Poincaré duality for loop spaces
aa r X i v : . [ m a t h . S G ] A ug POINCAR ´E DUALITY FOR LOOP SPACES
KAI CIELIEBAK, NANCY HINGSTON, AND ALEXANDRU OANCEA
Abstract.
We prove a Poincar´e duality theorem with productsbetween Rabinowitz Floer homology and cohomology, for bothclosed and open strings. This lifts to a duality theorem betweenopen-closed TQFTs. Specializing to the case of cotangent bun-dles, we define extended loop homology and cohomology and ex-plain from a unified perspective pairs of dual results which havebeen observed over the years in the context of the search for closedgeodesics. These concern critical levels, relations to the based loopspace, manifolds all of whose geodesics are closed, Bott index it-eration, level-potency, and homotopy invariance. We extend theloop cohomology product to include constant loops. We prove arelation conjectured by Sullivan between the loop product and theextended loop homology coproduct as a consequence of associativ-ity for the product on extended loop homology.
Contents
1. Introduction 21.1. Puzzles in string topology 21.2. Poincar´e duality for free loops 31.3. Closed TQFT structures 71.4. Puzzles resolved 81.5. Length filtration 101.6. Poincar´e duality for based loops 121.7. Open-closed TQFT structures 142. Applications of Poincar´e duality 162.1. Critical levels 162.2. Hopf-Freudenthal-Gysin formulas 182.3. Loop product with the point class 202.4. Manifolds all of whose geodesics are closed 212.5. String point invertibility and resonances for CROSS 262.6. Index growth 272.7. Level-Potency 312.8. Homotopy invariance 353. Poincar´e duality in Rabinowitz Floer homology 36
Date : September 1, 2020. SH ă ˚ p V, B V q via homotopies 78Appendix B. Grading conventions 82References 84 Introduction
Puzzles in string topology.
Let M be a closed connected n -dimensional manifold. Set S “ R { Z and denote byΛ “ Λ M “ C p S , M q the free loop space of M . If M is oriented, then both the homology H ˚ Λ and the cohomology H ˚ p Λ , Λ q relative to the subspace Λ Ă Λof constant loops carry natural products: ‚ the loop product ‚ on H ˚ Λ, which is graded commutative, associa-tive and unital of degree ´ n [15], and ‚ the cohomology product ⊛ on H ˚ p Λ , Λ q , which is graded commu-tative and associative of degree n ´ The loop product was defined by M. Chas and D. Sullivan in [15] and isalso known as the
Chas–Sullivan product . The homology coproduct dual to thecohomology product was first defined by D. Sullivan in [64]. The cohomologyproduct was extensively studied in [39] and is often referred to as the
Goresky–Hingston product . OINCAR´E DUALITY FOR LOOP SPACES 3
Subsequent studies of these products revealed striking similarities, butalso notable discrepancies. In particular, the following puzzles arose:(a) Why is ‚ defined on all of Λ and ⊛ only relative to the constantloops? Can this asymmetry be removed, e.g. by extending ⊛ to all ofΛ?(b) Heuristically, it looks as if “the cohomology product is the loopproduct on Morse cochains on the space of constant speed loops”. Canthis statement be given precise mathematical sense?(c) Sullivan [64] has conjectured the following relation between the loopproduct µ “ ‚ and the homology coproduct λ on H ˚ p Λ , Λ q dual to ⊛ :(1) λµ “ p b µ qp λ b q ` p µ b qp b λ q . Note that this is not a TQFT type relation, but it rather resemblesDrinfeld compatibility for Lie bialgebras. How is this relation to beinterpreted and proved?(d) The cohomology product has the flavour of a “secondary product”,e.g. its degree is shifted by 1 with respect to the loop product. Can ⊛ indeed be constructed as a secondary product derived from ‚ ?(e) Many results concerning ‚ and ⊛ arise in dual pairs. For example,the critical levels Cr p X q for X P H ˚ Λ and cr p x q for x P H ˚ p Λ , Λ q defined in [39] satisfy the dual inequalitiesCr p X ‚ Y q ď Cr p X q ` Cr p Y q , cr p x ⊛ y q ě cr p x q ` cr p y q . See Section 2 for more details on this and other pairs of dual resultsconcerning relations to the based loop space, manifolds all of whosegeodesics are closed, Bott index iteration, level-potency, and homotopyinvariance. Can each such dual pair be derived from one commonresult, with a unified proof, via some kind of duality?1.2.
Poincar´e duality for free loops.
It turns out that all the puz-zles get resolved at once by introducing a new player into string topol-ogy. Given a Riemannian metric on M we consider the unit spherebundle S ˚ M “ t p P T ˚ M : } p } “ u Ă T ˚ M , viewed as the contactboundary of the Liouville domain D ˚ M “ t p P T ˚ M : } p } ď u . Wedenote by q H ˚ Λ “ SH ˚ p S ˚ M q , q H ˚ Λ “ SH ˚ p S ˚ M q the Rabinowitz Floer (co)homology [16], or (V-shaped) symplectic (co)homology [18, 25], of S ˚ M with coefficients in a fixed principal idealdomain R . The invariance properties of symplectic homology implythat these homologies are independent of the choice of Riemannianmetric on M . In the following we call q H ˚ Λ and q H ˚ Λ the extended loop(co)homology . KAI CIELIEBAK, NANCY HINGSTON, AND ALEXANDRU OANCEA
It was proved in [25] that q H ˚ Λ is a Z -graded unital commutative al-gebra, with a product of degree ´ n . We will show in this paper that q H ˚ Λ is also a Z -graded unital commutative algebra, with product ofdegree n ´
1, and we have
Theorem 1.1 (Poincar´e duality for free loop spaces) . There is a canon-ical isomorphism of unital algebras
P D : q H ˚ Λ » ÝÑ q H ´˚` Λ . The extended loop homology q H ˚ Λ is related to loop space homologyand cohomology by a long exact sequence(2) ¨ ¨ ¨ H ´˚ Λ ε / / H ˚ Λ ι / / q H ˚ Λ π / / H ´˚ Λ ¨ ¨ ¨ where ι is a ring map with respect to the loop product on H ˚ Λ. Here thecoefficients of loop space (co)homology are twisted by a suitable localsystem, which allows us to drop the orientability assumption of M (seeRemark 1.3 below). This exact sequence follows from the symplectichomology exact sequence of the pair p D ˚ M, S ˚ M q in [18, 25], togetherwith the following isomorphism between symplectic (co)homology of D ˚ M and loop space (co)homology. Theorem 1.2 ([68, 2, 5, 3, 58, 57, 23, 49, 7]) . There are isomorphismsof R -modules SH ˚ p D ˚ M q » H ˚ Λ , SH ˚ p D ˚ M q » H ˚ Λ . The first isomorphism intertwines the pair-of-pants product on sym-plectic homology with the loop product on loop space homology. (cid:3)
Remark 1.3 (Grading and local systems) . (a) Extended loop homol-ogy q H ˚ Λ is graded by the Conley–Zehnder indices of 1-periodic orbitsof Hamiltonians defining V-shaped symplectic homology. These are de-fined using the canonical trivializations of the tangent bundle of T ˚ M along loops defined in [7], with degrees shifted down by 1 along loopswhich are orientation reversing (i.e., along which the pullback bundle T M is nonorientable). With this grading the maps ε and ι in (2) aredegree preserving.(b) In this paper we use loop space homology with coefficients twistedby suitable local systems described in [7] (see also [21, Appendix A]).Denote ev : Λ Ñ M the evaluation map at the starting point of aloop and O the orientation local system on M . The loop space Λcarries two canonical local systems: the orientation local system r O “ ev ˚ O on components of orientation preserving loops (which is trivial iff M is orientable), and the spin local system σ (which is trivial iff thesecond Stiefel–Whitney class of M vanishes). Then all results in thispaper hold in the following two situations (with corresponding twistsin cohomology): OINCAR´E DUALITY FOR LOOP SPACES 5 (i) H ˚ Λ is twisted by σ b r O and SH ˚ p D ˚ M q is untwisted;(ii) H ˚ Λ is twisted by r O and SH ˚ p D ˚ M q is twisted by σ ´ .The algebraic structures related by the second isomorphism in Theo-rem 1.2 have not been properly understood so far. As a by-productof results in this paper we will show that the second isomorphism alsopreserves suitably defined ring structures in the case when the Eulercharacteristic χ p M q is zero.By Lemma 4.13 below (see also [18] and [21, Appendix A]), the map ε in (2) lives only in degree 0 and factors as(3) ε : H Λ ÝÑ H p M ; O q e ÝÑ H p M ; O q ÝÑ H Λ . If M is orientable the map e is multiplication by the Euler character-istic χ p M q . If M is nonorientable then H p M ; O q » R , the 2-torsionpart of R , and H p M ; O q » R { R , and the map e : R Ñ R { R ismultiplication by the Euler characteristic mod 2. Let us define the “reduced” loop (co)homology groups H ˚ Λ : “ coker ε, H ˚ Λ : “ ker ε. For example, with (twisted) coefficients in a field K , we have H ˚ Λ “ H ˚ p Λ , ˚q if M is orientable and χ p M q ‰ K ,H ˚ Λ if χ p M q “ K ,H ˚ Λ “ H ˚ p Λ , ˚q if M is orientable and χ p M q ‰ K ,H ˚ Λ if χ p M q “ K . Note that the long exact sequence (2) induces a short exact sequence(4) 0 / / H ˚ Λ ι / / q H ˚ Λ π / / H ´˚ Λ / / . Theorem 1.4 (Canonical splitting) . If M is orientable, the short exactsequence (4) canonically splits (5) 0 / / H ˚ Λ ι / / q H ˚ Λ p r r π / / H ´˚ Λ i r r / / via maps p, i satisfying pι “ id and πi “ id . Moreover, i is a ring mapwith respect to a product ⊛ on H ˚ Λ extending the cohomology product.If M is non-orientable, the short exact sequence (4) admits such asplitting if the following two conditions hold:— the map R Ñ R { χ R induced by the inclusion R ã Ñ R is surjective.— the map R { χ R Ñ R { R induced by multiplication by χ is injective.In particular, such a splitting exists in each of the following situations:(i) χ “ in R , (ii) R is -torsion, e.g. R “ Z { . KAI CIELIEBAK, NANCY HINGSTON, AND ALEXANDRU OANCEA
On the level of modules this recovers the computation of q H ˚ Λ in [18].On the level of products it includes in particular the following corol-lary, whose part (1) follows from the fact that im ε “ ker ι Ă H ˚ Λ inthe exact sequence (2) is an ideal, and whose part (2) is contained inTheorem 1.4.
Corollary 1.5. (1) The loop product on H ˚ Λ descends to H ˚ Λ “ coker ε .(2) The cohomology product on H ˚ p Λ , Λ q has a canonical extension to H ˚ Λ “ ker ε . (cid:3) We point out that the extended cohomology product can have nontriv-ial contributions involving classes of constant loops. This happens forexample for the loop space of S , see [21].Theorem 1.4 provides a canonical splitting(6) q H ˚ Λ “ H ˚ Λ ‘ H ´˚ Λ , where the product on q H ˚ Λ restricts to the loop product on the subring H ˚ Λ, and to the cohomology product on the subring (not containingthe unit) H ´˚ Λ. More generally, let us denote the product on q H ˚ Λ r n s by m and its components with respect to the splitting by m `´` : H ˚ Λ r n s b H ´˚ Λ r n s Ñ H ˚ Λ r n s etc, where the upper indices denote the inputs, the lower index theoutput, ` corresponds to H ˚ Λ r n s , and ´ to H ´˚ Λ r n s . Each of thesemaps has degree 0, e.g. m `´` acts as H i ` n Λ b H ´ j ´ n Λ Ñ H i ` j ` n Λ.We denote the loop product by µ and the homology coproduct by λ ,both defined on H ˚ Λ r n s . The following result, which is proved in [24],expresses the product on q H ˚ Λ r n s in terms of µ and λ . In the statementwe use the following notation: given ¯ f P H ´˚ Λ r n s “ H ´˚´ n Λ, wedenote f P H ´˚ Λ r n s “ H ´˚´ n Λ the same element with degree shifteddown by 1, i.e. | f | “ | ¯ f | ´ Theorem 1.6 ([24, Theorem 1.2]) . Let n ě . The components of theproduct m on q H ˚ Λ r n s with respect to the splitting (6) are, for a, b P H ˚ Λ r n s and ¯ f , ¯ g P H ´˚ Λ r n s , given by(1) m ``` “ µ ;(2) m ´´´ “ ⊛ , i.e. x m ´´´ p ¯ f , ¯ g q “ p´ q | g | x f b g, λ p a qy ;(3) m ``´ “ and m ´´` “ ;(4) m ´`` p ¯ f , a q “ ´p´ q | f | x f b , λ p a qy , and m `´` p b, ¯ f q “ p´ q | b | x λ p b q , b f y ;(5) x m ´`´ p ¯ f , a q , b y “ x f, µ p a, b qy , and x a, m `´´ p b, ¯ f qy “ p´ q | b | x µ p a, b q , f y . OINCAR´E DUALITY FOR LOOP SPACES 7
The grading used in the previous statement has to be understood asfollows, see also [24, Appendix B]. For a homology class a P H ˚ Λ r n s the degree | a | is equal to the geometric degree shifted down by n , withrespect to which the product µ has degree 0. For a cohomology class f P H ´˚ Λ r n s “ H ´˚´ n Λ the degree ¯ f is minus (geometric degree shifted upby n ), and we denote by ¯ f P H ´˚ Λ r n s “ H ´˚´ n Λ the same class withdegree | ¯ f | “ | f | `
1, i.e. minus (geometric degree shifted up by n ´ H ´˚ Λ r n s b H ˚ Λ r n s Ñ R hasdegree 0 and the coproduct λ : H ˚ Λ r n s Ñ H ˚ Λ r n s b has odd degreeequal to 1 ´ n .Theorem 1.6 shows that the product on q H ˚ Λ has in general mixedterms with respect to the splitting (6), which is therefore not a directsum of rings.
