “ ψ p M q , Z “ Ψ p Z q . Moreover, the linear operators T b ψ : T b B Ñ T ψ p b q B and D z Ψ : T z F { T z Z Ñ T Ψ p z q F { T Ψ p z q Z induce for each b P M isomorphisms T b ψ : ker D b S Ñ ker D ψ p b q S , D S p b q Ψ : coker D b S Ñ coker D ψ p b q S such that the resulting isomorphism from det p S q to det p S q is ori-entation preserving. Proposition 4.3.
Each moduli problem p B , F , S , Z q has a canonical Euler class χ p B , F , S , Z q P H ind p S q p B ; Z q . Moreover, if p ψ, Ψ q is a morphism between moduli problems p B , F , S , Z q and p B , F , S , Z q , then ind p S q “ ind p S q and ψ ˚ ` χ p B , F , S , Z q ˘ “ χ p B , F , S , Z q P H ind p S q p B ; Z q . Proof.
This follows directly from the corresponding results in [15]. Toconstruct the Euler class, we compactly perturb S to a section r S whichis transverse to Z ; then Ă M “ r S ´ p Z q is a compact manifold of di-mension d “ ind p S q which inherits a canonical orientation and thusrepresents a class in H d p B ; Z q , and it is easy to see that this class isindependent of the choice of perturbation. The assertion about mor-phisms is obvious. (cid:3) A special case of a moduli problem arises if F “ E Ñ B is a Banach vector bundle and Z “ Z E is the zero section in E . In this case D b S is the vertical differential of S at b P M “ S ´ p q and we arrive atthe usual notion of a Fredholm section. This is the setup consideredin [15]; the general case can be reduced to this one (via a morphism ofmoduli problems) by passing to the normal bundle of Z .Consider now a moduli problem p B , F , S , Z q such that(i) M “ S ´ p Z q Ă B is a smooth submanifold, and(ii) ker p D b S q “ T b M for each b P M .Then the cokernels coker p D b S q fit together into the smooth obstructionbundle O : “ coker p D S q Ñ M whose rank is related to the Fredholm index of S bydim M “ ind p S q ` rk O . We thus obtain a finite dimensional moduli problem p M , O , , Z O q ,where 0 : M Ñ O denotes the zero section and Z O Ă O its graph. Lemma 4.4.
In the preceding situation there exists a canonical mor-phism of moduli problems p ι, exp q : p M , O , , Z O q Ñ p B , F , S , Z q , where ψ “ ι : M ã Ñ B is the inclusion and exp : O ã Ñ F is a fibrewiseexponential map.Proof. Choose N Ñ Z a smooth Banach vector bundle such that foreach z P Z , N z Ă T z F p p z q and T z F “ T z Z ‘ N z . Since N represents the normal bundle to Z in F , we can assume that D S takes values in N and O is a subbundle of N complementary toim D S . Pick a fibrewise Riemannian metric on F whose exponentialmap restricts to a fibre preserving embeddingexp : O ã Ñ F , O z ã Ñ F p p z q . Now it is easy to check that p ι, exp q with the inclusion ι : M ã Ñ B defines a morphism p M , O , , Z O q Ñ p B , F , S , Z q . (cid:3) In the situation of Lemma 4.4, the Euler class of p B , F , S , Z q is there-fore represented by the zero set η ´ p q of a section η : M Ñ O in theobstruction bundle which is transverse to the zero section. Concretely,keeping the notation from the proof, exp ˝ η defines a section of thefibre bundle F | M Ñ M . We extend the bundle O Ñ M to a bundle r O Ñ r B on a neighbourhood r B Ă B of M and η to a section r η of thebundle r O Ñ r B vanishing near the boundary of r B . Then the perturbedsection r S “ S ` exp ˝ r η of F Ñ B is transverse to Z and its solutionset r S ´ p Z q represents the Euler class of p B , F , S , Z q . Remark 4.5 (orientations) . In the situation of Lemma 4.4 we are givenan orientation of(14) det p S q “ Λ top T M b Λ top O ˚ . Let now η : M Ñ O be a section transverse to the zero section. Itszero set A : “ η ´ p q Ă M is a submanifold and at each b P A thelinearization D b η : T b M Ñ O b is surjective with kernel ker D b η “ T b A ,so we get a canonical isomorphism of line bundlesΛ top T M | A – Λ top T A b Λ top O | A Ñ A . Combined with (14) this yields a canonical isomorphismΛ top T A – det p S q| A , so an orientation of det p S q induces an orientation of A . In the caseind p S q “ O “ dim M and an orientation of det p S q induces an isomorphismΛ top T M – Λ top O . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 41
For b P η ´ p q we define the sign σ p b q to be ` D b η : T b M – ÝÑ O b preserves orientations, and ´ χ p O q “ ÿ b P η ´ p q σ p b q is the Euler number of the obstruction bundle O Ñ M .Finally, consider a moduli problem p B , F , S , Z q which splits as follows: ‚ p “ p p , p q : F “ F ˆ B F Ñ B ; ‚ Z “ Z ˆ B Z ; ‚ S “ S ˆ S for sections S i : B i Ñ F i such that S is transverse to Z . Lemma 4.6.
In the situation above there exists a reduced moduli prob-lem p B , F , S , Z q “ ` S ´ p Z q , F | B , S | B , Z | B ˘ and a morphism p ψ, Ψ q of moduli problems from p B , F , S , Z q to p B , F , S , Z q , with ψ : B ã Ñ B the inclusion and Ψ p f q “ ` f , S ˝ p p f q ˘ .Proof. Since S is transverse to Z , it follows that B Ă B is a submani-fold and p B , F , S , Z q defines a moduli problem. Now it follows directlyfrom the definitions that p ψ, Ψ q as in the lemma induces for b P B thecanonical identities T b ψ : ker D b S “ ker D b S X ker D b S “ ker D ψ p b q S ,D S p b q Ψ : coker D b S “ coker p D b S | ker D b S q “ coker D ψ p b q S , hence it defines a morphism of moduli problems. (cid:3) Constant Floer annuli.
In this subsection we apply the resultsof the previous subsection to moduli spaces of annuli. We begin witha rather general setup. Let p Σ , j q be a compact Riemann surfacewith boundary, and p V, J q be an almost complex manifold with a half-dimensional totally real submanifold L Ă V . For m P N and p P R with mp ą B “ W m,p ` p Σ , B Σ q , p V, L q ˘ and the Banach space bundle E Ñ B whose fibre over u P B is E u “ W m ´ ,p ` Σ , Hom , p T Σ , u ˚ T V q ˘ . Denote Z E the zero section. The Cauchy-Riemann operator B u “ p du q , “ ` du ` J p u q ˝ du ˝ j ˘ defines a Fredholm section B : B Ñ E . Assuming a setup in whichthe space of solutions B ´ p Z E q is compact (e.g. if J is tamed by an exact symplectic structure on V , the totally real submanifold L is exactLagrangian, and Σ has a compact group of automorphisms), we obtaina moduli problem p B , E , ¯ B , Z E q . Constant annuli of positive modulus.
Now we apply the preced-ing discussion to the moduli space of constant annuli appearing in theprevious subsection. Consider a fixed annulus p Σ , j q of finite conformalmodulus R ą
0, equipped with a 1-form β as above satisfying dβ ď β “ dt in cylindrical coordinates near the two (negative) bound-ary loops. Let K be the nonnegative Hamiltonian from § B K u : “ p du ´ X K b β q , defines a Fredholm section inthe appropriate bundle E Ñ B over the Banach manifold B “ W m,p ` p Σ , B Σ q , p T ˚ M, M q ˘ . We denote its zero set by M : “ B ´ K p q . For u P M the usual energyestimate (see e.g. [30]) gives E p u q “ ż Σ | du ´ X K p u q b β | vol Σ ď ´ A K p u | B Σ q “ , where the Hamiltonian action of u | B Σ vanishes because both the Li-ouville form and the Hamiltonian K vanish on the zero section M .This implies that du ´ X K p u q b β ”
0. Since X K vanishes near thezero section, it follows that du ” B Σ and therefore, by uniquecontinuation, u is constant equal to a point in M . Hence the modulispace M “ M consists of points in M , viewed as constant maps Σ Ñ M . Since X K vanishes near the zero section, the Floer operator B K agrees with theCauchy-Riemann operator B near M , so we can and will replace B K by B in the following discussion of obstruction bundles.We identify Σ with the standard annulus r , R s ˆ R { Z and its trivialtangent bundle T Σ “ Σ ˆ C . Consider a point u P M , viewed as aconstant map u : Σ Ñ M . We identify T ˚ u M “ R n , T u M “ i R n , T u p T ˚ M q “ C n . Then we have T u B “ W m,p ` p Σ , B Σ q , p C n , i R n q ˘ , E u “ W m ´ ,p ` Σ , Hom , p C , C n q ˘ “ W m ´ ,p p Σ , C n q , where for the last equality we use the canonical isomorphismHom , p C , C n q – ÝÑ C n , η ÞÑ η pB s q . With these identifications, the linearized Cauchy-Riemann operatorreads D u B : W m,p ` p Σ , B Σ q , p C n , i R n q ˘ Ñ W m ´ ,p p Σ , C n q , ξ ÞÑ B s ξ ` i B t ξ. OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 43
An easy computation using Fourier series (see [12]) shows thatker p D u Bq “ i R n “ T u M, coker p D u Bq “ R n “ T ˚ u M. So the Cauchy-Riemann operator, and thus the Floer operator, satisfiesconditions (i) and (ii) in the previous subsection with the obstructionbundle O “ coker p D B K q – T ˚ M Ñ M “ M , and Lemma 4.4 implies Corollary 4.7.