Remark 1.7.
The product m on q H ˚ Λ r n s is graded commutative (i.e. m “ m ˝ τ with τ the graded flip of factors in a tensor product) as a gen-eral consequence of graded commutativity of the pair-of-pants productin Hamiltonian Floer theory. From this perspective, the two formu-las in Theorem 1.6(4) are equivalent to each other as a consequenceof m ´`` “ m `´` ˝ τ , and so are the two formulas in Theorem 1.6(5)as a consequence of m ´`´ “ m `´´ ˝ τ . However, these formulas admitcounterparts for based loops and, in that context, no such automaticequivalence holds because the product on extended based loop homol-ogy is not graded commutative. See also § Remark 1.8.
The proof of Theorem 1.4 adapts in order to provide acanonical splitting of the cohomological counterpart of (2), and there-fore a decomposition q H ´˚ Λ “ H ´˚ Λ ‘ H ˚ Λ. The proof of Theo-rem 1.1 extends in order to show that the Poincar´e duality isomorphism
P D : q H ˚ Λ “ H ˚ Λ ‘ H ´˚ Λ » ÝÑ H ´˚ Λ ‘ H ˚ Λ “ q H ´˚ Λsimply flips the two factors. We will not give the details of this proof.In spite of this simple behavior with respect to the canonical splitting H ˚ Λ ‘ H ´˚ Λ, the Poincar´e duality isomorphism is far from beingtrivial. Indeed, as we will explain next, it exchanges the ring structureon q H ˚ Λ, which is of primary nature, with the ring structure on q H ´˚ Λ,which is of secondary nature.1.3.
Closed TQFT structures.
Recall that a closed TQFT structure on an R -module V is a collection of operations ψ S : V b q Ñ V b p associ-ated to homeomorphism classes S of compact connected oriented sur-faces with q ě p ě closed noncompact TQFT KAI CIELIEBAK, NANCY HINGSTON, AND ALEXANDRU OANCEA structure differs from this only by the requirement that each surfacemust have at least one outgoing boundary component. Ritter [57] hasproved that the symplectic homology of any Liouville domain carriesa closed noncompact TQFT structure, and by [25] the same holds forthe symplectic homology of any Liouville cobordism, in particular for q H ˚ Λ. The following result, which is proved in §
5, describes this TQFTstructure in more detail and relates it to Poincar´e duality.
Theorem 1.9 (Closed TQFT structures) . (a) q H ˚ Λ carries a primary closed noncompact TQFT structure with degree ´ n product m and co-product . Dually, q H ´˚ Λ carries a primary closed noncompact TQFTstructure with product and degree ´ n coproduct m ˚ algebraically dualto m .(b) q H ˚ Λ carries a secondary closed noncompact TQFT structure withproduct and degree ´ n coproduct c . Dually, q H ´˚ Λ carries a sec-ondary closed noncompact TQFT structure with degree ´ n product c ˚ algebraically dual to c and coproduct .(c) Poincar´e duality P D : q H ˚ Λ – ÝÑ q H ´˚ Λ sends m to c ˚ and c to m ˚ . The situation is summarized in the following table, where algebraicduality interchanges the columns, while Poincar´e duality interchangesthe rows as well as columns and shifts degrees by 1. q H ˚ Λ q H ´˚ Λprimary product m product 0coproduct 0 coproduct m ˚ secondary product 0 product c ˚ coproduct c coproduct 01.4. Puzzles resolved.
Now we can resolve the puzzles.Puzzle (a) is resolved by Corollary 1.5: the cohomology product hasa canonical extension from H ˚ p Λ , Λ q to H ˚ Λ. With coefficients in afield K , if M is K -orientable this equals H ˚ p Λ , ˚q if χ p M q ‰ K , and H ˚ Λ if χ p M q “ K . If M is not K -orientable (hence char p K q ‰ H ˚ p Λ q if χ p M q “ K . If M is not K -orientable, such anextension may not exist if χ p M q ‰ K .Puzzle (b) is resolved by Theorem 1.6: the loop product has a canonicalextension from H ˚ Λ (Morse homology on Λ) to a product m on q H ˚ Λ “ H ˚ Λ ‘ H ´˚ Λ whose restriction to the second summand H ´˚ Λ (Morsecohomology of Λ) is the cohomology product.
OINCAR´E DUALITY FOR LOOP SPACES 9
Puzzle (c) is resolved by the observation that
Sullivan’s relation followswith field coefficients from associativity of the product m on q H ˚ Λ r n s . Indeed, the ``´` component of the associativity relation m p b m ´ m b q “ “ m ``` p b m `´` q ` m `´` p b m `´´ q ´ m `´` p m ``` b q ´ m ´´` p m ``´ b q , where the first and the third summands correspond to splitting along H ˚ Λ r n s , and the second and the fourth summands to splitting along H ´˚ Λ r n s . By Theorem 1.6(2) the last summand vanishes. The re-maining three summands are shown in Figure 1, where inputs from H ˚ Λ r n s and outputs in H ´˚ Λ r n s are drawn as positive ends, and out-puts in H ˚ Λ r n s and inputs from H ´˚ Λ r n s as negative ends. Now weevaluate each term on inputs a, b P H ˚ Λ r n s and ¯ f P H ´˚ Λ r n s .Using Theorem 1.6(1) and (4) the first term becomes m ``` p b m `´` qp a b b b ¯ f q “ µ p a b m `´` p b b ¯ f qq“ p´ q | b | µ p a b x λ p b q , b f yq“ p´ q | b | ¨ p´ q | a | xp µ b qp b λ qp a b b q , b f y . There is no sign involved in the first equality because m `´` has degree0. The sign p´ q | b | appears in the second equality via formula (4) for m `´` . The sign p´ q | a | in the third equality appears because λ hasdegree odd degree (equal to 1 ´ n ) as a coproduct on H ˚ Λ r n s .Using Theorem 1.6(4) and (5) the second term becomes m `´` p b m `´´ qp a b b b ¯ f q “ m `´` p a b m `´´ p b b ¯ f qq“ p´ q | a | x λ p a q , b m `´´ p b b ¯ f qy“ p´ q | a | ¨ p´ q | b | xp b µ qp λ b qp a b b q , b f y . There is no sign involved in the first equality because m `´´ has degree0, whereas the signs p´ q | a | and p´ q | b | appear by the formulas (4) and(5) for m `´` and m `´´ .Using Theorem 1.6(1) and (4) the third term becomes m `´` p m ``` b qp a b b b ¯ f q “ m `´` p µ p a b b q b ¯ f q“ p´ q | a |`| b | x λµ p a b b q , b f y . The sign p´ q | a |`| b | appears by formula (4) for m `´` .Summing up the three terms we find Sullivan’s relation applied to a b b and inserted into 1 b f . If we use field coefficients, this implies Sullivan’srelation. Remark 1.10.
The above proof infers Sullivan’s relation from the asso-ciativity of the extended loop homology product with field coefficients.In [21] we prove that it holds with arbitrary coefficients. In the context of [24] Sullivan’s relation can be shown to be part of the chain level A -structure for the product on the cone. ¯ fba ba ` a b ´ a b ¯ f ¯ f ¯ f Figure 1.
Sullivan’s relation.Puzzle (d) is resolved by Theorem 1.9 restricted to the first summand ofthe splitting q H ˚ Λ “ H ˚ Λ ‘ H ´˚ Λ: the cohomology product on H ˚ Λ isthe secondary product derived from a closed noncompact TQFT withvanishing primary product.Puzzle (e) is resolved in a somewhat unexpected way: The loop product ‚ on H ˚ Λ and the cohomology product ⊛ on H ˚ Λ extend to products m on q H ˚ Λ and c ˚ on q H ˚ Λ which are dual to each other via Poincar´eduality. However, as explained in Remark 1.8 the Poincar´e dualityisomorphism matches the H ˚ Λ summands with their loop products onboth sides, rather than relating the loop and cohomology products.On the other hand, the ring p q H ˚ Λ , m q contains both p H ˚ Λ , ‚q and p H ´˚ Λ , ⊛ q as subrings, so each result proved for the product m impliesa dual pair of results for ‚ and ⊛ . We will discuss such examples inSection 2. The discussion requires various extensions of the precedingresults which we will now describe.1.5. Length filtration.
Let us fix a Riemannian metric on M anddenote by Λ ă a Ă Λ ď a Ă Λ the subspaces of loops of length ă a resp. ď a . For a ă b we set H p a,b q˚ Λ : “ H ˚ p Λ ă b , Λ ď a q , H ˚p a,b q Λ : “ H ˚ p Λ ă b , Λ ď a q . These groups form a double filtration in the sense that we have canonicalmaps H p a,b q˚ Λ Ñ H p a ,b q˚ Λ for a ď a and b ď b with the obviousproperties, and similarly on cohomology. Theorem 1.11 (Filtered Poincar´e duality for free loop spaces) . Achoice of Riemannian metric on M induces double filtrations q H p a,b q˚ Λ on q H ˚ Λ and q H ˚p a,b q Λ on q H ˚ Λ which are compatible with OINCAR´E DUALITY FOR LOOP SPACES 11 ‚ the products on homology and cohomology in the sense that m : q H p a,b q˚ Λ b q H p a ,b q˚ Λ Ñ q H p max t a ` b ,a ` b u ,b ` b q˚ Λ ,c ˚ : q H ˚p a,b q Λ b q H ˚p a ,b q Λ Ñ q H ˚p a ` a , min t a ` b ,a ` b uq Λ; ‚ Poincar´e duality in the sense that it induces isomorphisms
P D p a,b q : q H p a,b q˚ Λ » ÝÑ q H ´˚p´ b, ´ a q Λ; ‚ the canonical splitting (6) in the sense that q H p a,b q˚ Λ “ H p a,b q˚ Λ ‘ H ´˚p´ b, ´ a q Λ . For ε ą M the long exact sequence (2) becomes (by [25] and Poincar´e duality on M ) ¨ ¨ ¨ H ´˚p´ ε,ε q Λ ε / / – (cid:15) (cid:15) H p´ ε,ε q˚ Λ ι / / – (cid:15) (cid:15) q H p´ ε,ε q˚ Λ π / / – (cid:15) (cid:15) H ´˚p´ ε,ε q Λ ¨ ¨ ¨ – (cid:15) (cid:15) ¨ ¨ ¨ H ´˚ p M ; O q Yr e s / / H n ´˚ p M q p ˚ / / H n ´˚ p S ˚ M q p ˚ / / H ´˚ p M ; O q ¨ ¨ ¨ where the bottom row is the Gysin sequence of the sphere bundle p : S ˚ M Ñ M with Euler class r e s and O is the orientation local systemon M . The products on H p´ ε,ε q˚ Λ and q H p´ ε,ε q˚ Λ translate into the cupproducts on H n ´˚ p M q and H n ´˚ p S ˚ M q , for which p ˚ is a ring map.Poincar´e duality becomes q H p´ ε,ε q˚ Λ P D p´ ε,ε q – / / – (cid:15) (cid:15) q H ´˚` p´ ε,ε q Λ – (cid:15) (cid:15) H n ´˚ p S ˚ M q P D – / / H ˚` n ´ p S ˚ M q where the bottom horizontal arrow is the classical Poincar´e dualityisomorphism. From this perspective, the Poincar´e duality theorem 1.1should be seen as generalizing Poincar´e duality for S ˚ M (which is al-ways an orientable manifold), and not as generalizing Poincar´e dualityon M itself. Remark 1.12.
For M orientable, ε ą R -coefficientsthe splitting map i : H ´˚ p M q – H ´˚p´ ε,ε q Λ Ñ q H p´ ε,ε q˚ Λ – H n ´˚ p S ˚ M q from Theorem 1.4 can be described as follows. Pick a closed n -form e on M representing the Euler class r e s with support in a small ball B Ă M (if r e s “ e “ ψ P Ω n ´ p S ˚ M q in the sense of [12, §
11] with dψ “ ´ p ˚ e . Then i : H ´˚ p M q Ñ H n ´˚ p S ˚ M q is induced by the wedge product with ψ . Note that this map makes sense: if r e s “ ψ is closed, so wedge with ψ induces a map H ´˚ p M q Ñ H n ´˚ p S ˚ M q ; if r e s ‰ dψ has support in p ´ p B q , so wedge with ψ induces a map H ´˚ p M q “ H ´˚ p M, B q Ñ H n ´˚ p S ˚ M q . The product of a, b P H ´˚ p M q is givenby p ˚ ` i p a q Y i p b q ˘ .A similar discussion holds if M is nonorientable, but the global angularform must then be twisted by the orientation line bundle of M .1.6. Poincar´e duality for based loops.
The above constructionshave counterparts for the based loop spaceΩ “ Ω q M “ t γ P Λ | γ p q “ q u at a basepoint q P M . Given a Riemannian metric on M considerthe corresponding unit disc/sphere cotangent bundles D ˚ M , S ˚ M andtheir fibres D ˚ q M , S ˚ q M at the basepoint. We denote by q H ˚ Ω : “ SH ˚` n p S ˚ q M q , q H ˚ Ω : “ SH ˚` n p S ˚ q M q the Lagrangian Rabinowitz Floer (co)homology, or Lagrangian symplec-tic (co)homology of the Legendrian n ´ -sphere S ˚ q M as defined in [25](see also [54]). Here Lagrangian symplectic homology and cohomol-ogy are graded by the Conley-Zehnder index of Hamiltonian chords,cf. Appendix B. We will also refer to q H ˚ Ω and q H ˚ Ω as extended basedloop homology/cohomology . Following [25], the homology group q H ˚ Ω isa unital algebra with product of degree 0. In contrast to the free loopsituation, this algebra is not commutative. The arguments in the proofof Theorem 1.1 adapt verbatim to prove that q H ˚ Ω has the structure ofa unital algebra with product of degree n ´
1, and moreover we have:
Theorem 1.13 (Poincar´e duality for based loop spaces) . There is acanonical isomorphism of unital algebras
P D : q H ˚ Ω » ÝÑ q H ´ n ´˚ Ω . The connection between Floer theory and the topology of based loopsis provided by the following isomorphism theorem, which is the La-grangian analogue of Theorem 1.2. The isomorphism holds in either ofthe following two situations: (i) wrapped Floer homology is untwistedand based loop homology is twisted by σ | Ω , the restriction of the spinlocal system from Remark 1.3 to Ω, and (ii) wrapped Floer homologyis twisted by σ | ´ and based loop homology is untwisted. Theorem 1.14 ([1, 3, 5, 6]) . There are isomorphisms of R -modules SH ˚` n p D ˚ q M q » H ˚ Ω , SH ˚` n p D ˚ q M q » H ˚ Ω . The first isomorphism intertwines the half-pair-of-pants product onwrapped Floer homology with the Pontrjagin product on based loop ho-mology. (cid:3)
OINCAR´E DUALITY FOR LOOP SPACES 13
As a consequence of results in this paper combined with [21], we showthat the second isomorphism also preserves suitably defined ring struc-tures. More precisely: (i) the group SH ˚` n p D ˚ q M q carries a secondaryproduct which extends the one on SH ˚` n ą p D ˚ q M q defined in [25], (ii)the group H ˚ Ω carries a secondary product which extends the one on H ˚ p Ω , t q uq defined in [39], and (iii) the second isomorphism intertwinesthese two products.By [25], q H ˚ Ω is related to the homology and cohomology of the basedloop space by a long exact sequence ¨ ¨ ¨ H ´ n ´˚ Ω ε / / H ˚ Ω ι / / q H ˚ Ω π / / H ´ n ´˚ Ω ¨ ¨ ¨ where ι is a ring map with respect to the Pontrjagin product on H ˚ Ω.If the coefficients of q H ˚ Ω are twisted by the spin local system σ | Ω , thenso are the coefficients of H ˚ Ω and H ˚ Ω.Note that the map ε vanishes for degree reasons, so the long exactsequence is actually a short exact sequence(7) 0 / / H ˚ Ω ι / / q H ˚ Ω π / / H ´ n ´˚ Ω / / . The next result is the based loop counterpart of Theorems 1.4 and 1.11.
Theorem 1.15 (Canonical splitting for based loop spaces) . (a) Theshort exact sequence (7) canonically splits to give (8) q H ˚ Ω “ H ˚ Ω ‘ H ´ n ´˚ Ω , where the product on q H ˚ Ω restricts to the Pontrjagin product on thesubring H ˚ Ω , and to the based cohomology product on the subring (notcontaining the unit) H ´ n ´˚ Ω .(b) All the structures are compatible with the length filtrations on q H ˚ Ω and q H ˚ Ω . The based loop analogue of Remark 1.8 holds. The cohomological coun-terpart of (7) admits a canonical splitting which induces a decomposi-tion q H ´ n ´˚ Ω “ H ´ n ´˚ Ω ‘ H ˚ Ω. The Poincar´e duality isomorphismfrom Theorem 1.13 flips the two factors in the splitting.The splitting (8) and its dual generalize the computation of q H ˚ Ω and q H ˚ Ω done by Merry [54] from Z { Corollary 1.16 (Merry [54]) . For n ě we have q H ˚ Ω » $&% H ˚ Ω , ˚ ě , , ´ n ` ă ˚ ă ,H ´˚` ´ n Ω , ˚ ď ´ n ` , and similarly q H ˚ Ω » $&% H ˚ Ω , ˚ ě , , ´ n ` ă ˚ ă ,H ´˚` ´ n Ω , ˚ ď ´ n ` . (cid:3) By [25], for ε ą / / H p´ ε,ε q˚ Ω ι / / – (cid:15) (cid:15) q H p´ ε,ε q˚ Ω π / / – (cid:15) (cid:15) H ´ n ´˚p´ ε,ε q Ω – (cid:15) (cid:15) / / / / H ´˚ pt q uq p ˚ / / H ´˚ p S ˚ q M q p ˚ / / H ´ n ´˚ pt q uq / / p : S ˚ q M Ñ q . The products on H p´ ε,ε q˚ Ω and q H p´ ε,ε q˚ Ω translate intothe cup products on H ´˚ pt q uq and H ´˚ p S ˚ q M q , for which p ˚ is a ringmap. Poincar´e duality becomes q H p´ ε,ε q˚ Ω P D p´ ε,ε q – / / – (cid:15) (cid:15) q H ´ n ´˚p´ ε,ε q Ω – (cid:15) (cid:15) H ´˚ p S ˚ q M q P D – / / H n `˚´ p S ˚ q M q where the bottom horizontal arrow is the classical Poincar´e dualityisomorphism.1.7. Open-closed TQFT structures.
Finally, we extend the resultsfrom Section 1.3 to include the based loop space. The following defini-tion is taken from [50].
Definition 1.17.
Let ´ Cob ` be the cobordism category with objectsthe compact -dimensional oriented manifolds with boundary and mor-phisms the -dimensional oriented cobordisms with nonempty outgoingboundary. Let ´ Cob ` ,c , ´ Cob ` ,o be the full subcategories withobjects given by finite collections of oriented circles, resp. finite collec-tions of oriented intervals. (“c” stands for “closed”, and “o” standsfor “open”.)An open-closed (resp. closed, resp. open) noncompact TQFT is a mono-idal (i.e., mapping disjoint unions to tensor products) functor from ´ Cob ` (resp. ´ Cob ` ,c , resp. ´ Cob ` ,o ) to the category of graded R -modules. Convention.
It is on purpose that we did not specify in the definitionwhether the functor is covariant or contravariant. In our convention,covariance corresponds to homology and contravariance corresponds tocohomology.
OINCAR´E DUALITY FOR LOOP SPACES 15
The boundary of a cobordism in 2 ´ Cob ` consists of an incomingpart, an outgoing part, and some free boundary part meeting the in-coming/outgoing boundaries in corners. See Figure 2 for an example.The adjective “noncompact” in the definition refers to the fact that alloperations are required to have at least one output. free boundary ` `´ Figure 2.
Morphism of 2 ´ Cob ` with 1 outgoing and2 incoming boundary components.At the level of objects, an open-closed TQFT is determined by themodule C associated to the oriented circle, and by the module O asso-ciated to the oriented interval. We refer in this case to the pair p O , C q as defining an open-closed noncompact TQFT.Our next theorem addresses open-closed TQFT structures from theperspective of Poincar´e duality. It can be seen as a generalization ofprevious work of Ritter [57]. Proofs are given in § Theorem 1.18 (Open-closed TQFT structures) . (1) The pair p q H ˚ Ω , q H ˚ Λ q defines an open-closed noncompact TQFT.(2) The pair p q H ˚ Ω , q H ˚ Λ q defines an open-closed noncompact TQFT.(3) The Poincar´e duality isomorphisms for based and free loops deter-mine a canonical isomorphism between these two TQFTs. Structure of the paper and relation to other papers.
This isthe “master paper” of a series of six papers on Poincar´e duality for loopspaces. It is related to the other papers [24, 21, 19, 22, 20] as follows.In the present paper we introduce extended loop homology and es-tablish its basic properties: Poincar´e duality, primary and secondaryTQFT structures, relation to loop product and cohomology product,extension to based loops. All these results are proved in Sections 3–5,with two exceptions: Theorem 1.6 is proved in [24] as a consequenceof a systematic study of product structures on cones of chain maps;and the last assertion in Theorem 1.4 is proved in [21] where we relatevarious constructions of secondary coproducts.In Section 2 of this paper we prove several applications of Poincar´eduality. Further applications are contained in the three sequel papers.In [19] we prove homotopy invariance of the extended loop homology algebra, which simultaneously yields homotopy invariance of the loopproduct and of the cohomology product. In [22] we compute the BValgebra structures on loop homology of compact rank one symmetricspaces, with applications to resonances and a conjecture of Viterbo.In [20] we study index growth and level nilpotency on extended loophomology, unifying earlier results on loop homology and cohomology.
Acknowledgements.
The second author is grateful for support overthe years from the Institute for Advanced Study, and in particular dur-ing the academic year 2019-20. The third author acknowledges mem-bership and financial support from the Institute for Advanced Studyduring the first half of the year 2017, when this project got started.The third author also acknowledges current financial support from theAgence Nationale de la Recherche under the grants MICROLOCALANR-15-CE40-0007 and ENUMGEOM ANR-18-CE40-0009. The au-thors thank Matthias Schwarz, who long, long ago pointed out theconcept and importance of nilpotence of products in the context of thepair-of-pants product.2.
Applications of Poincar´e duality
In this section we present evidence for Poincar´e duality that was col-lected over the past years by the second author (and got this projectstarted). In each case, we present a pair of theorems valid in the clas-sical setting of homology/cohomology of the free loop space. We thenpresent an extension of that pair of theorems to q H ˚ and q H ˚ and weexplain that, on the one hand, the extended statements are related viaPoincar´e duality, and on the other hand, the classical statements areimplied by the extended statements.2.1. Critical levels.
Recall from Section 1.5 the length filtered ho-mology and cohomology groups H p a,b q˚ Λ “ H ˚ p Λ ă b , Λ ď a q and H ˚p a,b q Λ “ H ˚ p Λ ă b , Λ ď a q with respect to some Riemannian metric on M . In par-ticular, we obtain an increasing filtration on H ˚ Λ by H p´8 ,a q˚ Λ “ H ˚ p Λ ă a q , and a decreasing filtration on H ˚ Λ by H ˚p a, Λ “ H ˚ p Λ , Λ ď a q . Definition 2.1 (Critical levels) . (1) Given a homology class X P H ˚ Λ denote Cr p X q “ inf t a P R : X P im p H p´8 ,a q˚ Λ Ñ H ˚ Λ qu . In other words, Cr p X q is the infimum of the values of a such that X is represented by a cycle contained in Λ ă a , i.e., X is supported in Λ ă a . OINCAR´E DUALITY FOR LOOP SPACES 17 (2) Given a cohomology class x P H ˚ Λ denote Cr p x q “ sup t a P R : x P im p H ˚p a, Λ Ñ H ˚ Λ qu . In other words, Cr p x q is the supremum of the values of a such that x isrepresented by a cochain that vanishes on all chains contained in Λ ď a ,i.e., x is supported in Λ ą a . Theorem 2.2 (Goresky-Hingston [39]) . (1) For any two homologyclasses X, Y P H ˚ Λ we have Cr p X ‚ Y q ď Cr p X q ` Cr p Y q . (2) For any two cohomology classes x, y P H ˚ p Λ , Λ q we have Cr p x ⊛ y q ě Cr p x q ` Cr p y q . We now consider an extension of the previous theorem to the setting of q H ˚ and q H ˚ . Recall from Theorem 1.11 that our choice of Riemannianmetric determines an increasing filtration q H p´8 .a q˚ Λ on q H ˚ Λ, and adecreasing filtration q H ˚p a, Λ on q H ˚ Λ. As such the following definitionis analogous to Definition 2.1 above.