In the preceding situation there exists a canonical mor-phism of moduli problems p ι, I q : p M, T ˚ M, , Z T ˚ M q Ñ p B , E , B , Z E q , where ι : M ã Ñ B is the inclusion as constant maps and I converts acotangent vector into a constant p , q -form. (cid:3) Note in particular that B K has index zero. A section in the obstruc-tion bundle transverse to the zero section corresponds under the iso-morphism O – T ˚ M to a 1-form η on M with nondegenerate zeroes p , . . . , p k , and the zero set of the perturbed Floer operator B K ` r η consists of p , . . . p k viewed as constant maps Σ Ñ M . Having chosenthe orientation of det p ¯ Bq to be induced by the canonical isomorphism T M – T ˚ M , we obtain that the signed count k ÿ i “ σ p p i q “ χ p T ˚ M q agrees with the Euler number of T ˚ M . Note that the Euler numberof T ˚ M equals the Euler characteristic of M (this follows from thecanonical isomorphism T ˚ M – T M and the Poincar´e-Hopf theorem).
Constant annuli of modulus zero.
Annuli of conformal moduluszero can be viewed as moduli problems in two equivalent ways. For thefirst view, we take as domain the compact region A Ă C bounded bytwo circles touching at one point, the node. Given p V, J q and L Ă V asabove, we therefore obtain a moduli problem p B A , E A , S A , Z E A q with B A “ W m,p ` p A, B A q , p V, L q ˘ , E Au “ W m ´ ,p ` A, Hom , p T A, u ˚ T V q ˘ , the Cauchy-Riemann operator S A “ B A , and the zero section Z E A Ă E A .For the second view, we take as domain the closed unit disk D Ă C with ˘ i viewed as nodal points which are identified. This gives rise to a moduli problem p B D , F D , S D , Z D q with B D “ W m,p ` p D, B D q , p V, L q ˘ , F D “ E D ˆ p L ˆ L q , E Du “ W m ´ ,p ` D, Hom , p T D, u ˚ T V q ˘ , S D “ B D ˆ ev : B D Ñ E D ˆ p L ˆ L q , ev p u q “ ` u p i q , u p´ i q ˘ , Z D “ Z E D ˆ ∆ , ∆ “ tp q, q q | q P L u Ă L ˆ L. Note that the indices of the two moduli problems agree,ind p S D q “ ind pB D q ´ n “ ind p S A q . Let φ : D Ñ A be a continuous map which maps ˘ i onto the nodalpoint and is otherwise one-to-one, and which is biholomorphic in theinterior. Then composition with φ defines a diffeomorphism B D Ą ev ´ p ∆ q – B A (where we use as area form on A the pullback under φ of an area formon D ). Since ev : B D Ñ L ˆ L is transverse to the diagonal ∆, weare in the situation of Lemma 4.6. We conclude that there exists amorphism of moduli problems p ψ, Ψ q : p B A , E A , S A , Z E A q Ñ p B D , F D , S D , Z D q , where ψ : B A “ ev ´ p ∆ q ã Ñ B D is the inclusion and Ψ p u ; η q “ ` u ; η, ev p u q ˘ .Now we specialize to the case p V, L q “ p T ˚ M, M q with its canonical al-most complex structure J . Then both solution spaces M A “ pB A q ´ p q and M D “ p S D q ´ p Z D q “ pB D q ´ p q “ M consist of constant mapsto M . Morover, in view of the preceding discussion and the fact thatthe Cauchy-Riemman operator B D : B D Ñ E D over the disk is trans-verse to the zero section, they both satisfy the hypotheses (i) and (ii)of Lemma 4.4, so combined with the preceding discussion we obtain Corollary 4.8.
There exists a commuting diagram of morphisms ofmoduli problems p B A , E A , S A , Z E A q p ψ, Ψ q / / p B D , F D , S D , Z D qp M, T ˚ M, T ˚ M , Z T ˚ M q p ι A , Ψ A q O O p id , exp q / / p M, M ˆ M, ev , ∆ q p ι D , Ψ D q O O where ι A : M ã Ñ B A and ι D : M ã Ñ B D are the inclusions as constantmaps, the bundle M ˆ M Ñ M is given by projection onto the first We may construct φ as a composition φ “ ϑ ˝ log ˝ ψ where ψ is the M¨obiustransformation sending D onto the upper halfplane H with ψ p´ i q “ ψ p i q “ 8 ,log is the logarithm sending H onto the strip S “ t z P C | ď Im z ď π u , and ϑ isthe M¨obius transformation sending S onto A . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 45 factor, and exp : T ˚ M Ñ M ˆ M is the composition of the isomor-phism T ˚ M – T M induced by a metric on M with the exponential map T M Ñ M ˆ M . Thus the Euler class of each of these moduli problemsis represented by the nondegenerate zeroes p , . . . , p k of a -form η on M (or equivalently, of a vector field v on M ), with signs that add up(up to a global sign) to the Euler characteristic χ of M . (cid:3) Poincar´e duality coproduct equals loop coproduct.
In thissubsection we prove Theorem 4.1.For x P F C ˚ p K q consider the moduli space P p x q of Floer annuli de-scribed in § B : P p x q Ñ Λ ˆ Λ.Pick a 1-form η on M with nondegenerate zeroes p , . . . , p k . As in § η as a section of the obstruction bundle over the vertical sidesof the hexagon in Figure 11. We extend this section by a cutoff functionto a section r η over the whole hexagon and add it as a right hand sideto the Floer equation. We choose the data such that the moduli space P p x q is transversely cut out, and thus defines a compact manifold withcorners of dimension CZ p x q ` ´ n .We may assume without loss of generality that M is connected. Wepick a C -small Morse function V : M Ñ R with a unique maximumat q P M such that p , . . . , p k flow to q under the positive gradientflow of V . Let M C ˚ p S q denote the Morse complex of the perturbedenergy functional S : Λ Ñ R , S p q q : “ ż ` | q | ´ V p q q ˘ dt (note that there is no factor 1 { | q | ). For x P F C ˚ p K q and a, b P M C ˚ p S q we define P p x ; a, b q : “ tp τ, u q P P p x q | ev B p u q P W ` p a q ˆ W ` p b qu , where W ` p a q is the stable manifold of a with respect to the negativegradient flow of S . Recall that the boundary evaluation map involvesreparametrization of the boundary loops proportionally to arclength.For generic choices, these are manifolds of dimensiondim P p x ; a, b q “ CZ p x q ´ ind p a q ´ ind p b q ` ´ n. If the dimension is 0 these spaces are compact and their signed countsΘ p x q : “ ÿ a,b P dim “ p x ; a, b q a b b define a degree 2 ´ n mapΘ : F C ˚ p K q Ñ M C ˚ p S q b M C ˚ p S q . Next we consider a 1-dimensional moduli space P dim “ p x ; a, b q and com-pute its boundary. Besides splitting off index 1 Floer cylinders and neg-ative gradient flow lines, which give rise to the term B Θ ` Θ B F , there are contributions from the sides of the hexagon in Figure 11 which weanalyze separately. Note that the indices now satisfyCZ p x q ´ ind p a q ´ ind p b q “ n ´ . Vertical left side: Here the broken curves consist of a half-cylinder at-tached at a boundary node to an annulus without interior puncture,where the two boundary loops flow into a, b under the negative gra-dient flow of S . By the discussion in § r ,
8s ˆ η ´ p q , where r , encodes the conformal modulus and η ´ p q consists of the points p , . . . , p k (with signs σ p p i q ). In partic-ular, we must have b “ q and therefore ind p b q “ ind p q q “
0. Thehalf-cylinders belong to the moduli space M p x ; a q “ t u : r ,
8q ˆ S Ñ T ˚ M | p du ´ X K b β q , “ r η,u p8 , ¨q “ x, u p , ¨q P W ` p a qu . They carry a boundary nodal point which is aligned with the boundarymarked point p , q and the puncture at , and is therefore given by p , { q . The evaluation at the nodal point defines an evaluation mapev { : M p x ; a q Ñ M, u ÞÑ u p , { q . For the broken curve to exist this evaluation map must meet one of theconstant annuli, i.e. one of the points p , . . . , p k P M , which genericallydoes not happen becausedim M p x ; a q “ CZ p x q ´ ind p a q “ n ´ . Hence the vertical left side gives no contribution to the boundary.Vertical right side: Similarly, the vertical right side gives no contribu-tion to the boundary.Lower left side: Here the broken curves consist of a disc with two inte-rior punctures, one positive and one negative, attached at its negativepuncture to the positive puncture of a half-cylinder along an orbit in
F C ˚ p´ K q , where the two boundary loops flow into a, b under the nega-tive gradient flow of S . By choosing the 1-form β equal to dt on a longcylindrical piece of the half-cylinder, we can achieve that these half-cylinders are in one-to-one correspondence with broken curves consist-ing of a cylinder with weights p´ , q and a half-cylinder with weights p , q , see Figure 12. As such, their count corresponds to the composi-tion F C ˚ p´ K q c ÝÑ F C ˚ p K q Ψ ÝÑ M C ˚ p S q of the continuation map c from § § c is Rχq and Ψ p q q “ q (viewed as a constantloop), this shows that the contribution from the lower left side landsin M C ˚ p S q b Rχq . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 47
Figure 12.
Degenerating the half-cylindersLower right side: Similarly, the contribution from the lower right sidelands in
Rχq b M C ˚ p S q .Let us draw some conclusion from the discussion so far. For this,note that for x “ q the punctured annuli in P p q ; a, b q are constantequal to q , so P p q ; a, b q can only be nonempty if a “ b “ q . Sincedim P p q ; q , q q “ ´ n is zero iff n “
2, this shows that Θ p q q “ q b q if n “
2, and Θ p q q “ Rχq to Rχq b Rq and thus descends to a map between the reduced chaincomplexes Θ : F C ˚ p K q Ñ M C ˚ p S q b M C ˚ p S q . The preceding discussion shows that this map satisfies B Θ ` Θ B F “ Θ top ´ Θ bottom , where Θ top and Θ bottom are the degree 1 ´ n maps arising from the con-tributions of the top and bottom sides of the hexagon to the boundaryof P dim “ p x ; a, b q which we discuss next.Bottom side: The family of broken curves on the bottom side can bedeformed in an obvious way to the family of broken curves shown inFigure 13. Since the half-cylinders with weights p , q define the mapΨ and the family of 3-punctured spheres above them defines the con-tinuation coproduct λ cont from § bottom is chainhomotopic to p Ψ b Ψ q λ cont . Figure 13.