Definition 2.3. (1) Given a homology class X P q H ˚ Λ denote Cr p X q “ inf t a P R : X P im p ˇ H p´8 ,a q˚ Λ Ñ ˇ H ˚ Λ qu . (2) Given a cohomology class x P ˇ H ˚ Λ denote Cr p x q “ sup t a P R : x P im p ˇ H ˚p a, Λ Ñ ˇ H ˚ Λ qu . Let us denote the products on q H ˚ Λ and q H ˚ Λ by µ p X, Y q “ X ‚ Y and c ˚ p x, y q “ x ⊛ y , respectively. The following extension of Theorem 2.2is now an immediate consequence of the compatibility of these productswith the filtrations in Theorem 1.11. Theorem 2.4. (1) For any two homology classes
X, Y P ˇ H ˚ Λ we have Cr p X ‚ Y q ď Cr p X q ` Cr p Y q . (2) For any two cohomology classes x, y P ˇ H ˚ Λ we have Cr p x ⊛ y q ě Cr p x q ` Cr p y q . Note that each of the statements (1) and (2) is a consequence of theother one via the Filtered Poincar´e Duality Theorem 1.11. Moreover,the Splitting Theorem 1.4 ensures that, except in the homological range t , . . . , n u , Theorem 2.2 is a consequence of either of the statements (1)or (2) in Theorem 2.4. Hopf-Freudenthal-Gysin formulas. Here we discuss the relations between the product structures on the freeand based loop spaces. We denote the Pontrjagin product (of degree 0)on H ˚ Ω by ‚ Ω , and the based cohomology product (of degree n ´
1) on H ˚ Ω by ⊛ Ω . The inclusion i : Ω ã Ñ Λ induces pushforward/pullbackmaps i ˚ : H ˚ Ω Ñ H ˚ Λ and i ˚ : H ˚ Λ Ñ H ˚ Ω, as well as shriek maps(induced by intersection with the codimension n submanifold Ω Ă Λ) i ! : H ˚ Λ Ñ H ˚´ n Ω and i ! : H ˚ Ω Ñ H ˚` n Λ. Theorem 2.5 (Goresky-Hingston [39]) . (1) For all A, B P H ˚ Λ and C P H ˚ Ω we have i ! p A ‚ B q “ i ! A ‚ Ω i ! B, p i ˚ C q ‚ A “ i ˚ p C ‚ Ω i ! A q . (2) For all a, b P H ˚ Λ and c P H ˚ Ω we have i ˚ p a ⊛ b q “ i ˚ a ⊛ Ω i ˚ b, p i ! c q ⊛ a “ i ! p c ⊛ Ω i ˚ a q . The above theorem extends to ˇ H ˚ and ˇ H ˚ as follows. We denote theproducts on ˇ H ˚ Λ by ‚ (of degree ´ n ), on ˇ H ˚ Ω by ‚ Ω (of degree 0),on ˇ H ˚ Λ by ⊛ (of degree n ´ H ˚ Ω by ⊛ Ω (of degree n ´ p ˇ H ˚ Ω , ˇ H ˚ Λ q and p ˇ H ˚ Ω , ˇ H ˚ Λ q from Theorem 1.18 include in particular operations i ! : ˇ H ˚ Λ Ñ ˇ H ˚´ n Ω , i ! : ˇ H ˚ Ω Ñ ˇ H ˚` n Λ ,i ˚ : ˇ H ˚ Ω Ñ ˇ H ˚ Λ , i ˚ : ˇ H ˚ Λ Ñ ˇ H ˚ Ω . Here i ! is the closed-open map defined by the zipper, i ˚ is the open-closed map defined by the cozipper, and i ! , i ˚ are their algebraic duals.See Figure 3. Historical note.
The name “Gysin formulas” seems to have been coined byFulton in his book on intersection theory [33] in connection with the geometricinterpretation of shriek maps in fibered setups. Gradually, this name came todesignate a variety of algebraic relations involving shriek maps. The ones thatwe prove in this section should more appropriately be called “Hopf-Freudenthalformulas”. The situation is clearly summarized in the Introduction of [27] (but thereference to Hopf’s paper is wrong). Hopf [48] first defined a “reverse” ( umkehr )homomorphism in singular homology associated to a map between manifolds of thesame dimension, and Freudenthal [32] made the connection to Poincar´e duality andhenceforth extended the definition to maps between manifolds of any dimension.In particular, Hopf [48, (4)] and Freudenthal [32, (V)] proved the finite dimensionalanalogues of the second formulas in Theorem 2.5(1) and (2) below. While these alsoappear in the later—and unique—paper of Gysin [41, (11.1)], the new contributionof that paper is to construct the “Gysin long exact sequence” associated to a spherebundle and its salient point is a geometric interpretation of an umkehr map in afibered context. This long exact sequence happens to play an important role inour paper. The umkehr homomorphisms of Hopf-Freudenthal are also referred tonotationally as “shriek” maps.
OINCAR´E DUALITY FOR LOOP SPACES 19 zipper cozipper ` `´´
Figure 3.
The zipper and the cozipper.
Theorem 2.6. (1) For all
A, B P ˇ H ˚ Λ and C P ˇ H ˚ Ω we have p .i q i ! p A ‚ B q “ i ! A ‚ Ω i ! B, p .ii q p i ˚ C q ‚ A “ i ˚ p C ‚ Ω i ! A q . (2) For all a, b P ˇ H ˚ Λ and c P ˇ H ˚ Ω we have p .i q i ˚ p a ⊛ b q “ i ˚ a ⊛ Ω i ˚ b, p .ii q p i ! c q ⊛ a “ i ! p c ⊛ Ω i ˚ a q . Proof.
Part (1) is a consequence of the open-closed noncompact TQFTstructure on p ˇ H ˚ Ω , ˇ H ˚ Λ q , see Figure 4: Relation (1.i) holds becauseboth sides are operations defined by the same punctured surface, namelythe 2-disc with two incoming punctures, both interior, and one outgo-ing boundary puncture. Relation (1.ii) holds because both sides areoperations defined by the same punctured surface, namely the 2-discwith two incoming punctures, one interior and one on the boundary,and one outgoing interior puncture.Part (2) is a consequence of the open-closed noncompact TQFT struc-ture on p ˇ H ˚ Ω , ˇ H ˚ Λ q : Relation (2.i) holds because both sides are oper-ations defined by the same punctured surface, namely the 2-disc withone positive boundary puncture (cohomologically outgoing) and twonegative interior punctures (cohomologically incoming). Relation (2.ii)holds because both sides are operations defined by the same puncturedsurface, namely the 2-disc with one positive interior puncture (coho-mologically outgoing), and two negative punctures (cohomologicallyincoming), one in the interior and one on the boundary. (cid:3) By part (3) of the TQFT Theorem 1.18, statements (1) and (2) in The-orem 2.6 imply one another via Poincar´e duality. Theorem 2.6 impliesTheorem 2.5 due to Lemma 5.7 below, which asserts that suitable re-strictions of the maps defined by the zipper and cozipper at the levelof ˇ H ˚ and ˇ H ˚ coincide with their topological counterparts through theisomorphisms of Theorems 1.4 and 1.15. i ! p A ‚ B q “ i ! A ‚ Ω i ! B ” ”` ` ` ` ` `´” ”´ ´ ´```´ ` `` ´p i ˚ C q ‚ A “ i ˚ p C ‚ Ω i ! A q Figure 4.
TQFT proof of the Hopf-Freudenthal-Gysin formulas.2.3.
Loop product with the point class.
In the following discussionwe assume that M is oriented and we use Z -coefficients. The long exactsequence (2) and the fact that ι is a ring map imply that im ε “ ker ι isan ideal. By the description (3) of the map ε we have im ε “ Z χ p M qr q s ,where r q s P H Λ is the class of the constant loop at the basepoint q P M .Since the loop product with r q s is given by the composition H ˚` n Λ i ! ÝÑ H ˚ Ω i ˚ ÝÑ H ˚ Λ , we obtain the following refinement of Corollary 1.5(1). Corollary 2.7 (Tamanoi [65]) . If χ p M q ‰ , then Z χ p M qr q s is anideal in the ring H ˚ Λ . Thus χ p M q i ˚ i ! a “ χ p M qr q s ‚ a is an integermultiple of χ p M qr q s P H Λ for each a P H ˚ Λ , in particular it vanisheswhenever deg a ‰ n or a lives in a nontrivial path component of Λ . The corollary is derived in [65] from the TQFT structure on H ˚ Λ. Notethat we always have the nontrivial product r q s ‚ r M s “ r q s . Example 2.8.
In this example we use Z -coefficients and the computa-tions of the loop homology rings in [26] . OINCAR´E DUALITY FOR LOOP SPACES 21 (a) For M “ C P n we have H n `˚ Λ C P n “ Λ r w s b Z r c, s s{x c n ` , p n ` q c n s, wc n y with | w | “ ´ , | c | “ ´ and | s | “ n . The point class is r q s “ c n , theEuler characteristic is p n ` q , and we see that Z ¨ p n ` q c n is an ideal.Note that Z ¨ c n is not an ideal.(b) For M “ S n with n even we have H n `˚ Λ S n “ Λ r b s b Z r a, s s{x a , ab, as y with | a | “ ´ n , | b | “ ´ and | s | “ n ´ . The point class is r q s “ a ,the Euler characteristic is , and we see that Z ¨ a is an ideal. Notethat Z ¨ a is not an ideal.(c) For M “ S n with n ě odd we have H n `˚ Λ S n “ Λ r a s b Z r u s “ H n `˚ p S n q b H ˚ Ω S n with | a | “ ´ n and | u | “ n ´ . The point class is r q s “ a and the Eulercharacteristic is . It follows that in the composition Λ r a s b Z r u s i ! ÝÑ Z r u s i ˚ ÝÑ Λ r a s b Z r u s the first map is the canonical projection and the second one multiplica-tion with a , so the map i ˚ i ! is the projection (with infinite dimensionalimage) H n `˚ p S n q b H ˚ Ω S n Ñ aH ˚ Ω S n – H ˚ Ω S n . (d) Let M “ G be a compact Lie group. Then the Euler characteristicis zero, since there always exists a nowhere vanishing left-invariantvector field. To compute the image of the map i ˚ i ! note that we have Λ G – G ˆ Ω G , so the K¨unneth formula gives an injection H ˚ G b H ˚ Ω G ã Ñ H ˚ Λ G. It follows that in the composition H n `˚ Λ G i ! ÝÑ H ˚ Ω G i ˚ ÝÑ H ˚ Λ G thefirst map is surjective (with right inverse a ÞÑ r G s b a ) and the secondone is the canonical inclusion, so the map i ˚ i ! is the projection (withinfinite dimensional image) H n `˚ Λ G Ą r M s b H ˚ Ω G Ñ H ˚ Ω G. Manifolds all of whose geodesics are closed.
Manifolds all ofwhose geodesics are closed have a long history of study, see [10]. In [39]it was observed that the loop and cohomology products have specialproperties on such manifolds.
Theorem 2.9. [39]
Let M be a closed n -dimensional Riemanian man-ifold all of whose geodesics are simple and closed of the same primitivelength. Let λ denote their Morse index and set b : “ n ´ ` λ . (a) Let Θ P H n ´ ` λ Λ be the homology class of the cycle consisting ofall simple closed geodesics. Then the loop product with Θ defines aninjective map ‚ Θ : H ˚ p Λ , Λ q Ñ H ˚` b p Λ , Λ q . (b) Let ω P H λ p Λ , Λ q be the Morse cohomology class generated by onesimple closed geodesic. Then the cohomology product with ω defines aninjective map ⊛ ω : H ˚ p Λ , Λ q Ñ H ˚` b p Λ , Λ q . A common generalization of this pair of results arises from the followingspecial case of a theorem of P. Uebele on Rabinowitz Floer homology.
Theorem 2.10 (Uebele [67]) . Consider a Liouville domain W with c p W q “ such that the Reeb flow on B W is T -periodic. (Here T isthe minimal common period, but there can be Reeb orbits of smallerperiods.) Let s P SH n ` b pB W q be the class of a principal orbit, corre-sponding to the maximum on the Bott manifold of Reeb orbits of period T . Suppose that b ą and all closed Reeb orbits on B W have Conley–Zehnder index ą ´ n . Then b is even and the following hold withcoefficients in a field K .(a) The class s is invertible and makes Rabinowitz Floer homology SH ˚ pB W q a free and finitely generated module over the ring of Lau-rent polynomials K r s, s ´ s .(b) This module is (not necessarily freely) generated by the Morse–Bottclasses corresponding to Reeb orbits of period at most T .(c) SH ˚ pB W q and SH ˚ p W q are finitely generated as K -algebras. Remark 2.11.