Degenerating the curves on the bottom sideTop side: The family on the top side of the hexagon consists of punc-tured annuli of modulus 0, i.e., punctured discs with two nodal pointson the boundary that are identified to a node. Moreover, the bound-ary carries two marked points that are separated by the nodal pointsand aligned with the interior puncture. We wish to relate this familyto the loop coproduct, but for this we face two problems: First, theboundary loops carry two marked points whereas the loops for the ho-mology coproduct carry only one (the initial time t “ above the dashed line connecting the two boundary markedpoints; we could equally well have taken the mirror hexagon where theinterior puncture moves below the dashed line. OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 49
Figure 14.
Floer annuli of modulus zeroThe hexagon in Figure 14 defines a deformation from the bottom(black) side to the top side (drawn in red). The configurations inthis figure are to be interpreted as follows. ‚ Each configuration has two boundary loops obtained by going aroundin the counterclockwise direction: the first loop from the bottom to thetop nodal point, and the second one from the top to the bottom nodalpoint. Each boundary loop carries a marked point. As before, eachboundary loop of the zero section is reparametrized proportionally toarclength and then flown into a critical point on Λ under the negativegradient flow of the functional S : Λ Ñ R . ‚ In each configuration the unique component carrying the interiorpuncture (which may be nonconstant) is drawn as a large disc, so thesmall discs are all constant. In particular, each small disc carrying thetwo nodal points is a constant annulus of modulus zero. Under the per-turbation of the Cauchy-Riemann equation described in Corollary 4.8,such a component lands on the transverse zeroes p , . . . , p k of a 1-form η and thus flows into the basepoint q . Since the signs add up to ˘ χ ,this shows that all configurations on the upper and lower left sides landin Rχq b M C ˚ p S q , while those on the upper and lower right sides landin M C ˚ p S q b Rχq . In particular, all contributions from the upper andlower left and right sides vanish in the reduced Morse complex. Thus Figure 15.
Interpreting the curves on the top sidethe hexagon in Figure 14 provides a chain homotopy on reduced com-plexes from Θ top (defined by the bottom side) to the operation r Θ top defined by the top side. ‚ Consider now the top side. Since both marked points and the blacknodal point lie on the same constant component, we can remove thiscomponent and replace the three points by one nodal/marked point asshown in Figure 15. The boundary of these configurations consists ofloops q : r , s Ñ M with one (black) marked/nodal point at time 0and an additional (red) nodal point at time s which moves from 0 to 1as we traverse the side from left to right. In view of Corollary 4.8 andRemark 4.5, the map r Θ top : F C ˚ p K q Ñ M C ˚ p S q b M C ˚ p S q is definedby counting isolated configurations consisting of punctured discs asin the definition of the moduli spaces M p x q from § S starting at the de-concatenated loops. Now we deform r Θ top once moreby inserting a negative gradient trajectory of S of finite length T ě T Ñ 8 this becomesthe chain map Ψ :
F C ˚ Ñ M C ˚ p Λ q followed by the Morse theoreticcoproduct λ . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 51
Altogether, we obtain a chain homotopy on reduced complexes fromΘ top to λ Ψ. Together with the preceding discussion this concludes theproof of Theorem 4.1. (cid:3) Relation to other Floer-type coproducts
In this section we restrict the continuation coproduct λ cont of the pre-vious section to positive action symplectic homology SH ą ˚ p V q . Wespecialize to the case of a unit cotangent bundle V “ D ˚ M and werelate it to the Abbondandolo–Schwarz coproduct λ F defined in [4].In [4], Abbondandolo and Schwarz defined the ring isomorphism Ψ ˚ : SH ˚ p D ˚ M q – ÝÑ H ˚ Λ and they asserted [4, Theorem 1.4] that its re-duction modulo constant loops Ψ ą ˚ : SH ą ˚ p D ˚ M q – ÝÑ H ˚ p Λ , Λ q in-tertwines the coproduct λ F with the homology coproduct. Since noproof of this result has appeared, we give a proof in this section. Wewill actually give two proofs: the first one uses Theorem 4.1 and theidentification λ cont “ λ F , the second one uses a direct argument andsuitable interpolating moduli spaces.This section is structured as follows. In § λ w , which coincides with λ cont [16, Lemma 7.2] and which can be more easily related to λ F .In § λ F . In § λ F is equal to λ w . In § λ F corresponds to the homology coproduct λ under theisomorphism SH ą ˚ p D ˚ M q – H ˚ p Λ , Λ q .The situation is summarized in the following diagram.(15) λ § . . λ F § . λ w [16] λ cont . Remark 5.1.
The chain of equalities (15) proves Theorem 4.1 on pos-itive action homology, and our first attempt at the full proof was byextending this chain of equalities to reduced homology. While thismay be possible, we ultimately gave up on it and instead resorted tothe direct proof presented in § M is orientedand we use untwisted coefficients in a commutative ring R ; the neces-sary adjustments in the nonorientable case and with twisted coefficientsare explained in Appendix A. We denote S : “ R { Z and Λ : “ W , p S , M q . Varying weights coproduct.
We recall here the definition ofthe varying weights coproduct λ w on SH ą ˚ p V q from [16, § SH ă ˚ p V, B V q ,we will recap in some detail the necessary notation and arguments. Theconstruction goes back to Seidel, see also [21]. We work with a Liou-ville domain V of dimension 2 n , the symplectic completion is denoted p V “ V Y r ,
8q ˆ B V and the radial coordinate in the positive sym-plectization r ,
8q ˆ B V is denoted r .Let Σ be the genus zero Riemann surface with three punctures, one ofthem labeled as positive χ ` and the other two labeled as negative υ ´ , ζ ´ , endowed with cylindrical ends r ,
8q ˆ S at the positive punctureand p´8 , s ˆ S at the negative punctures. Denote p s, t q , t P S theinduced cylindrical coordinates at each of the punctures. Consider asmooth family of 1-forms β τ P Ω p Σ q , τ P p , q satisfying the followingconditions: ‚ (nonpositive) dβ τ ď ‚ (weights) β τ “ dt near each of the punctures; ‚ (interpolation) we have β τ “ τ dt on r´ R p τ q , s ˆ S in thecylindrical end near υ ´ , and β τ “ p ´ τ q dt on r´ R p ´ τ q , s ˆ S inthe cylindrical end near ζ ´ , for some smooth function R : p , q Ñ R ą . In other words, the family t β τ u interpolates between a 1-formwhich varies a lot near υ ´ and very little near ζ ´ , and a 1-formwhich varies a lot near ζ ´ and very little near υ ´ ; ‚ (neck stretching) we have R p τ q Ñ `8 as τ Ñ τ close to 0 we have β τ “ f τ p s q dt in the cylindrical end at the negative puncture υ ´ , with f τ ď f τ “ ´8 , and f τ “ τ on r´ R p τ q , s , and similarly for τ close to 1 in the cylindrical end at the negative puncture ζ ´ .Let H : p V Ñ R be a convex smoothing localized near B V of a Hamil-tonian which is zero on V and linear with respect to r with positiveslope on r ,
8q ˆ B V . The Hamiltonian H further includes a smalltime-dependent perturbation localized near B V , so that all 1-periodicorbits are nondegenerate. Assume the slope is not equal to the periodof a closed Reeb orbit. Denote P p H q the set of 1-periodic orbits of H .The elements of P p H q are contained in a compact set close to V .Let J “ p J ζτ q , ζ P Σ, τ P p , q be a generic family of compatiblealmost complex structures, independent of τ and s near the punctures,cylindrical and independent of τ and ζ in the symplectization r ,
8q ˆ
OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 53 B V . For x, y, z P P p H q denote M p x ; y, z q : “ p τ, u q ˇˇ τ P p , q , u : Σ Ñ p V , p du ´ X H b β τ q , “ , lim s Ñ`8 ζ “p s,t qÑ χ ` u p ζ q “ x p t q , lim s Ñ´8 ζ “p s,t qÑ υ ´ u p ζ q “ y p t q , lim s Ñ´8 ζ “p s,t qÑ ζ ´ u p ζ q “ z p t q ( . In the symplectization r ,
8q ˆ B V we have H ě 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 ; y, z q “ CZ p x q ´ CZ p y q ´ CZ p z q ´ n ` . When it has dimension zero the moduli space M “ p x ; y, z q is com-pact. When it has dimension 1 the moduli space M “ p x ; y, z q admitsa natural compactification into a manifold with boundary B M “ p x ; y, z q “ ž CZ p x q“ CZ p x q´ M p x ; x q ˆ M “ p x ; y, z q> ž CZ p y q“ CZ p y q` M “ p x ; y , z q ˆ M p y ; y q> ž CZ p z q“ CZ p z q` M “ p x ; y, z q ˆ M p z ; z q> M τ “ p x ; y, z q > M τ “ p x ; y, z q . Here M τ “ p x ; y, z q and M τ “ p x ; y, z q denote the fibers of the first pro-jection M “ p x ; y, z q Ñ p , q , p τ, u q ÞÑ τ near 1, respectively near0. (By a standard gluing argument the projection is a trivial fibrationwith finite fiber near the endpoints of the interval p , q .)Consider the degree ´ n ` λ w : F C ˚ p H q Ñ F C ˚ p H q b F C ˚ p H q defined on generators by λ w p x q “ ÿ CZ p y q` CZ p z q“ CZ p x q´ n ` M “ p x ; y, z q y b z, where M “ p x ; y, z q denotes the signed count of elements in the0-dimensional moduli space M “ p x ; y, z q . Consider also the degree ´ n operations λ wi : F C ˚ p H q Ñ F C ˚ p H q b F C ˚ p H q , i “ , λ wi p x q “ ÿ CZ p y q` CZ p z q“ CZ p x q´ n M τ “ i p x ; y, z q y b z, where M τ “ i p x ; y, z q denotes the signed count of elements in the 0-dimensional moduli space M τ “ i p x ; y, z q .Denote by B F the Floer differential on the Floer complex of H . Theformula for B M “ p x ; y, z q translates into the algebraic relation(16) B F λ w ` λ w pB F b id ` id b B F q “ λ w ´ λ w . We now claim thatIm p λ w q Ă F C “ ˚ p H q b F C ˚ p H q , Im p λ w q Ă F C ˚ p H q b F C “ ˚ p H q . To prove the claim for λ w , note that this map can be expressed as acomposition p c b id q˝ λ , where λ : F C ˚ p H q Ñ F C ˚ p τ H qb F C ˚ p H q is apair-of-pants coproduct with τ ą c : F C ˚ p τ H q Ñ F C ˚ p H q is a continuation map. Taking into account that τ H has no nontrivial1-periodic orbits for τ small, and because the action decreases alongcontinuation maps, we obtain c p F C ˚ p τ H qq Ă F C “ ˚ p H q , which provesthe claim. The argument for λ w is similar.It follows that λ w induces a degree ´ n ` λ w : F C ą ˚ p H q Ñ F C ą ˚ p H q b F C ą ˚ p H q . Passing to the limit as the slope of H goes to `8 we obtain the degree ´ n ` varying weights coproduct λ w on SH ą ˚ p V q . Proposition 5.2 ([16, Lemma 7.2]) . The continuation coproduct andthe varying weights coproduct coincide on SH ą ˚ p V q : λ cont “ λ w . (cid:3) Abbondandolo–Schwarz coproduct.
In this subsection we re-call from [4] the definition of a secondary pair-of-pants product onFloer homology of a cotangent bundle, which we will refer to as the
Abbondandolo–Schwarz coproduct λ F . We recall the notation and con-ventions from § § H p q, p q “ ε | p | ` V p q q for a small ε ą V : M Ñ R such that all nonconstant critical points of A H have action larger than min V .For x, y, z P Crit p A H q set (see Figure 16) M ,F p x ; y, z q : “ p τ, u, v, w q ˇˇ τ P r , s , u : r ,
8q ˆ S Ñ T ˚ Mv, w : p´8 , s ˆ S Ñ T ˚ M, B H u “ B H v “ B H w “ ,u p`8 , ¨q “ x, v p´8 , ¨q “ y, w p´8 , ¨q “ z,v p , t q “ u p , τ t q , w p , t q “ u p , τ ` p ´ τ q t q ( . Note that the matching conditions imply u p , τ q “ u p , q . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 55 w τ u xzy v
Figure 16.
The moduli spaces M ,F p x ; y, z q . Lemma 5.3 ([4, § . For generic choices of Hamiltonian and almostcomplex structure the space M ,F p x ; y, z q is a transversely cut out man-ifold of dimension dim M ,F p x ; y, z q “ CZ p x q ´ CZ p y q ´ CZ p z q ´ n ` . (cid:3) The dimension of M ,F p x ; y, z q is calculated in [4] using an equivalentdescription of the moduli space as follows. Define r v, r w : p´8 , s ˆr , s Ñ T ˚ M by r v p s, t q “ v p s, t q and r w p s, t q “ w p s, t q , and also r y, r z : r , s Ñ T ˚ M by r y p t q “ y p t q and r z p t q “ z p t q . Then there is a canonicalidentification between elements of M ,F p x ; y, z q and elements of Ă M ,F p x ; r y, r z q : “ p τ, u, r v, r w q ˇˇ τ P r , s , u : r ,
8q ˆ S Ñ T ˚ M r v, r w : p´8 , s ˆ r , s Ñ T ˚ M, B H u “ B H r v “ B H r w “ ,u p`8 , ¨q “ x, r v p´8 , ¨q “ r y, r w p´8 , ¨q “ r z, `r v p s, q , C r v p s, q ˘ P N ˚ ∆ , ` r w p s, q , C r w p s, q ˘ P N ˚ ∆ , r v p , t q “ u p , τ t q , r w p , t q “ u p , τ ` p ´ τ q t q ( . Here C : T ˚ M Ñ T ˚ M is the antisymplectic involution p q, p q ÞÑp q, ´ p q , ∆ Ă M ˆ M is the diagonal, and N ˚ ∆ Ă T ˚ p M ˆ M q its conor-mal bundle. The space Ă M ,F p x ; r y, r z q is a moduli space with jumpingLagrangian boundary conditions as in [3], so for generic H and J it is atransversely cut out manifold. Its dimension is given by the Fredholmindex of the linearized problem [4, (37)].If M ,F p x ; y, z q has dimension zero it is compact and defines a map λ F : F C ˚ Ñ p
F C b F C q ˚´ n ` , x ÞÑ ÿ y,z M ,F dim “ p x ; y, z q y b z. If it has dimension 1 it can be compactified to a compact 1-dimensionalmanifold with boundary B M ,F dim “ p x ; y, z q “ ž CZ p x q“ CZ p x q´ M p x ; x q ˆ M ,F dim “ p x ; y, z q> ž CZ p y q“ CZ p y q` M ,F dim “ p x ; y , z q ˆ M p y ; y q> ž CZ p z q“ CZ p z q` M ,F dim “ p x ; y, z q ˆ M p z ; z q> M ,Fτ “ p x ; y, z q > M ,Fτ “ p x ; y, z q . Here the first three terms correspond to broken Floer cylinders and thelast two terms to the intersection of M ,F p x ; y, z q with the sets t τ “ u and t τ “ u , respectively. So we have(18) pB F b id ` id b B F q λ F ` λ F B F “ λ F ´ λ F , where for i “ , λ Fi : F C ˚ Ñ p
F C b F C q ˚´ n , x ÞÑ ÿ y,z M ,Fτ “ i p x ; y, z q y b z. Let us look more closely at the map λ F . For τ “ M ,F p x ; y, z q imply that w p , t q “ u p , q is a constant loop.For action reasons z must then be a critical point, so that Im p λ F q Ă F C ˚ p H qb F C “ ˚ p H q . Similarly we have Im p λ F q Ă F C “ ˚ p H qb F C ˚ p H q ,and therefore λ F descends to a chain map(19) λ F : F H ą ˚ Ñ p
F H ą b F H ą q ˚´ n ` with F C ą ˚ “ F C ˚ p H q{ F C “ ˚ p H q . Note that we have F H ą ˚ p H q – SH ą ˚ p D ˚ M q for the quadratic Hamiltonians considered in this section.5.3. Varying weights coproduct equals Abbondandolo–Schwarzcoproduct.Proposition 5.4.
Let M be a closed connected oriented manifold. Thesecondary coproducts λ w defined via (17) and λ F defined via (19) agreeon SH ą ˚ p D ˚ M q .Proof. We assume without loss of generality that the Hamiltonian usedin the definition of the coproduct λ F is the same as the one used inthe definition of the coproduct λ w , i.e. a convex smoothing of a Hamil-tonian which vanishes on D ˚ M and is linear with respect to the radialcoordinate r “ | p | outside of D ˚ M . The point of the proof is to ex-hibit the Floer problem defining the moduli spaces M ,F p x ; y, z q for λ F as a limiting case of the Floer problem defining the moduli spaces M p x ; y, z q for λ w .Note first that for 0-dimensional moduli spaces M ,F dim “ p x ; y, z q we canrestrict τ to p , q . Given τ P p , q a triple p u, v, w q as in the definition OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 57 of M ,F p x ; y, z q can be interpreted as a single map ˜ u : Σ Ñ T ˚ M satisfying p d ˜ u ´ X H b β τ q , “
0, where Σ is a Riemann surface and β τ is a 1-form explicitly described as follows. The Riemann surface isΣ “ R ˆ r´ τ, s > R ˆ r , ´ τ s { „ with p s, ´ τ q „ p s, ´ τ q , p s, ´ q „ p s, ` q for s ě , p s, ´ τ q „ p s, ´ q , p s, ` q „ p s, ´ τ q for s ď . (We use the notation p s, ´ q for points in R ˆ t u Ă Bp R ˆ r´ τ, sq ,and p s, ` q for points in R ˆ t u Ă Bp R ˆ r , ´ τ sq .) This is a smoothRiemann surface with canonical cylindrical ends r ,
8q ˆ S at thepositive puncture and p´8 , s ˆ R { τ Z and p´8 , s ˆ R {p ´ τ q Z at thenegative punctures. See Figure 17. p , q R {p ´ τ q Z R { ZR { τ Z , τ P p , q Figure 17.
A pair-of-pants Σ with large cylindrical ends.A conformal parametrization of Σ near the point p , q is induced fromthe map C Ñ C , z ÞÑ z . The Riemann surface Σ carries a canonicalsmooth closed 1-form dt . Upon identifying the cylindrical ends at thenegative punctures with p´8 , s ˆ S , this canonical 1-form becomesequal to τ dt , respectively p ´ τ q dt at those punctures. The 1-form β τ is defined to be the discontinuous dt on the cylindricalend r ,
8q ˆ S at the positive puncture, equal to τ dt on the cylindri-cal end p´8 , s ˆ R { τ Z at the first negative puncture, and equal to ´ τ dt on the cylindrical end p´8 , s ˆ R {p ´ τ q Z at the second neg-ative puncture. Equivalently, upon normalizing the cylindrical ends at Consider the half-pair-of-pants Σ “ R ˆ r´ τ, s > R ˆ r , ´ τ s { „ , where p s, ´ q „ p s, ` q for s ě
0. A conformal parametrization near p , q is given bythe map z ÞÑ z defined in a neighborhood of 0 P t Re z ě u . This map actu-ally establishes a global conformal equivalence between H “ t z P C : Re z ě , p Re z qp Im z q P r´ τ, ´ τ su and Σ . The Riemann surface Σ admits a naturalpresentation as the gluing of two copies of Σ . Accordingly, it can be identifiedto H Y ´ H { „ where the equivalence relation „ stands for suitable identifica-tions of boundary components. The map z ÞÑ z defined in a neighborhood of0 P H Y ´ H { „ provides a conformal parametrization of Σ near the point p , q . Read through the identification of Σ with H Y ´ H { „ , this is 2 d p xy q in aneighborhood of 0. the negative punctures into p´8 , s ˆ S , the 1-form β τ is simply dt .This discontinuous 1-form β τ can be interpreted as a limit of 1-formswhich are obtained by interpolating from τ dt and p ´ τ q dt (near 0)towards dt (near ´8 ) in the normalized cylindrical ends at the nega-tive punctures, where the interpolation region shrinks and approaches s “
0. It was noted in § Ă M ,F p x ; r y, r z q with jumping Lagrangian boundary conditions.The Fredholm problem before the limit is naturally phrased in termsof the Riemann surface Σ without boundary, but it can be reinter-preted as a problem with Lagrangian boundary conditions by cuttingΣ open along t s “ u . As such, it converges in the limit to the Fred-holm problem with jumping Lagrangian boundary conditions describedabove. By regularity and compactness, the two Fredholm problems areequivalent near the limit, and the corresponding counts of elements in0-dimensional moduli spaces are the same. (cid:3) Abbondandolo–Schwarz coproduct equals loop coproduct.