In [67] the result is stated with Z -coefficients and un-der the additional hypothesis π pB W q “
0. The hypothesis π pB W q “ K -coefficients is straightforwardusing coherent orientations in Floer theory. The restriction to field co-efficients is essential because the proof uses the fact that K r s, s ´ s is aprincipal ideal domain. Corollary 2.12.
Let M be a closed n -dimensional Riemanian manifoldall of whose geodesics are closed of (not necessarily primitive) length ℓ .Let s P q H n ` b Λ be the class of a principal closed geodesic, correspondingto the maximum on the Bott manifold of Reeb orbits of period ℓ . Sup-pose that all closed geodesics have index ą ´ n . Then b is even andthe following hold with coefficients in a field K .(a) The class s is invertible and makes extended loop homology q H ˚ Λ afree and finitely generated module over the ring of Laurent polynomials K r s, s ´ s .(b) This module is (not necessarily freely) generated by the Morse–Bottclasses corresponding to closed geodesics of length at most ℓ . OINCAR´E DUALITY FOR LOOP SPACES 23 (c) In the splitting q H ˚ Λ “ H ˚ Λ ‘ H ´˚ Λ , the summand H ˚ Λ inheritsthe structure of a free and finitely generated K r s s -module, and H ´˚ Λ inherits the structure of a free and finitely generated K r s ´ s -module.(d) q H ˚ Λ , H ˚ Λ and H ´˚ Λ are finitely generated as K -algebras.Proof. We apply Theorem 2.10 to the unit disk cotangent bundle W “ D ˚ M . Note that c p D ˚ M q “
0, and by assumption all closed geodesicshave Conley–Zehnder index ( “ Morse index) ě ą ´ n . Moreover, b “ n ´ ` λ ą
0, where λ ą ´ n is the Morse index of a principalclosed geodesic. So the hypotheses of Theorem 2.10 are satisfied, andparts (a) and (b) follow immediately.For part (c) consider the induced splitting(9) H ´˚ p S ˚ M q “ H n `˚ p M q ‘ H ´ n ´˚ p M q . on the constant loops, where the first summand is contained in H ˚ Λand the second one in H ´˚ Λ. Denote by H p ,ℓ s˚ Λ Ă q H ˚ Λ the subspacegenerated by the positively traversed Reeb orbits of period (or action)in p , ℓ s . We claim that H ˚ Λ is the K r s s -submodule of q H ˚ Λ generatedby the K -vector space V ˚ : “ H n ´˚ p M q ‘ H p ,ℓ s˚ Λ . To see this, we use that by construction of the splitting in Section 4.7the summand H ˚ Λ is generated by the positively traversed Reeb orbitsand the constant orbits generating H n ´˚ p M q , while H ˚ Λ is generatedby the negatively traversed Reeb orbits and the constant orbits gen-erating H ´ n `˚ p M q . In particular, s P H ˚ Λ. Since H ˚ Λ Ă q H ˚ Λ isa subring, this proves the inclusion K r s s V ˚ Ă H ˚ Λ. For the converseinclusion, we use an argument from [67]. By Theorem 2.10(b), the K -vector space q H ˚ Λ is generated by elements of the form s k a with k P Z and a P H p ,ℓ s˚ Λ. If s k a P H ˚ Λ, then by action reasons we must have k ě ´
1. If k ě
0, then s k a P K r s s H p ,ℓ s˚ Λ Ă K r s s V ˚ . If k “ ´
1, then s k a must belong to the constant part H n ´˚ p M q Ă V ˚ and the claim isproved.By the claim, H ˚ Λ is finitely generated as a K r s s -submodule. It istorsion free because q H ˚ Λ is torsion free as a K r s, s ´ s -module. Since K r s s is a principal ideal domain, it follows that the K r s s -module H ˚ Λis free. This proves the assertion on H ˚ Λ. An analogous argumentgives the assertion on H ´˚ Λ, which is the K r s ´ s -submodule of q H ˚ Λgenerated by the K -vector space V ˚ : “ H r´ ℓ, q˚ Λ ‘ H ´ n `˚ p M q . Part (d) is an immediate consequence of part (c). (cid:3)
Corollary 2.12 requires neither that the geodesics have the same prim-itive length, nor that they are simple. In order to describe the alge-bra structure in examples, suppose now that all geodesics are closedwith the same primitive length ℓ . Then the spaces V ˚ and V ˚ in theproof of Corollary 2.12 can be replaced by H n `˚ p M q ‘ s H ´ n ´˚ p M q and s ´ H n `˚ p M q ‘ H ´ n ´˚ p M q , respectively, and we extract from theproof the following statements: ‚ The K r s, s ´ s -module q H ˚ Λ is (not necessarily freely) generated by H ´˚ p S ˚ M q “ H n `˚ p M q ‘ H ´ n ´˚ p M q . ‚ The K r s s -submodule H n `˚ Λ is (not necessarily freely) generated by r V ˚ : “ H n `˚ p M q ‘ s H ´ n ´˚ p M q . ‚ The K r s ´ s -submodule H ´ n ´˚ Λ is (not necessarily freely) gener-ated by r V ˚ : “ s ´ H n `˚ p M q ‘ H ´ n ´˚ p M q . Let us introduce the degree shifted algebra q H ˚ Λ : “ q H ˚` n Λ “ H ˚` n Λ ‘ H ´ n ´˚ Λ , graded by the shifted degree | γ | “ CZ p γ q ´ n . Then the product hasdegree zero and is graded commutative, and the class s above has degree | s | “ b “ n ´ ` λ ą , where λ is the Morse index of a principal closed geodesic. Example 2.13 (Spheres) . For the loop space of S n the loop productand the cohomology product have been computed in [26] and [39] , respec-tively. Corollary 2.12 provides a simple way to derive one product fromthe other in the case n ě . For this, note first that in this case eachclosed geodesic has index at least λ “ n ´ ą ´ n , so Corollary 2.12is applicable with the generator s of shifted degree | s | “ n ´ ` λ “ n ´ . Now we distinguish two cases.
The case n ě In this case the K r s, s ´ s -module q H ˚ Λ S n isgenerated by the graded vector space H ´˚ p S ˚ S n q “ span K t , a, b, ab u in degrees | | “ , | a | “ ´ n, | b | “ ´ n, | ab | “ ´ n, where , a generate the first summand and b, ab the second one in thesplitting (9) . For degree reasons there can be no nontrivial relations OINCAR´E DUALITY FOR LOOP SPACES 25 involving different powers of s in span K t , a, b, ab u b K K r s, s ´ s and weconclude that q H ˚ Λ S n “ span K t , a, b, ab u b K K r s, s ´ s as a K r s, s ´ s -module. The preceding discussion then gives H n `˚ Λ S n “ span K t , a, u, au u b K K r s s , u : “ sb, | u | “ n ´ as a K r s s -module, and H ´ n ´˚ Λ S n “ s ´ span K t , a, u, au u b K K r s ´ s as a K r s ´ s -module. Here the reduced (co)homologies are the samebecause χ p S n q “ . To determine the ring structure we use an inputfrom [26] , where it is shown that the ring structure on H n `˚ Λ S n hasonly one additional relation u “ s , hence H n `˚ Λ S n “ K r a, u s{x a y as a K -algebra. Since any relation in q H ˚ Λ S n gives rise under multi-plication by a large negative power of s to a relation in H n `˚ Λ S n andvice versa, it follows that q H ˚ Λ S n “ K r a, u, u ´ s{x a y as a K -algebra. This in turn implies that H ´ n ´˚ Λ S n “ u ´ K r a, u ´ s{x a y as a K -algebra. Since the classes u ´ , u ´ a correspond to the constantloops, the cohomology relative to the constant loops becomes H ´ n ´˚ p Λ S n , Λ S n q “ u ´ K r a, u ´ s{x a y as a K -algebra, in accordance with [39] . The case n ě If K has characteristic the (co)homology ringsare exactly as in the case n odd. Suppose now that K has characteristic ‰ . Then the K r s, s ´ s -module q H ˚ Λ S n is generated by the gradedvector space H ´˚ p S ˚ S n q “ span K t , c u in degrees | | “ , | c | “ ´ n, where generates the first summand and c the second one in the split-ting (9) . Again there can be non nontrivial relations and we concludethat q H ˚ Λ S n “ span K t , c u b K K r s, s ´ s as a K r s, s ´ s -module, H n `˚ Λ S n “ span K t , b u b K K r s s , b : “ sc, | b | “ ´ as a K r s s -module, and H ´ n ´˚ Λ S n “ s ´ span K t , b u b K K r s ´ s as a K r s ´ s -module. Note that the non-reduced loop space homology H n `˚ Λ S n “ ´ span K t , b u b K K r s s ¯ ‘ K a, | a | “ ´ n is not free as a K r s ´ s -module because sa “ . The ring structure isagain determined in [26] , where it is shown that H n `˚ Λ S n “ K r a, b, s s{x a , ab, b , sa y as a K -algebra. (Here the factor can be dropped because it is invertiblein K , but the homology as written also gives the correct answer for K replaced by Z .) From this we again deduce the K -algebras q H ˚ Λ S n “ K r b, s, s ´ s{x b y ,H n `˚ Λ S n “ K r b, s s{x b y , H ´ n ´˚ Λ S n “ s ´ K r b, s ´ s{x b y ,H ´ n ´˚ p Λ S n , Λ S n q “ s ´ K r b, s ´ s{x b y , the last one in accordance with [39] . String point invertibility and resonances for CROSS.
Ex-ample 2.13 can be generalized to all compact rank one symmetric spaces(CROSS) , i.e., the projective spaces R P n , C P n , H P n , and the Cayleyplane CaP . This is carried out in joint work with E. Shelukhin [22]where we compute for each CROSS the extended loop homology ringtogether with its BV operator (see Section 6), and thus by restrictionthe BV algebra structures on its loop homology and loop cohomology.Moreover, we apply these computations to the following two questions. String point invertibility.
Consider a closed manifold M and denote by t¨ , ¨u the Chas-Sullivan loop bracket on H ˚ Λ. For any given class a P H ˚ Λ, consider the operator P a : H ˚ p M q Ñ H ˚` deg a ´ n ` p M q definedby P a “ ev ˚ ˝ t¨ , a u ˝ i ˚ , with i : M ã Ñ Λ the inclusion of constant loops, and ev : Λ Ñ M theevaluation. We call M string point invertible if there exists a coeffi-cient field K such that M is K -orientable, and a collection of classes a , . . . , a N P H ˚ Λ such that r M s “ P a N ˝ ¨ ¨ ¨ ˝ P a pr pt sq . This property was introduced by Shelukhin [62], who derived from itthe following conjecture of Viterbo for string point invertible manifolds:
The spectral norm of the pair consisting of the zero-section inside T ˚ M and its image under a Hamiltonian diffeomorphism supported in theunit disc bundle is uniformly bounded. Moreover, Shelukhin provedthat spheres are string point invertible, and string point invertibility ispreserved under taking products. Generalizing this, we have This is defined by t A, B u “ p´ q | A | ∆ p AB q ´ p´ q | A | p ∆ A q B ´ A ∆ B , with ∆the BV-operator on H ˚ Λ “ H ˚` n Λ (cf. § OINCAR´E DUALITY FOR LOOP SPACES 27
Theorem 2.14. [22] (a) Let M be a CROSS modelled on C P d , H P d , or CaP (set d “ in this last case). Then M is string point invertibleif and only if the Euler characteristic χ p M q “ d ` is prime (withcoefficient field K “ Z {p d ` q Z q .(b) Let M be a CROSS modelled on R P n , n ě . Then M is not stringpoint invertible with Z { -coefficients.Resonances. Consider a closed Riemannian manifold M and fix a coef-ficient ring R . To (co)homology classes X P H k p Λ M q and x P H k p Λ M q we associate their degrees deg p X q “ k , deg p x q “ k and critical levelsCr p X q , Cr p x q as defined in Section 2.1. We say that M is resonant with R -coefficients if there exists a constant α ą p X q ´ α Cr p X q and deg p x q ´ α Cr p x q are uniformly bounded for all X P H ˚ p Λ M ; R q and x P H ˚ p Λ M ; R q .This property is introduced in [46] and its implications for indices andlengths of closed geodesics are discussed. Moreover, it is proved in [46]that spheres of dimension at least 3 are resonant with field coefficients.Generalizing this, we have Theorem 2.15. [22]
The string point invertible CROSS from Theo-rem 2.14(a) are resonant with coefficients in the field Z {p d ` q Z . Index growth.
Consider the following result on the index growthof an iterated closed geodesic.