In order to distinguish them from the corresponding operations on theMorse complex, we will decorate the operations B F , λ F on the Floercomplex by an upper index F .Recall the Hamiltonian H : S ˆ T ˚ M Ñ R from § L : S ˆ T M Ñ R from § § M C ˚ of the action func-tional S L which we will use freely. In particular, B denotes the Morseboundary operator and r λ the coproduct from Remark 2.5.We assume that M is oriented and we use the Morse complex twistedby the local system σ obtained by transgressing the second Stiefel-Whitney class.Following [4], for x P Crit p A H q and a P Crit p S L q we define M p x q : “ t u : r ,
8q ˆ S Ñ T ˚ M | B H u “ ,u p`8 , ¨q “ x, u p , ¨q Ă M u and(20) M p x ; a q : “ t u P M p x q | u p , ¨q P W ` p a qu , wherer W ` p a q is the stable manifold of a for the negative gradient flowof S L . See Figure 6.For generic H these are manifolds of dimensionsdim M p x q “ CZ p x q , dim M p x ; a q “ CZ p x q ´ ind p a q . The signed count of 0-dimensional spaces M p x ; a q defines a chain mapΨ : F C ˚ Ñ M C ˚ , a ÞÑ ÿ ind p a q“ CZ p a q M p x ; a q a. OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 59
The induced map on homology is an isomorphismΨ ˚ : F H ˚ – ÝÑ M H ˚ – H ˚ p Λ; σ q intertwining the pair-of-pants product with the loop product. Proposition 5.5.
The map Ψ descends to an isomorphism on homol-ogy modulo the constant loops Ψ ˚ : F H ą ˚ – ÝÑ M H ą ˚ – H ˚ p Λ , Λ ; σ q which intertwines the Abbondandolo–Schwarz coproduct λ F with the ho-mology coproduct λ .Proof. For x P Crit p A H q and b, c P Crit p S L q define M ` p x q : “ p σ, τ, u, v, w q ˇˇ σ P r , , τ P r , s ,u : r σ,
8q ˆ S Ñ T ˚ M, v, w : r , σ s ˆ S Ñ T ˚ M, B H u “ B H v “ B H w “ ,u p`8 , ¨q “ x, v p , t q P M, w p , t q P M,v p σ, t q “ u p σ, τ t q , w p σ, t q “ u p σ, τ ` p ´ τ q t q ( , M ` p x ; b, c q : “ tp σ, τ, u, v, w q P M ` p x q | v p , ¨q P W ` p b q , w p , ¨q P W ` p c qu , M ´ p x ; b, c q : “ p σ, τ, u, α, β, γ q ˇˇ σ P p´8 , s , τ P r , s , u P M p x q ,α “ φ ´ σ p u p , ¨qq , β P W ` p b q , γ P W ` p c q ,β p t q “ α p τ t q , γ p t q “ α p τ ` p ´ τ q t q ( , where M p x q was defined above and φ s : Λ Ñ Λ for s ě ´ ∇ S L . Note that α, β, γ in the definition of M ´ p x ; b, c q are actually redundant and just included to make the definition moretransparent. As above it follows that for generic H these spaces aretransversely cut out manifolds of dimensions dim M ` p x q “ CZ p x q ´ n ` M ` p x ; b, c q “ dim M ´ p x ; b, c q “ CZ p x q ´ ind p b q ´ ind p c q ´ n ` . We set M p x ; b, c q : “ M ` p x ; b, c q > M ´ p x ; b, c q . If this space has dimension zero it is compact and defines a mapΘ :
F C ˚ Ñ p
M C b M C q ˚´ n ` , x ÞÑ ÿ b,c M “ p x ; b, c q b b c. If it has dimension 1 it can be compactified to a compact 1-dimensionalmanifold with boundary B M “ p x ; b, c q “ ž CZ p x q“ CZ p x q´ M p x ; x q ˆ M “ p x ; b, c q> ž ind p b q“ ind p b q` M “ p x ; b , c q ˆ M p b ; b q> ž ind p c q“ ind p c q` M “ p x ; b, c q ˆ M p c ; c q> ž y,z M “ p x ; y, z q ˆ M p y ; b q ˆ M p z ; c q> ž a M p x ; a q ˆ Ă M “ p a ; b, c q> M τ “ p x ; b, c q > M τ “ p x ; b, c q , where Ă M p a ; b, c q are the moduli spaces in Remark 2.5 defining thecoproduct r λ with f t “ id. Here the first terms corresponds to splittingoff of Floer cylinders, the second and third ones to splitting off of Morsegradient lines, the fourth one to σ “ `8 , the fifth one to σ “ ´8 ,and the last two terms to the intersection of M p x ; b, c q with the sets t τ “ u and t τ “ u , respectively. The intersections of M ˘ p x ; b, c q with the set t σ “ u are equal with opposite orientations and thuscancel out. So we have(21) pB b id ` id b Bq Θ ` Θ B F “ p Ψ b Ψ q λ F ´ r λ Ψ ` Θ ´ Θ , where for i “ , i : F C ˚ Ñ p
M C b M C q ˚´ n ` , x ÞÑ ÿ b,c M τ “ i p x ; b, c q b b c. Arguing as in the previous subsection, we see that the Θ has imagein M C “ ˚ b M C ˚ , and Θ has image in M C ˚ b M C “ ˚ . Together withequation (21) this shows that Θ descends to a mapΘ : F C ą ˚ Ñ p
M C ą b M C ą q ˚´ n ` between the positive chain complexes which is a chain homotopy be-tween p Ψ b Ψ q λ F and r λ Ψ, which concludes the proof. (cid:3)
Appendix A. Local systems
We describe in this section the loop product and the homology coprod-uct with general twisted coefficients. This allows us in particular to dis-pose of the usual orientability assumption for the underlying manifold.To the best of our knowledge, the Chas-Sullivan product on loop spacehomology was constructed for the first time on non-orientable manifoldsby Laudenbach [27], and the BV algebra structure by Abouzaid [6]. In
OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 61 this appendix we extend the definitions to more general local systems,we take into account the coproduct, and we discuss the adaptationsto reduced homology and cohomology groups H ˚ Λ and H ˚ Λ. We alsodiscuss the formulation and properties of the isomorphism betweensymplectic homology and loop homology with twisted coefficients.A.1.
Conventions.
We use the following conventions from [6, § V , its determinant line is the 1-dimensional real vector space det V “ Λ max V . We view it asbeing a Z -graded real vector space supported in degree dim R V . To any1-dimensional graded real vector space L we associate an orientationline | L | , which is the rank 1 graded free abelian group generated bythe two possible orientations of L , modulo the relation that their sumvanishes. The orientation line | L | is by definition supported in the samedegree as L . When L “ det V we denote its orientation line | V | .Given a Z -graded line ℓ (rank 1 free abelian group), its dual line ℓ ´ “ Hom Z p ℓ, Z q is by definition supported in opposite degree as ℓ . There isa canonical isomorphism ℓ ´ b ℓ – Z induced by evaluation.Given a Z -graded object F , we denote F r k s the Z -graded object ob-tained by shifting the degree down by k P Z , i.e. F r k s n “ F n ` k . Forexample, the shifted orientation line | V |r dim V s is supported in degree0. A linear map f : E Ñ F between Z -graded vector spaces or freeabelian groups has degree d if f p E n q Ă F n ` d for all n . In an equiva-lent formulation, the induced map f r d s : E Ñ F r d s has degree 0. Forexample, the dual of a vector space or free abelian group supportedin degree k is supported in degree ´ k . This is compatible with thegrading convention for duals of Z -graded orientation lines. Given a Z -graded rank 1 free abelian group ℓ , we denote ℓ the same abeliangroup with degree set to 0. For example | V | “ | V |r dim V s .Given two oriented real vector spaces U and W , we induce an orienta-tion on their direct sum U ‘ W by defining a positive basis to consist ofa positive basis for U followed by a positive basis for W . This definesa canonical isomorphism at the level of orientation lines | U | b | W | – | U ‘ W | . Given an exact sequence of vector spaces0 Ñ U Ñ V Ñ W Ñ V out of orientations of U and W bydefining a positive basis to consist of a positive basis for U followed bythe lift of a positive basis for W . This defines a canonical isomorphism | U | b | W | – | V | . The following example will play a key role in the sequel.
Example A.1 (normal bundle to the diagonal) . Let M be a manifoldof dimension n . Consider the diagonal ∆ Ă M ˆ M and denote ν ∆ its normal bundle. Let p , : M ˆ M Ñ M be the projections on thetwo factors, so that we have a canonical isomorphism T p M ˆ M q – p ˚ T M ‘ p ˚ T M . When restricted to ∆ the projections coincide with thecanonical diffeomorphism p : ∆ » ÝÑ M . We obtain an exact sequenceof bundles Ñ T ∆ Ñ p ˚ T M ‘ p ˚ T M Ñ ν ∆ Ñ . This gives rise to a canonical isomorphism | ∆ | b | ν ∆ | – p ˚ | M | b p ˚ | M | and, because p ˚ | M | b p ˚ | M | is canonically trivial, we obtain a canonicalisomorphism | ∆ | – | ν ∆ | . Explicitly, this isomorphism associates to the equivalence class of abasis pp v , v q , . . . , p v n , v n qq , v i P T q M of T p q,q q ∆ the equivalence classof the basis prp , v qs , . . . , rp , v n qsq of ν p q,q q ∆ . A.2.