Theorem 2.16 ([39, Proposition 6.1]) . Let γ be a closed geodesic withindex λ and (transverse) nullity ν on a manifold of dimension n . Let λ m and ν m denote the index and nullity of the m -fold iterate γ m . Then ν m ď p n ´ q and (10) | λ m ´ mλ | ď p m ´ qp n ´ q , (11) | λ m ` ν m ´ m p λ ` ν q| ď p m ´ qp n ´ q . These inequalities follow from standard properties of the Bott function S Ñ Z determined by the linearization of the geodesic flow along γ ,see [11, 39]. In the context of the present paper, we wish to explainthat (10) and (11) are dual statements. We proceed as in the previ-ous sections: first generalize each of these inequalities to a symplecticsetting, then prove a duality theorem for the generalized statements.The linearization of the geodesic flow along γ determines a path inSp p p n ´ qq starting at Id, canonically defined up to conjugation.Based on Bott [11], Long [52] developed an index iteration theory forgeneral paths of symplectic matrices, not necessarily obtained as lin-earizations of geodesic flows. To any path P : r , s Ñ Sp p N q suchthat P p q “ Id is assigned its
Bott-Long index i p P q P Z . See [52, Definitions 5.2.7 and 5.4.1]. (In the notation of Long [52] wehave i p P q “ i p P q .) This is defined to be the Conley-Zehnder index ofthe concatenation P P p q ξ where ξ is a “ minus curve”, i.e. a pathof the form t t ÞÑ exp p tJ S q : t P r , ε su with S a symmetric negativedefinite matrix and ε ą The nullity of such a path is ν p P q “ ν p P p qq “ dim ker p P p q ´ Id q . The key property that is of interest to us regarding the Bott-Long indexis that, if P “ P γ : r , s Ñ Sp p p n ´ qq is the linearized transversegeodesic flow along a given geodesic γ , then i p P γ q equals the Morseindex of γ , cf. [52, Theorem 7.3.4] and [11, Theorem A]. Similarly, ν p P γ q equals the nullity of γ (in the transverse direction).To formulate the iteration inequalities for the Bott-Long index, define P m : r , m s Ñ Sp p N q by P m p t q “ P p t ´ j q P p q j , j ď t ď p j ` q , j “ , . . . , m ´ . If P “ P γ : r , s Ñ Sp p p n ´ qq is the linearized transverse geodesicflow along a given geodesic γ , then P m is the linearized transversegeodesic flow along the m -th iterate γ m .The following generalization of Theorem 2.16 is just a reformulation ofa result by Liu and Long. It specializes to Theorem 2.16 if P “ P γ isthe linearized transverse geodesic flow along some geodesic γ . Theorem 2.17 (Liu-Long [51]) . Let P : r , s Ñ Sp p N q be a contin-uous path with P p q “ Id . Then for all m ě we have (12) | i p P m q ´ mi p P q| ď p m ´ q N, (13) ˇˇ p i p P m q ` ν p P m qq ´ m p i p P q ` ν p P qq ˇˇ ď p m ´ q N. Proof.
According to [51, Theorem 1.2] (see also [52, Theorem 10.1.3]),the following inequalities hold for all m ě m ` i p P q ` ν p P q ´ N ˘ ` N ´ ν p P qď i p P m qď m ` i p P q ` N ˘ ´ N ´ p ν p P m q ´ ν p P qq . Theorem 2.17 follows directly from these using the additional obvi-ous inequalities 2 N ě ν p P m q ě ν p P q ě
0. For example, the secondinequality implies i p P m q ´ mi p P q ď p m ´ q N ´ p ν p P m q ´ ν p P qq ď p m ´ q N. (cid:3) Remark 2.18.
The proof of Theorem 2.17 ultimately relies on prop-erties of the Bott function determined by the path P . As an example, we have i p P ” Id q “ ´ N . OINCAR´E DUALITY FOR LOOP SPACES 29
The key definition for the duality statement is the following.
Definition 2.19.
Given a path P : r , s Ñ Sp p N q with P p q “ Id ,the reverse path P : r , s Ñ Sp p N q is defined by P p t q “ P p ´ t q P p q ´ , t P r , s . Note that P p q “ Id. The motivation for the definition is the following.Consider a 1-periodic compactly supported Hamiltonian H : R { Z ˆ R N Ñ R and denote ϕ t , t P R the flow of the Hamiltonian vectorfield X tH , t P R { Z , which solves the equation ddt ϕ t “ X tH ˝ ϕ t withinitial condition ϕ “ Id. The 1-periodicity of the Hamiltonian implies ϕ t ˝ ϕ “ ϕ ˝ ϕ t “ ϕ ` t for all t P R . The reverse flow ϕ t “ ϕ ´ t satisfies the equation ϕ t ˝ ϕ “ ϕ ´ t , and its linearization satisfies theequation dϕ t “ dϕ ´ t ˝ dϕ ´ . Thus, reversing the time direction for aHamiltonian flow corresponds at the linearized level to reversal of thepath as in Definition 2.19. Proposition 2.20.
Given a path P : r , s Ñ Sp p N q with P p q “ Id ,the index of the reverse path P is (14) i p P q “ ´ i p P q ´ ν p P q . Before proving the proposition we first state and prove our dualitytheorem for the index.
Theorem 2.21 (Duality for the index) . Let P : r , s Ñ Sp p N q be apath with P p q “ Id and P be the reverse path.(i) The index inequality (12) for P is equivalent to the index+nullityinequality (13) for P .(ii) The index+nullity inequality (13) for P is equivalent to the indexinequality (12) for P .Proof. Since taking the reverse of a path is an involutive operation, as-sertions (i) and (ii) are equivalent. To prove (ii) we use Proposition 2.20and the equality P m “ P m to get mi p P q ´ p m ´ q N ď i p P m q ď mi p P q ` p m ´ q N ô m p´ i p P q ´ ν p P qq ´ p m ´ q N ď ´ i p P m q ´ ν p P m q ď m p´ i p P q ´ ν p P qq ` p m ´ q N ô m p i p P q ` ν p P qq ` p m ´ q N ě i p P m q ` ν p P m q ě m p i p P q ` ν p P qq ´ p m ´ q N. (cid:3) Proof of Proposition 2.20.
We use the
Long index for paths P : r , s Ñ Sp p N q which do not necessarily start at the identity, cf. [52, Defini-tion 6.2.9]. We denote it i L p P q P Z . The Long index is defined from the Bott-Long index by attaching to P an arbitrary path ξ starting at Id and ending at P p q and setting i L p P q “ i p ξ P q ´ i p ξ q . Among the properties of the Long index we will use additivity under concatenations, vanishing on paths for whichthe nullity is constant along the path, and homotopy invariance withfixed endpoints. Taking into account that i p P ” Id q “ ´ N , we seethat the Long index and the Bott-Long index for paths P which startat the identity are related by the equation i L p P q “ i p P q ` N. In particular, equation (14) is equivalent to(15) i L p P q “ ´ i L p P q ´ ν p P q ` N. To prove (15) we use a method from [18, Lemma 2.3]. Denote P ´ p t q “ P p ´ t q , so that P “ P ´ P p q ´ . We start from the relation P ´ P ´ ´ ” Id. Given two paths
Q, R starting at the identity, their product QR : t ÞÑ Q p t q R p t q is homotopic to the concatenation Q Q p q R . As a par-ticular case, the path P ´ P ´ ´ “ P P p q P ´ ´ is homotopic to the concate-nation of P ´ P p q ´ “ P and P p q P p q P ´ ´ “ P ´ ´ . From i L p Id q “ i L p P q “ ´ i L p P ´ ´ q . Since the concatenation P P ´ is homotopic with fixed endpoints toId, we obtain that i L p P ´ q “ ´ i L p P q . On the other hand, we provebelow that(16) i L p R ´ q “ ´ i L p R q ´ ν p R p qq ` ν p R p qq for any path R : r , s Ñ Sp p N q . Then (15) follows: i L p P q “ ´ i L p P ´ ´ q“ ´p´ i L p P ´ q ´ ν p P ´ p qq ` ν p P ´ p qqq“ ´p i L p P q ´ ν p P p qq ` ν p P p qqq“ ´ i L p P q ´ ν p P q ` N. Equation (16) follows directly from the definition of the Long index i L and the equation(17) i p R ´ q “ ´ i p R q ´ ν p R q for paths R : r , s Ñ Sp p N q with R p q “ Id. We are thus left toprove (17).Denote P ˚ p N q “ t R : r , s Ñ Sp p N q : R p q “ Id , det p R p q ´ Id q ‰ u . We first observe that (17) holds for paths R P P ˚ p N q :the Bott-Long index i p R q is equal to the Conley-Zehnder index on P ˚ p N q [52, Definition 5.2.7], and the definition of the Conley-Zehnderindex in terms of the canonical extension ρ : Sp p N q Ñ U p q of thedeterminant function det : U p N q Ñ U p q from [59, Theorem 3.1] im-plies the equality i p R ´ q “ ´ i p R q for all paths R P P ˚ p N q since ρ p A ´ q “ ρ p A q for every A P Sp p N q . Since ν p R q “ R P P ˚ p N q , this proves (17). OINCAR´E DUALITY FOR LOOP SPACES 31
To prove (17) for arbitrary paths we use the following minimizing char-acterization of the Bott-Long index, which combines Theorem 5.4.1,Definition 5.4.2, Corollary 6.1.9 and Definition 6.1.10 from [52]: i p R q “ sup U P N p R q inf t i p B q : B P U X P ˚ p N qu“ inf U P N p R q sup t i p B q : B P U X P ˚ p N qu ´ ν p R q . Here N p R q is the set of neighbourhoods of R in the space of paths r , s Ñ Sp p N q starting at Id. We obtain i p R ´ q “ sup U P N p R ´ q inf t i p B q : B P U X P ˚ p N qu“ sup U P N p R q inf t i p B ´ q : B P U X P ˚ p N qu“ sup U P N p R q inf t´ i p B q : B P U X P ˚ p N qu“ ´ inf U P N p R q sup t i p B q : B P U X P ˚ p N qu“ ´ i p R q ´ ν p R q . The third equality makes use of (17) for paths in P ˚ p N q . (cid:3) Level-Potency.
Let M be a closed Riemannian manifold of di-mension n whose nonconstant closed geodesics are isolated. Let γ be aclosed geodesic of length L ą
0, and denote by Sγ “ S ¨ γ Ă Λ the satu-ration of the geodesic with respect to the circle action. By local level ho-mology/cohomology at γ we mean H ˚ p Sγ q “ lim U Ą Sγ H ˚ p Λ X U, Λ ă L X U q and H ˚ p Sγ q “ lim U Ą Sγ H ˚ p Λ X U, Λ ă L X U q , where U is an open setin Λ containing Sγ . We will denote by H L ˚ p Λ q “ H ˚ p Λ ă L ` ε , Λ ă L q andby H ˚ L p Λ q “ H ˚ p Λ ă L ` ε , Λ ă L q for ε ą the level homol-ogy/cohomology at level L . See [39, p. 119, 131, 188], especially Prop.A.6.Consider the following conditions, where “F” stands for “fast” and“S” for “slow”, and powers are taken with respect to the loop andcohomology products, respectively. Condition 1F.
There is a nontrivial cohomology class x P H ˚ p Λ , Λ q that is level-potent; that is, cr p x m q “ m cr p x q for all m ě . Condition 1S.
There is a homology class X P H ˚ p Λ q with Cr p X q ą that is level-potent; that is, Cr p X m q “ m Cr p X q for all m ě . Condition 2F.
For some L ą and k ě there is a level cohomologyclass x P H kL p Λ q that is level-potent. That is, the level class x m P H mk `p m ´ qp n ´ q mL p Λ q is nonzero for all m ě . Condition 2S.
For some L ą and k ě there is a level homologyclass X P H Lk p Λ q that is level-potent. That is, the level class X m P H mLmk ´p m ´ q n p Λ q is nonzero for all m ě . Condition 3F. M carries an isolated closed geodesic γ that is a sym-plectically degenerate minimum (SDmin) ; that is, it has fastest possibleindex growth, and it has nontrivial local level cohomology H ˚ p Sγ q with Z { -coefficients in the dimension of its index. Condition 3S. M carries an isolated closed geodesic γ that is a sym-plectically degenerate maximum (SDMax) ; that is, it has slowest pos-sible index+nullity growth, and it has nontrivial local level homology H ˚ p Sγ q with Z { -coefficients in the dimension of its index+nullity+1. Condition 4. M has infinitely many closed geodesics. Theorem 2.22 ([20]) . Let M be a compact orientable Riemannianmanifold. Assume that all closed geodesics on M are isolated. Thenthe following implications hold.1F ñ
2F 1S ñ ô
3F 2S ô ñ ñ We will see that Theorem 2.22 is a consequence of Theorem 2.24 be-low. The implications 3S ñ ñ S -equivariant local level homology [38],which explains the slight discrepancy in indices. A comparison of thesevarious points of view will appear in [20].Our goal here is to explain the S- and F-sides of Theorem 2.22 asbeing dual. We proceed as in the previous sections: first generalize thetheorem to a symplectic setting, then prove a duality theorem for thegeneralized statement. The key definition is the following. OINCAR´E DUALITY FOR LOOP SPACES 33
Definition 2.23.
Let p X, ξ q be a contact manifold and α P Ω p X q adefining contact form with Reeb vector field R α . A generalized closedReeb orbit for α is either a constant, or a closed orbit of R α or ´ R α . By convention, the period of a generalized closed Reeb orbit is negativefor closed orbits of ´ R α , zero for the constants, and positive for closedorbits of R α . The generalized period spectrum of α is the set of allperiods of generalized closed Reeb orbits.Let ˜ γ be a nonconstant generalized closed Reeb orbit, i.e. a closedorbit of ˘ R α . Fix a trivialization τ of the contact distribution along˜ γ . The index of ˜ γ with respect to the trivialization τ is denoted i p ˜ γ : τ q and is defined to be the Bott-Long index of the symplectic pathgiven by the linearization of the flow of ˘ R α along ˜ γ , cf. § X and such atrivialization has been fixed, we simply denote the index by i p ˜ γ q .We focus here on X “ S ˚ M with its canonical contact form α . Thenonconstant generalized closed Reeb orbits are of two kinds: lifts ofclosed geodesics on M on the one hand (these are the closed orbits of R α ), and their backward parametrizations (closed orbits of ´ R α ). Justlike closed Reeb orbits are “seen” by Hamiltonians used in the definitionof symplectic homology SH ˚ p D ˚ M q » H ˚ Λ, generalized closed Reeborbits are “seen” by Hamiltonians used in the definition of RabinowitzFloer homology SH ˚ p S ˚ M q “ q H ˚ Λ. We assume that the nonconstantclosed geodesics of M are isolated, or equivalently that the nonconstantgeneralized closed Reeb orbits on S ˚ M are isolated.Let ˜ γ be a nonconstant generalized closed Reeb orbit of period L ‰ local level extended homology/cohomology , denoted q H ˚ p S ˜ γ q and q H ˚ p S ˜ γ q , we mean symplectic homology/cohomology of S ˚ M localizednear the isolated set S ˜ γ in the following sense: we choose a Hamil-tonian H as in the definition of SH ˚ p S ˚ M q with negative slope ´ µ and positive slope τ such that L P p´ µ, τ q , and consider local Floer(co)homology of the isolated fixed point set which corresponds to S ˜ γ in the convex region of H . This is a mild variation on McLean’s defi-nition of local symplectic homology in [53]. We twist the Floer homol-ogy groups by the local system η discussed in the Introduction. Seealso [20].By level extended homology/cohomology at level L we mean q H L ˚ Λ “ q H p L ´ ε,L ` ε q˚ Λ and q H ˚ L Λ “ q H ˚p L ´ ε,L ` ε q Λ, where ε ą η .Consider the following conditions (“s” stands for “symplectic”). Condition 1Fs.
There is a cohomology class x P q H ˚ Λ with cr p x q ‰ that is level-potent; that is, cr p x m q “ m cr p x q for all m ě . Condition 1Ss.
There is a homology class X P q H ˚ Λ with Cr p X q ‰ that is level-potent; that is, Cr p X m q “ m Cr p X q for all m ě . Condition 2Fs.
For some L ‰ and k P Z there is a level cohomologyclass x P q H kL Λ that is level-potent. That is, the level class x m P q H mk `p m ´ qp n ´ q mL Λ is nonzero for all m ě . Condition 2Ss.
For some L ‰ and k P Z there is a level homologyclass X P q H Lk Λ that is level-potent. That is, the level class X m P q H mLmk ´p m ´ q n Λ is nonzero for all m ě . Condition 3Fs. S ˚ M carries an isolated nonconstant generalizedclosed Reeb orbit ˜ γ which is a symplectically degenerate minimum (SD-min) ; that is, it has fastest possible index growth, and it has nontriviallocal level extended cohomology q H ˚ p S ˜ γ q with Z { -coefficients in the di-mension of its index. Condition 3Ss. S ˚ M carries an isolated nonconstant generalizedclosed Reeb orbit ˜ γ which is a symplectically degenerate maximum (SD-Max) , that is, it has slowest possible index+nullity growth, and it hasnontrivial local level extended homology q H ˚ p S ˜ γ q with Z { -coefficientsin the dimension of its index+nullity+1. Condition 4s. S ˚ M has infinitely many closed Reeb orbits. Theorem 2.24 ([20]) . Let M be a compact orientable Riemannianmanifold. Assume that all closed geodesics on M are isolated. Thenthe following implications hold.1Fs ñ ñ ô ô ñ
4s 3Ss ñ (cid:3) We prove below that the F-column in Theorem 2.24 is equivalent tothe S-column. In turn, the S-column admits purely symplectic proofs,see [20]. The implication 3Ss ñ
4s is akin to [38, Theorem 1.2], whichuses the methods of [35]. The proof of the equivalence 2Ss ô §
4] and [36, § ñ L “ Cr p X q . OINCAR´E DUALITY FOR LOOP SPACES 35
Theorem 2.24 implies Theorem 2.22.
This is immediate in view of thesplitting q H ˚ Λ “ H ˚ Λ ‘ H ´˚ Λ from Theorem 1.4. Indeed, each of theconditions involved in Theorem 2.22 is strictly generalized within thecorresponding condition from Theorem 2.24. (cid:3)
Theorem 2.25.
Under the Poincar´e duality isomorphism, the F-columnin Theorem 2.24 is equivalent to the S-column.Proof.
It is enough to prove that the three conditions labeled “-Fs” areequivalent to the corresponding conditions labeled “-Ss”.The equivalences 1FS ô ô ô γ satisfies condition 3Fs if and only if the general-ized closed Reeb orbit ˜ γ , defined as ˜ γ traversed backwards, satisfiescondition 3Ss. This follows from the following two ingredients: (i)Proposition 2.20 ensures that their indices and nullities are related bythe formula i p ˜ γ q “ ´ i p ˜ γ q ´ ν p ˜ γ q , so that1 ´ i p ˜ γ q “ i p ˜ γ q ` ν p ˜ γ q ` , and (ii) the fact that Poincar´e duality specializes to an isomorphism q H ˚ p S ˜ γ q » q H ´˚ p S ˜ γ q . (cid:3) Homotopy invariance.
Ever since the definition of string topol-ogy operations in [15] people have wondered what structure on a man-ifold they actually depend on, e.g., whether they depend only on thehomotopy type. Note that homotopy equivalent manifolds must nec-essarily have the same dimension (since the latter is detected by thetop homology group), so the string topology operations have the samedegree. For the same reason, the manifolds are either both orientable,or both nonorientable.In 2008, Cohen, Klein and Sullivan proved that the loop product andthe string bracket are homotopy invariants [28] (see also [40, 29]), whilefor other string topology operations the question has remained open.Using the methods of the present paper, we prove in [19] the followingresult.
Theorem 2.26 ([19]) . Let M and N be closed connected manifoldswhich are homotopy equivalent. Then the rings q H ˚ p Λ M q and q H ˚ p Λ N q are isomorphic. In view of the Splitting Theorem 1.4, this implies
Corollary 2.27 ([19]) . The rings ` H ˚ p Λ M q , ‚ ˘ and ` H ˚ p Λ M q , ⊛ ˘ de-pend only on the homotopy type of M . Here Theorem 2.26 perfectly fulfils its role as a bridge between the loopproduct and the cohomology product, since it allows us to prove in onesweep the homotopy invariance of both. Let us mention that homotopyinvariance of the cohomology product was also proved using differentmethods by Hingston and Wahl in [47].3.
Poincar´e duality in Rabinowitz Floer homology
Poincar´e duality H ˚ p Σ q – H m ´˚ p Σ q for an m -dimensional closed ori-ented manifold Σ is known to be induced by a canonical chain isomor-phism M C ˚ p f q – M C m ´˚ p´ f q between the Morse chain complex ofa Morse function f : Σ Ñ R and the Morse cochain complex of ´ f .In this section we show that this isomorphism canonically extends toRabinowitz Floer homology if Σ is the boundary of a Liouville domain.3.1. Rabinowitz Floer homology.
Recall from [16] the definitionof Rabinowitz Floer homology. Consider the completion p p V , λ q of aLiouville domain p V, λ q with boundary Σ “ B V . We abbreviate by L “ C p S , p V q the free loop space of p V , where S “ R { Z . A definingHamiltonian for Σ is a smooth function H : p V Ñ R with regularlevel set Σ “ H ´ p q whose Hamiltonian vector field X H (defined by i X H ω “ ´ dH ) has compact support and agrees with the Reeb vectorfield R along Σ. Given such a Hamiltonian, the Rabinowitz actionfunctional is defined by A H : L ˆ R Ñ R ,A H p x, η q : “ ż x ˚ λ ´ η ż H p x p t qq dt. Critical points of A H are pairs p x, η q such that x : S Ñ Σ solves theequation x “ ηX H p x q “ ηR p x q . So there are three types of critical points: closed Reeb orbits on Σwhich are positively parametrized and correspond to η ą
0, closedReeb orbits on Σ which are negatively parametrized and correspond to η ă
0, and constant loops on Σ which correspond to η “
0. The actionof a critical point p x, η q is A H p x, η q “ η .Pick a smooth family p J t q t P S of compatible almost complex structureson p V that are cylindrical at infinity. It induces a metric g “ g J on L ˆ R which at a point p x, η q P L ˆ R and two tangent vectors p ˆ x , ˆ η q , p ˆ x , ˆ η q P T p x,η q p L ˆ R q “ Γ p S , x ˚ T p V q ˆ R is given by g p x,η q ` p ˆ x , ˆ η q , p ˆ x , ˆ η q ˘ “ ż ω ` ˆ x p t q , J t p x p t qq ˆ x p t q ˘ dt ` ˆ η ¨ ˆ η . OINCAR´E DUALITY FOR LOOP SPACES 37
Positive gradient flow lines of the Rabinowitz action functional A H withrespect to this metric are solutions p x, η q P C p R ˆ S , p V q ˆ C p R , R q of the Rabinowitz Floer equation(18) " B s x ` J t p x q ` B t x ´ ηX H p x q ˘ “ , B s η ` ş H p x p t qq dt “ . We fix action values ´8 ă a ă b ă 8 outside the action spectrum of A H , and we pick an additional small Morse function f on the criticalmanifold Crit p A H q . Thus f consists of a Morse function f Σ : Σ Ñ R and Morse functions f γ k : im p γ q Ñ R for each simple closed Reeb orbit γ and k P Z zt u , where we assume the latter to have unique minima m γ k and maxima M γ k . Then the chain group RF C p a,b q˚ p H, f ; J q is thefree abelian group generated by the critical points of f with action in p a, b q and the boundary operator B : RF C p a,b q˚ p H, f ; J q Ñ RF C p a,b q˚´ p H, f ; J q counts cascades as in [16]. They combine the negative gradient flowof A H with respect to the metric g and the negative gradient flow of f with respect to some metric on Crit p A H q . As grading we use the half-integer grading defined in [16]. The resulting filtered RabinowitzFloer homology groups RF H p a,b q˚ p Σ , λ q : “ RF H p a,b q˚ p H, f ; J q , are well-defined and do not depend on the choice of J , H and f , thoughthey do depend on the contact form λ | Σ . The Rabinowitz Floer homol-ogy of Σ is defined as the limit
RF H ˚ p Σ q : “ lim ÝÑ b lim ÐÝ a RF H p a,b q˚ p Σ , λ q , a Ñ ´8 , b
Ñ 8 . By [17, Theorem A], this definition is equivalent to the original onein [16]. By similar direct-inverse limits one defines as in [25] the groups
RF H ♥ ˚ p Σ q , ♥ P t ∅ , ą , ě , “ , ď , ă u , with the meaning that RF H ∅ ˚ “ RF H ˚ .We define the Rabinowitz Floer cohomology groups by a similar pro-cedure following [25, § RF C ˚p a,b q p H, f ; J q “ RF C p a,b q˚ p H, f ; J q _ . The filtered Rabinowitz Floer cohomology groupsare defined as RF H ˚p a,b q p Σ , λ q : “ RF H ˚p a,b q p H, f ; J q , and the Rabinowitz Floer cohomology of Σ is the limit
RF H ˚ p Σ q : “ lim ÝÑ a lim ÐÝ b RF H ˚p a,b q p Σ , λ q , with variants RF H ˚ ♥ p Σ q for ♥ P t ∅ , ą , ě , “ , ď , ă u . These Rabinowitz Floer co/homology groups depend on the contactstructure on Σ (though no longer on the contact form) as well as theLiouville filling V (up to Liouville homotopy). As in [25] we do notindicate the filling in the notation; the filling will always be clear fromthe context.3.2. Poincar´e duality.