Homology with local systems. By local system we mean a lo-cal system of Z -graded rank 1 free Z -modules. On each path-connectedcomponent of the underlying space we think of such a local system inone of the following three equivalent ways: either as the data of theparallel transport representation of the fundamental groupoid, or as thedata of the monodromy representation from the fundamental group π to the multiplicative group t˘ u together with the data of an integer(the degree), or as the data of a Z -graded Z r π s -module which is freeand of rank 1 as a Z -module. Isomorphism classes of local systems ona path connected space X are thus in bijective correspondence with H p X ; Z { q ˆ Z , where the first factor corresponds to the monodromyrepresentation and the second factor to the grading. Here and in the se-quel we identify the multiplicative group t˘ u with the additive group Z {
2. We refer to [7] for a comprehensive discussion with emphasis onlocal systems on free loop spaces. One other point of view on localsystems describes these as locally constant sheaves [25], but we willonly marginally touch upon it in § A.3.Given a local system ν , we can change the coefficients to any commu-tative ring R by considering ν R “ ν b Z R . The monodromy of such alocal system still takes values in t˘ u , and this property characterizeslocal systems of rank 1 free R -modules spaces which are obtained fromlocal systems of rank 1 free Z -modules by tensoring with R .Let X be a path connected space admitting a universal cover ˜ X . De-note its fundamental group at some fixed basepoint π “ π p X q . Inter-preting a local system ν on X as a Z r π s -module, one defines singularhomology/cohomology with coefficients in ν in terms of singular chainson ˜ X as H ˚ p X ; ν q “ H ˚ p C ˚ p ˜ X ; Z q b Z r π s ν q , OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 63 H ˚ p X ; ν q “ H ˚ p Hom Z r π s p C ˚ p ˜ X ; Z q , ν qq . The homology/cohomology with local coefficients extended to a com-mutative ring R are the R -modules H ˚ p X ; ν R q “ H ˚ p C ˚ p ˜ X ; Z q b Z r π s ν R q ,H ˚ p X ; ν R q “ H ˚ ` Hom Z r π s p C ˚ p ˜ X ; Z q , ν R q ˘ . In our grading convention the cohomology with constant coefficientsis supported in nonpositive degrees and equals the usual cohomologyin the opposite degree. The induced differential on the dual groupHom Z r π s p C ˚ p ˜ X ; Z q , ν R q has degree ´ tensor product ν b ν of two local systems is again a local system.Its Z r π s -module structure is the diagonal one and its degree is the sumof the degrees of the factors. Note that viewing ν , ν as elements in H p X ; Z { qˆ Z , their tensor product is given by their sum ν ` ν . Oper-ations like cap or cup product naturally land in homology/cohomologywith coefficients in the tensor product of the coefficients of the factors.Homology/cohomology with local coefficients behave functorially in thefollowing sense. Given a continuous map f : X Ñ Y and a local system ν on Y described as a Z r π p Y qs -module, the pullback local system f ˚ ν on X is defined by inducing a Z r π p X qs -module structure via f ˚ . Wethen have canonical maps f ˚ : H ˚ p X ; f ˚ ν q Ñ H ˚ p Y ; ν q , f ˚ : H ˚ p Y ; ν q Ñ H ˚ p X ; f ˚ ν q . The algebraic duality isomorphism with coefficients in a field K takesthe form H ´ k p X ; ν ´ K q – ÝÑ H k p X ; ν K q _ , k P Z . The map is induced by the canonical evaluation of cochains on chains.We check that degrees fit in the case of graded local systems: given alocal system ν K of degree d , and recalling our notation ν K “ ν K r d s and ν ´ K “ ν ´ K r´ d s , we have H k p X ; ν K q “ H k ´ d p X ; ν K q , H ´ k p X ; ν ´ K q “ H ´ k ` d p X ; ν ´ K q , so H k p X ; ν K q _ and H ´ k p X ; ν ´ K q both live in degree d ´ k .A.3. Poincar´e duality.
Consider a manifold M of dimension n . Wedenote by | M | the local system on M whose fiber at any point q P M is the orientation line | T q M | , supported by definition in degree n . Werefer to | M | as the orientation local system of M . The monodromyalong a loop γ is ` γ ˚ T M is orientable), and ´ | M | is trivial. A choiceof orientation is equivalent to the choice of one of the two possibleisomorphisms | M | » Z . The local system | M | b Z { Z { Suppose now that M is closed. Then it carries a fundamental class r M s P H n p M ; | M |q “ H p M ; | M | ´ q .For any local system ν on M , the cap product with a fundamental classdefines a Poincar´e duality isomorphism H ˚ p M ; ν q – ÝÑ H ˚ p M ; ν b | M | ´ q , α ÞÑ r M s X α. Remark A.2.
Here is a description of the fundamental class usingthe interpretation of local systems as locally constant sheaves [25, § M p Ñ M be the orientation double cover. Giventhe constant local system Z on M , the pushforward p ˚ Z to M hasrank 2 and can be decomposed as | M | ‘ Z (the map Z ‘ Z Ñ Z ‘ Z , p x, y q ÞÑ p y, x q fixes the diagonal and acts by ´ Id on the anti-diagonal).The composition H ˚ p M ; Z q p ˚ Ñ H ˚ p M ; p ˚ Z q » Ñ H ˚ p M ; | M |q‘ H ˚ p M ; Z q is an isomorphism because p ˚ is an isomorphism. Since H n p M ; Z q “ M is nonorientable, we obtain that H n p M ; | M |q has rank 1. A genera-tor is the image of a generator in H n p M ; Z q via the above composition.A.4. Thom isomorphism and Gysin sequence.
Let E p ÝÑ X bea real vector bundle of rank r , and denote E the complement of thezero section. Let | E | be the local system on X whose fiber at a point x P X is by definition the orientation line | E x | of the fiber of E at x .The local system | E | is called the orientation local system of E and issupported in degree r . The Thom class is a generator τ P H ´ r p E, E ; p ˚ | E |q “ H p E, E ; p ˚ | E |q . The
Thom isomorphism takes the form H k p E, E q » ÝÑ H k ´ r p X ; | E |q “ H k p X ; | E |q , k P Z (cap product with τ ), respectively H k p X ; | E | ´ q “ H k ` r p X ; | E | ´ q » ÝÑ H k p E, E q , k P Z (cup product with τ ). More generally, for any local system ν on X wehave isomorphisms H k p E, E ; p ˚ ν q » ÝÑ H k ´ r p X ; ν b | E |q “ H k p X ; ν b | E |q ,H k p X ; ν b | E | ´ q “ H k ` r p X ; ν b | E | ´ q » ÝÑ H k p E, E ; p ˚ ν q . Pulling back the Thom class under the inclusion i : X Ñ E of the zerosection yields the Euler class e “ i ˚ τ P H ´ r p X ; | E |q “ H p X ; | E |q . Denote by S Ă E the sphere bundle with projection π “ p | S : S Ñ X .Then the long exact sequence of the pair p E, E q fits into the commuting OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 65 diagram ¨ ¨ ¨ H k p E, E q / / H k p E q i ˚ – (cid:15) (cid:15) / / H k p E q / / H k ´ p E ; E q ¨ ¨ ¨¨ ¨ ¨ H k p X ; | E | ´ q Y τ – O O Y e / / H k p X q π ˚ / / H k p S q π ˚ / / H k ´ p X ; | E | ´ q ¨ ¨ ¨ Y τ – O O where the lower sequence is the Gysin sequence . More generally, foreach local system ν on X we get a Gysin sequence ¨ ¨ ¨ H k p X ; ν b | E | ´ q Y e ÝÑ H k p X ; ν q π ˚ ÝÑ H k p S ; π ˚ ν q π ˚ ÝÑ H k ´ p X ; ν b | E | ´ q ¨ ¨ ¨ A.5.
Spaces of loops with self-intersection.
Let M be a manifoldof dimension n , Λ “ Λ M its space of free loops of Sobolev class W , ,and ev s : Λ Ñ M the evaluation of loops at time s . We define F “ tp γ, δ q P Λ ˆ Λ | γ p q “ δ p qu Ă Λ ˆ Λ(pairs of loops with the same basepoint), and F s “ t γ P Λ | γ p s q “ γ p qu Ă Λ , s P p , q (loops with a self-intersection at time s ). Denoting f : Λ ˆ Λ Ñ M ˆ M , f “ ev ˆ ev and f s : Λ Ñ M ˆ M , f s “ p ev , ev s q , we can equivalentlywrite F “ f ´ p ∆ q , F s “ f ´ s p ∆ q . The maps f and f s are smooth and transverse to the diagonal ∆, sothat F and F s are Hilbert submanifolds of codimension n . Denoting ν F and ν F s their normal bundles we obtain canonical isomorphisms ν F – f ˚ ν ∆ , ν F s – f ˚ s ν ∆ . In view of Example A.1 we infer canonical isomorphisms(22) | ν F | – f ˚ | ∆ | – ev ˚ | M | , | ν F s | – f ˚ s | ∆ | – ev ˚ | M | , where, in the first formula, ev : F Ñ M is the evaluation of pairs ofloops at their common origin.Denote i : F ã Ñ Λ ˆ Λ and i s : F s ã Ñ Λ the inclusions. Recall therestriction maps (4). Define the cutting map at time sc s : F s Ñ F , c s p γ q “ p γ | r ,s s , γ | r s, s q and the concatenation map at time sg s : F Ñ F s , g s p γ , γ qp t q “ " γ p ts q , t P r , s s ,γ p t ´ s ´ s q , t P r s, s . The maps c s and g s are smooth diffeomorphisms inverse to each other.The situation is summarized in the diagramΛ ˆ Λ F i o o g s „ / / F sc s o o i s / / Λ . Lemma A.3.
Let ν be a local system (of rank free abelian groups)on Λ supported in degree . Denote p , : Λ ˆ Λ Ñ Λ the projectionson the two factors. The following two conditions are equivalent: (23) c ˚ s p p ˚ ν b p ˚ ν q| F » ν | F s , and (24) p p ˚ ν b p ˚ ν q| F » g ˚ s p ν | F s q . Proof.
The first condition is c ˚ s i ˚ p p ˚ ν b p ˚ ν q » i ˚ s ν . Since g s is a home-omorphism, this is equivalent to g ˚ s c ˚ s i ˚ p p ˚ ν b p ˚ ν q » g ˚ s i ˚ s ν. In view of c s g s “ Id F s , this is the same as the second condition. (cid:3) Definition A.4.
A degree local system ν on Λ is compatible withproducts if it satisfies the equivalent conditions of Lemma A.3. A local system ν which is compatible with products must necessarilyhave degree 0 (and rank 1). Also, ν | M must be trivial: restricting bothsides of (23) or (24) to the constant loops yields ν | M b ν | M » ν | M . Remark A.5.
Local systems which are compatible with products playa key role in the sequel definition of the loop product and loop co-product with local coefficients. Condition (23) is the one that ensuresthe coproduct is defined with coefficients twisted by ν , whereas condi-tion (24) is the one that ensures the product is defined with coefficientstwisted by ν . That the two conditions are equivalent can be seen asyet another instance of Poincar´e duality for free loops.We refer to Remark A.11 for an additional condition on the isomor-phisms (24) which is needed for the associativity of the product andcoassociativity of the coproduct. Example A.6 (Transgressive local systems) . Let Λ “ \ α Λ α M be thedecomposition of the free loop space into connected components, indexedby conjugacy classes α in the fundamental group. We view loops γ : S Ñ Λ as maps γ ˆ S : S ˆ S Ñ M , p u, t q ÞÑ γ p u qp t q . Thisinduces a map π p Λ α M q Ñ H p Λ α M ; Z q Ñ H p M ; Z q , r γ s ÞÑ r γ ˆ S s . Dually, and specializing to Z { -coefficients, any cohomology class c P H p M ; Z { q determines a cohomology class τ c P H p Λ; Z { q “ ś α Hom p π p Λ α M q ; Z { q via x τ c , r γ sy “ x c, r γ ˆ S sy . We denote the corresponding local system on Λ also by τ c . Degree local systems obtained in this way are called transgressive [6] .Transgressive local systems are compatible with products. Indeed, theidentity (24) is a direct consequence of the equality r g s p γ , γ q ˆ S s “r γ ˆ S s ` r γ ˆ S s , which holds in H p M ; Z q for all p γ , γ q : S Ñ F . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 67
The transgressive local system (25) σ “ τ w defined by the second Stiefel-Whitney class w P H p M ; Z { q will playa special role in the sequel. Example A.7.
Following Abouzaid [6] , define for each loop γ P Λ the shift w p γ q “ " , if γ preserves the orientation , ´ , if γ reverses the orientation . Define the local system (26) ˜ o “ ev ˚ | M | ´ w to be trivial on the components where γ preserves the orientation, andequal to ev ˚ | M | on components where γ reverses the orientation.The local system ˜ o is compatible with products: the equality w p γ q ` w p γ q “ w p g s p γ , γ qq which holds in Z { for all p γ , γ q P F . Note thatthe local system ˜ o is not transgressive and, in case M is nonorientable,it is nontrivial on all connected components Λ α M whose elements re-verse orientation. Question A.8.
Characterize in cohomological terms the local systemson Λ which are compatible with products. For example, it followsfrom [7, Lemma 1] that, on a simply connected manifold, a local sys-tem ν is compatible with products if and only if ν | M is trivial. A mildgeneralization is given by [7, Proposition 10] . A.6.
Loop product with local coefficients.
Following [22], we viewthe loop product as being defined by going from left to right in thediagram Λ ˆ Λ Ð â F g ÝÑ Λ , where g “ i s g s for some fixed s P p , q . More precisely, the loop productwith integer coefficients is defined as the composition H i p Λ; Z q b H j p Λ; Z q ǫ ˆ / / H i ` j p Λ ˆ Λ; Z q / / H i ` j p ν F , ν F ; Z q » / / H i ` j p F ; ev ˚ | M |q g ˚ / / H i ` j p Λ; ev ˚ | M |q . The first map is the homology cross-product corrected by a sign ǫ “p´ q n p i ` n q ([24, Appendix B]), the second map is the composition ofthe map induced by inclusion Λ ˆ Λ ã Ñ p Λ ˆ Λ , Λ ˆ Λ z F q with excisionand the tubular neighbourhood isomorphism, and the third map is theThom isomorphism. In case M is not orientable the loop product doesnot land in homology with integer coefficients and thus fails to definean algebra structure on H ˚ p Λ; Z q . This can be corrected by using atthe source homology with local coefficients. Definition A.9.
Define on Λ the local system µ : “ ev ˚ | M | ´ . The archetypal loop product is the bilinear map ‚ : H i p Λ; µ q b H j p Λ; µ q Ñ H i ` j p Λ; µ q defined as the composition H i p Λ; µ q b H j p Λ; µ q ǫ ˆ / / H i ` j p Λ ˆ Λ; p ˚ µ b p ˚ µ q / / H i ` j p ν F , ν F ; p ˚ µ b p ˚ µ | ν F q » / / H i ` j p F ; p p ˚ µ b p ˚ µ q| F b ev ˚ | M |q g ˚ / / H i ` j p Λ; µ q . The description of the maps is the same as above, with ǫ “ p´ q ni because of the shift H i p Λ; µ q “ H i ` n p Λ; µ q . However, one still needs tocheck that the local systems of coefficients are indeed as written. Forthe first, second and third map the behavior of the coefficients followsgeneral patterns. For the last map we use that p p ˚ µ b p ˚ µ q| F b ev ˚ | M | » g ˚ µ, which is true for our specific µ “ ev ˚ | M | ´ .The archetypal loop product is associative, graded commutative, andit has a unit represented by the fundamental class r M s P H p M ; | M | ´ q “ H n p M ; | M |q from § A.3. With our grading conventions, the archetypal loop producthas degree 0 and the local system µ is supported in degree ´ n . In thecase where M is oriented we recover the usual loop product.More generally, the loop product can be defined with further twistedcoefficients. Definition A.10.
Let ν be a degree local system (of rank free Z -modules) on Λ which is compatible with products. The loop productwith coefficients twisted by ν is the bilinear map ‚ : H i p Λ; ν b µ q b H j p Λ; ν b µ q Ñ H i ` j p Λ; ν b µ q (with µ “ ev ˚ | M | ´ as above) defined as the composition H i p Λ; ν b µ q b H j p Λ; ν b µ q ǫ ˆ / / H i ` j p Λ ˆ Λ; p p ˚ ν b p ˚ ν q b p p ˚ µ b p ˚ µ qq / / H i ` j p ν F , ν F ; p p ˚ ν b p ˚ ν q b p p ˚ µ b p ˚ µ q| ν F q » / / H i ` j p F ; p p ˚ ν b p ˚ ν q b p p ˚ µ b p ˚ µ q| F b ev ˚ | M |q g ˚ / / H i ` j p Λ; ν b µ q . OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 69
As before, we have ǫ “ ni . For the last map we use the isomorphism p p ˚ µ b p ˚ µ q| F b ev ˚ | M | » g ˚ µ , and the isomorphism p p ˚ ν b p ˚ ν q| F » g ˚ ν which expresses the compatibility with products for ν .The loop product with twisted coefficients is graded commutative andunital. Recalling that the compatibility with products for ν forces itsrestriction to M to be trivial, the unit is again represented by thefundamental class r M s P H p M ; ν | M b | M | ´ q “ H p M ; | M | ´ q “ H n p M ; | M |q . Remark A.11.
Associativity of the loop product with twisted coeffi-cients depends on the following associativity condition on the isomor-phisms (23) and (24). Given s, s P p , q denote s “ p s ´ ss q{p ´ ss q ,so that g s ˝ p g s ˆ id q “ g ss ˝ p id ˆ g s q . Denoting Φ s : p p ˚ ν b p ˚ ν q| F » Ñ g ˚ s ν | F s the isomorphism from (24), we require the associativity condi-tion Φ ss ˝ p Id b Φ s q “ Φ s ˝ p Φ s b Id q . This holds for the transgressive local systems from Example A.6 andfor the local system in Example A.7.Also, because (23) and (24) are equivalent, this condition on (24) willguarantee co-associativity of the coproduct, see below.A.7.
Homology coproduct with coefficients.
Again following [22],we view the primary coproduct on loop homology as being defined bygoing from left to right in the diagramΛ Ð â F s c s ÝÑ Λ ˆ Λfor some fixed s P p , q , where c s stands for ic s in the notation of § A.5.We restrict in this section to coefficients in a field K and all local sys-tems are accordingly understood in this category. The reason for thisrestriction is explained below. The primary coproduct with constantcoefficients is defined as the composition H k p Λ; K q / / H k p ν F s , ν F s ; K q H k p F s ; ev ˚ | M |q » o o c s ˚ / / H k p Λ ˆ Λ; p ˚ ev ˚ | M |q AW / / À i ` j “ k H i p Λ; ev ˚ | M |q b H j p Λ; K q . The first map is the composition of the map induced by inclusionΛ
Ñ p Λ , Λ z F s q with the excision isomorphism towards the homologyrel boundary of a tubular neighbourhood of F s . The second map isthe Thom isomorphism. For the third map we use that c ˚ s p ˚ ev ˚ “ ev ˚ .The fourth map is the Alexander-Whitney diagonal map followed bythe K¨unneth isomorphism. As for the loop product, we see that if The Alexander-Whitney diagonal map [20, VI.12.26] takes values in H ˚ p C ˚ p Λ; ev ˚ | M |q b C ˚ p Λ qq with arbitrary coefficients. In order to further landin H ˚ p Λ; ev ˚ | M |q b H ˚ p Λ q we need to restrict to field coefficients so the K¨unnethisomorphism holds. M is nonorientable the primary coproduct fails to define a coalgebrastructure on H ˚ p Λ; K q . This is corrected by using homology with localcoefficients as follows. Definition A.12.
Define on Λ the local system o : “ ev ˚ | M | “ µ ´ . The archetypal primary coproduct is the bilinear map _ s : H k p Λ; o q Ñ à i ` j “ k H i p Λ; o q b H j p Λ; o q (for some fixed s P r , s ) defined as the composition H k p Λ; o q / / H k p ν F s , ν F s ; o | ν F s q H k p F s ; o b ev ˚ | M |q » o o c s ˚ / / H k p Λ ˆ Λ; p ˚ o b p ˚ o q AW / / À i ` j “ k H i p Λ; o q b H j p Λ; o q . With our grading conventions this coproduct has degree 0. Taking intoaccount that o “ ev ˚ | M | is supported in degree n , this results in thecoproduct having the usual degree ´ n in ungraded notation. In theorientable case it recovers the usual primary coproduct.Just like the product, the primary coproduct can be defined with fur-ther twisted coefficients. Definition A.13.
Let ν be a degree local system (of rank one K -vector spaces) on Λ which is compatible with products. The primarycoproduct with twisted coefficients is the bilinear map _ s : H k p Λ; ν b o q Ñ à i ` j “ k H i p Λ; ν b o q b H j p Λ; ν b o q (with o “ ev ˚ | M | as above and some fixed s P r , s ) defined as thecomposition H k p Λ; ν b o q / / H k p ν F s , ν F s ; ν b o | ν F s q H k p F s ; ν b o b q » o o c s ˚ / / H k p Λ ˆ Λ; p ˚ p ν b o q b p ˚ p ν b o qq AW / / À i ` j “ k H i p Λ; ν b o q b H j p Λ; ν b o q . In the definition we use c ˚ s p p ˚ o b p ˚ o q » o b o , and the condition c ˚ s p p ˚ ν b p ˚ ν q » ν | F s which is part of the condition of being compatiblewith products for ν .The arguments of Goresky-Hingston [22, §
8] which show that, in theorientable case, there is a secondary coproduct of degree ´ n ` H ˚ p Λ , Λ ; K q , apply verbatim in the current setupinvolving local coefficients. As an outcome, we obtain the following. OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 71
Definition-Proposition A.14.
For any degree local system ν ofrank one K -vector spaces on Λ which is compatible with products, thereis a well-defined (secondary) homology coproduct with twisted coeffi-cients (abbreviate o “ ev ˚ | M | ) _ : H k p Λ , Λ ; ν b o q Ñ à i ` j “ k ` H i p Λ , Λ ; ν b o q b H j p Λ , Λ ; ν b o q . (cid:3) As explained in [24], in order for this secondary coproduct to be coasso-ciative in the case of a constant local system ν we need to correct the ij -component of the secondary product induced by the previously definedprimary product by a sign ǫ “ p´ q p n ´ qp j ´ n q (see [24, Definition 1.7]and note the shift in grading H j p Λ , Λ ; ν b o q “ H j ´ n p Λ , Λ ; ν b o q ).With this correction the coproduct with constant ν is also graded co-commutative if gradings are shifted so that it has degree 0. However,it has no counit (this would contradict the infinite dimensionality ofthe homology of Λ).In the case of coefficients twisted by a local system ν which is non-constant, coassociativity further requires that the isomorphisms Φ s ex-pressing compatibility with products for ν satisfy the condition fromRemark A.11. The coproduct is then also co-commutative.One obtains dually a cohomology product [22, 24]. Note that, in con-trast to the homology coproduct, the dual cohomology product is de-fined with arbitrary coefficients because its definition does not requirethe K¨unneth isomorphism. Definition-Proposition A.15.
For any degree local system ν ofrank free abelian groups on Λ which is compatible with products, thereis a well-defined cohomology product with twisted coefficients (abbre-viate µ “ ev ˚ | M | ´ ) ⊛ : H i p Λ , Λ ; ν b µ q b H j p Λ , Λ ; ν b µ q Ñ H i ` j ´ p Λ , Λ ; ν b µ q . (cid:3) The cohomology product with twisted coefficients is associative. It isalso graded commutative when viewing it as a degree 0 product on H ˚´ p Λ , Λ ; ν b µ q .A.8. Loop product and coproduct on H ˚ Λ . Recall the previousnotation o “ ev ˚ | M | “ µ ´ , and let ν be a local system compatiblewith products. The arguments of § verbatim to give adescription of the homology product and homology coproduct in Morsehomology with local coefficients in ν b µ , respectively ν b o . For adefinition of Morse homology with local coefficients we refer to [6, § § The reduced groups
M H ˚ and M H ˚ are defined with local coefficientsas follows. Recall that ν | M is trivial. We consider the map ε given asthe composition H ˚ p Λ; ν b µ q ε / / (cid:15) (cid:15) H ˚ p Λ; ν b µ q H p M ; µ q ε / / H p M ; µ q O O where the vertical maps are restriction to, respectively inclusion ofconstant loops, and ε is induced by multiplication with the Eulercharacteristic. We then define M H ˚ p Λ; ν b µ q “ ker ε, M H ˚ p Λ; ν b µ q “ coker ε. Thus
M H ˚ p Λ; ν b µ q “ M H ˚´ n p Λ; ν b µ q and M H ˚ p Λ; ν b o q “ M H ˚` n p Λ; ν b µ q .The arguments of § verbatim in order to show that theloop product descends to M H ˚ p Λ; ν b µ q and the homology coproductextends to M H ˚ p Λ; ν b o q (in the latter case we use field coefficients asin § A.7). Interpreted dually as a product on cohomology, this is definedwith arbitrary coefficients on
M H ˚ p Λ; ν b µ q . With our grading con-ventions, the degrees are 0 for the loop product, ` ´ Isomorphism between symplectic homology and loop ho-mology.
We spell out in this section the isomorphism between thesymplectic homology of the cotangent bundle and the homology of thefree loop space with twisted coefficients.For the next definition, recall the local systems σ “ τ w , ˜ o “ ev ˚ | M | ´ w from (25) and (26), as well as the orientation local systems µ “ ev ˚ | M | ´ “ o ´ . Definition A.16 (Abouzaid [6]) . The fundamental local system forsymplectic homology of the cotangent bundle is the local system on Λ given by η “ σ b µ b ˜ o. The fundamental local system η is supported in degree ´ n . Our previ-ous discussion shows that the loop product is defined and has degree 0on H ˚ p Λ; η q , and the homology coproduct is defined and has degree ` H ˚ p Λ , Λ ; η ´ q . We can view the loop product as being defined on H ˚ p Λ; η q , where it has degree ´ n , and the homology coproduct as be-ing defined (with field coefficients) on H ˚ p Λ , Λ ; η q , where it has degree OOP COPRODUCT IN MORSE AND FLOER HOMOLOGY 73 ´ n . This point of view is useful when considering H ˚ p Λ; η q , theirnatural common space of definition (to which the product descends andthe coproduct extends).As proved in [5, 6], the chain map Ψ “ Ψ quadratic discussed in § F C ˚ p H q Ñ M C ˚ p S L ; η q and induces an isomorphism SH ˚ p D ˚ M q » ÝÑ H ˚ p Λ; η q . Given any localsystem ν , the same map acts as Ψ : F C ˚ p H ; ν q Ñ M C ˚ p S L ; ν b η q andinduces an isomorphism SH ˚ p D ˚ M ; ν q » ÝÑ H ˚ p Λ; ν b η q .Our filtered chain map Ψ “ Ψ linear from § F C ˚ p H q » ÝÑ M C ď µ ˚ p E { ; η q , with µ the slope of the Hamiltonian. Given any local system ν , we obtain afiltered chain isomorphism F C ˚ p H ; ν q » ÝÑ M C ď µ ˚ p E { ; ν b η q .In case the local system ν is compatible with products, the argumentsof [1, 3, 6] adapt in order to show that the map Ψ intertwines the pair-of-pants product on the symplectic homology side with the homologyproduct on the Morse side. The arguments of Theorem 4.1 adapt inorder to show that the map Ψ descends on reduced homology, where itintertwines the continuation coproduct with the homology coproduct(with field coefficients).To summarize: Theorem A.17 ([1, 3, 5, 6], Theorem 3.3, Corollary 3.4, Theorem 4.1) . Given any local system ν compatible with products, the filtered chainlevel map Ψ induces filtered isomorphisms Ψ ˚ : SH ˚ p D ˚ M ; ν q » ÝÑ H ˚ p Λ; ν b η q , Ψ ą ˚ : SH ą ˚ p D ˚ M ; ν q » ÝÑ H ˚ p Λ , Λ ; ν b η q , and Ψ ˚ : SH ˚ p D ˚ M ; ν q » ÝÑ H ˚ p Λ; ν b η q . Moreover:– Ψ ˚ intertwines the pair-of-pants product with the Chas–Sullivan loopproduct,– Ψ ą ˚ intertwines the continuation coproduct with the loop coproduct(with field coefficients),– Ψ ˚ intertwines both the pair-of-pants product and the continuationcoproduct with the loop product and coproduct on reduced homology(with field coefficients for the coproduct). (cid:3) Interpreting the coproduct on homology as a product in cohomology, weobtain the following dual statement: The algebraic dual of the filtered chain map Ψ induces a filtered isomorphismΨ ˚ : H ˚ p Λ; ν b η q » ÝÑ SH ˚ p D ˚ M ; ν q which intertwines the cohomology product on the Morse side with thecontinuation product on the symplectic cohomology side. References [1] A. Abbondandolo and M. Schwarz. On the Floer homology of cotangent bun-dles.
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J. Pure Appl. Algebra „ viterbo/FCFH.II.2003.pdf. Universit¨at AugsburgUniversit¨atsstrasse 14, D-86159 Augsburg, Germany
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