Poincar´e duality results from the followingobservation: Under the canonical involution L ˆ R Ñ L ˆ R , p x, η q ÞÑ p ¯ x, ¯ η q , ¯ x p t q “ x p´ t q , ¯ η “ ´ η the Rabinowitz action functional changes sign, A H p ¯ x, ¯ η q “ ´ A H p x, η q . It follows that the involution maps positive gradient lines of A H tonegative gradient lines of A H , provided that we also replace the family J t by ¯ J t : “ J ´ t (and the resulting metric accordingly). In other words,if p x, η q P C p R ˆ S , p V q ˆ C p R , R q solves the Rabinowitz Floer equa-tion (18) with positive asymptotic p x ` , η ` q and negative asymptotic p x ´ , η ´ q , then p ¯ x, ¯ η q defined by¯ x p s, t q : “ x p´ s, ´ t q , ¯ η p s q : “ ´ η p´ s q solves (18) with positive asymptotic p ¯ x ´ , ¯ η ´ q and negative asymptotic p ¯ x ` , ¯ η ` q .When applying the involution, we also replace the Morse functions f Σ : Σ Ñ R and f γ k : im p γ q Ñ R by¯ f Σ : “ ´ f Σ , ¯ f γ k : “ ´ f γ k . Then the preceding discussion shows that the involution p x, η q ÞÑ p ¯ x, ¯ η q defines a chain isomorphism between the Rabonowitz Floer chain andcochain groups RF C p a,b q˚ p H, f ; J q – ÝÑ RF C ´˚p´ b, ´ a q p H, ¯ f ; ¯ J q . Therefore, we have shown
Theorem 3.1 (Poincar´e duality in Rabinowitz Floer homology) . Theinvolution p x, η q ÞÑ p ¯ x, ¯ η q induces isomorphisms between filtered Rabi-nowitz Floer homology and cohomology groups P D : RF H p a,b q˚ p Σ , λ q – ÝÑ RF H ´˚p´ b, ´ a q p Σ , λ q , and between the Rabinowitz Floer homology and cohomology groups P D : RF H ♥ ˚ p Σ q – ÝÑ RF H ´˚´ ♥ p Σ q for ♥ P t ∅ , ą , ě , “ , ď , ă u . (cid:3) Given ♥ P t ∅ , ą , ě , “ , ď , ă u , the meaning of ´ ♥ is thatequalities are preserved and inequalities are reversed, e.g., if ♥ “ ∅ then ´ ♥ “ ∅ , and if ♥ “ “ ě
0” then ´ ♥ “ “ ď OINCAR´E DUALITY FOR LOOP SPACES 39
Remark 3.2.
Rabinowitz Floer homology has a Lagrangian versiondefined in terms of Lagrange multipliers as above, see [54]. Theorem 3.1has a straightforward counterpart in that setting.4.
Poincar´e duality in symplectic homology
In this section we discuss Poincar´e duality in (V-shaped) symplectichomology. While being more involved than the one in Rabinowitz Floerhomology, the description in symplectic homology has three additionalfetures which we establish in this section: it is directly related to loopspace (co)homology in the case of cotangent bundles; it is compatiblewith the pair-of-pants products; and it carries a canonical splitting inthe case of cotangent bundles.Much of this section works for general Liouville domains. When appliedto cotangent bundles it leads to the proofs of Theorems 1.1 and 1.11from the Introduction. For general background on symplectic homologywe refer to [25].4.1.
Recollections on Poincar´e duality and exact sequences.
Let V be a Liouville domain of dimension 2 n . The main result in [18]states that(19) RF H ♥ ˚´ { pB V q – SH ♥ ˚ pB V q , where SH ♥ ˚ pB V q denotes the symplectic homology of the trivial cobor-dism r , s ˆ B V in the sense of [25] (which equals the “V-shaped sym-plectic homology” in [18]). Here SH ♥ ˚ pB V q is graded as in [25, 18] byConley-Zehnder indices. Then Theorem 3.1 has the following formula-tion in symplectic homology. Theorem 4.1 (Poincar´e duality in symplectic homology of a triv-ial cobordism [25], Theorem 9.4) . There exist canonical isomorphismsbetween the symplectic homology and cohomology groups of a trivialcobordism
P D : SH ♥ ˚ pB V q – ÝÑ SH ´˚´ ♥ pB V q for ♥ P t ∅ , ą , ě , “ , ď , ă u . (cid:3) This isomorphism is constructed in [25] without relying on the connec-tion between symplectic homology and Rabinowitz Floer homology.We expect the isomorphisms of Theorems 3.1 and 4.1 to be compatiblewith the identification (19). Within the rest of the paper we will referto Poincar´e duality for a trivial cobordism as being the isomorphismin Theorem 4.1.
Remark 4.2 (grading) . Poincar´e duality in Theorem 3.1 exhibits abeautiful symmetry with respect to the half-integer grading from [16],which is destroyed for the integer grading in Theorem 4.1. On the other hand, the latter grading more easily relates to the gradings in singularcohomology and loop space homology, so we will use this one in therest of the paper.The Poincar´e duality isomorphism has the following properties, see [25]. (A) Compatibility with the exact sequence of the pair p V, B V q . Theorem 4.3 ([25], Theorem 9.5) . For every Liouville domain V and ♥ P t ∅ , ą , ě , “ , ď , ă u there exists a commuting diagram (20) . . . SH ♥ ˚ p V, B V q / / – PD (cid:15) (cid:15) SH ♥ ˚ p V q / / – PD (cid:15) (cid:15) SH ♥ ˚ pB V q / / – PD (cid:15) (cid:15) SH ♥ ˚´ p V, B V q . . . – PD (cid:15) (cid:15) . . . SH ´˚´ ♥ p V q / / SH ´˚´ ♥ p V, B V q / / SH ´˚´ ♥ pB V q / / SH ´˚´ ♥ p V q . . . where the rows are the long exact sequences of the pair p V, B V q from [25] and the vertical arrows are the Poincar´e duality isomorphisms fromTheorem 4.1 (the third one) and for pairs as defined in [25] (the otherones). Moreover, the Poincar´e duality isomorphisms are compatiblewith filtration exact sequences. (cid:3) (B) Relation to singular homology. Recall from [25] that at actionzero symplectic homology specializes to singular cohomology, SH “ ˚ p V q – H n ´˚ p V q , and similarly for the other versions. Therefore, we obtain Corollary 4.4 ([25], Corollary 9.7) . The commuting diagram in The-orem 4.3 specializes at action zero to (21) . . . H n ´˚ p V, B V q / / – PD (cid:15) (cid:15) H n ´˚ p V q / / – PD (cid:15) (cid:15) H n ´˚ pB V q / / – PD (cid:15) (cid:15) H n ´˚` p V, B V q . . . – PD (cid:15) (cid:15) . . . H n `˚ p V q / / H n `˚ p V, B V q / / H n `˚´ pB V q / / H n `˚´ p V q . . . where the rows are the long exact sequences of the pair p V, B V q and thevertical arrows are the Poincar´e duality isomorphisms for the closedmanifold B V (the third one) and the manifold-with-boundary V (theother ones). (cid:3) (C) Description of the first map in (20) . The following result is simplya restatement of [18, Proposition 1.3].
OINCAR´E DUALITY FOR LOOP SPACES 41
Proposition 4.5. (a) The map SH ˚ p V, B V q Ñ SH ˚ p V q in (20) (for ♥ “ ∅ ) fits into a commutative diagram (22) SH ˚ p V, B V q / / c ˚ (cid:15) (cid:15) SH ˚ p V q H n ´˚ p V, B V q / / H n ´˚ p V q c ˚ O O in which the bottom arrow is the restriction map and the vertical arrowsare compositions of action zero isomorphisms with action truncationmaps c ˚ : H n ´˚ p V q – SH “ ˚ p V q Ñ SH ě ˚ p V q “ SH ˚ p V q ,c ˚ : SH ˚ p V, B V q “ SH ď ˚ p V, B V q Ñ SH “ ˚ p V, B V q – H ˚` n p V, B V q . (b) The map SH ´˚ p V q Ñ SH ´˚ p V, B V q in (20) (for ♥ “ ∅ ) fits intoa commutative diagram (23) SH ´˚ p V q / / c ˚ (cid:15) (cid:15) SH ´˚ p V, B V q H n `˚ p V q / / H n `˚ p V, B V q c ˚ O O in which the bottom arrow is induced by inclusion and the vertical ar-rows are compositions of action zero isomorphisms with action trunca-tion maps c ˚ : H n `˚ p V, B V q – SH ´˚“ p V, B V q Ñ SH ´˚ď p V, B V q “ SH ´˚ p V, B V q ,c ˚ : SH ´˚ p V q “ SH ´˚ě p V q Ñ SH ´˚“ p V q – H n `˚ p V q . (c) The above two diagrams are isomorphic via the Poincar´e dualityisomorphism P D . (cid:3) Remark 4.6 (Lagrangian case) . The previous results have Lagrangiancounterparts, with symplectic homology replaced by Lagrangian sym-plectic homology, or wrapped Floer homology, cf. [25]. We spell outsome of these statements, mainly with the purpose of explaining theeffect of the grading convention described in Appendix B.Given an exact n -dimensional Lagrangian L which is conical near itsboundary B L , the Poincar´e duality isomorphisms from [25] read SH ˚ p L, B L q » SH n ´˚ p L q , and(24) SH ˚ pB L q » SH n ´˚` pB L q . This fixes the grading for the Lagrangian counterpart of (20).The action zero part of Lagrangian symplectic homology and cohomol-ogy is expressed in topological terms as SH “ ˚ p L q » H n ´˚ p L q , SH ˚“ p L q » H n ´˚ p L q , where L stands for any of the symbols L , p L, B L q , or B L . This fixes thegrading for the Lagrangian counterpart of (21).The Lagrangian analogue of (22) is SH ˚ p L, B L q / / c ˚ (cid:15) (cid:15) SH ˚ p L q H n ´˚ p L, B L q / / H n ´˚ p L q . c ˚ O O In particular, if L is a disc the bottom map vanishes for degree reasons,hence the map SH ˚ p L, B L q Ñ SH ˚ p L q vanishes as well.4.2. TQFT operations on Floer homology.
In this subsection werecall the definition of TQFT operations on Hamiltonian Floer ho-mology from [60, 57], see also [25, 30]. Consider a punctured Rie-mann surface S with p negative and q positive punctures and cho-sen cylindrical ends Z ´ i “ p´8 , s ˆ S near the negative punctures z ´ i , resp. Z ` j “ r ,
8q ˆ S near the positive punctures z ` j . Let H : S ˆ p V Ñ R be an S -dependent Hamiltonian on a completed Li-ouville domain p V which is linear outside a compact subset of p V , and S -independent equal to H ˘ ℓ near each puncture z ˘ ℓ of S . Pick positiveweights A ˘ ℓ ą β on S with the following properties:(i) d S p Hβ q ď β “ A ˘ ℓ dt in cylindrical coordinates p s, t q P Z ˘ ℓ near the punc-ture z ˘ ℓ .We consider maps u : S Ñ p V that are holomorphic in the sense that p du ´ X H b β q , “ E p u q “ ş S | du ´ X H b β | vol S . They converge at the punctures to 1-periodic orbits x ˘ ℓ of H ˘ ℓ and satisfy the energy estimate(25) 0 ď E p u q ď q ÿ j “ A A ` j H ` j p x ` j q ´ p ÿ i “ A A ´ i H ´ i p x ´ i q . The signed count of such holomorphic maps yields an operation ψ S : q â j “ F H ˚ p A ` j H ` j q Ñ p â i “ F H ˚ p A ´ i H ´ i q . of degree n p ´ g ´ p ´ q q which does not increase action. Theseoperations are graded commutative if degrees are shifted by ´ n andsatisfy the usual TQFT composition rules. As they respect the action,the operations descend to operations between suitable filtered Floerhomology groups and thus to sympletic homology. OINCAR´E DUALITY FOR LOOP SPACES 43
Suppose now that H : p V Ñ R is S -independent. Then β and theweights are related by Stokes’ theorem q ÿ j “ A ` j ´ p ÿ i “ A ´ i “ ż S dβ. Conversely, if the quantity on the left-hand side is nonnegative (resp. zero,resp. nonpositive), then we find a 1-form β with properties (i) and (ii)such that dβ ě “
0, resp. ď S -independent H wecan arrange conditions (i)–(ii) in the following situations:(a) H arbitrary, dβ ” p, q ě H ě dβ ď p ě H ď dβ ě q ě S is a pair-of-pants with twopositive punctures and one negative puncture. Then the operation ψ S induces a pair-of-pants product on filtered Floer homology ψ S : F H p a ,b q i p A ` H q b F H p a ,b q j p A ` H qÝÑ F H p max t a ` b ,a ` b u ,b ` b q i ` j ´ n p A ´ H q . By taking suitable inverse and direct limits (see [25]) this leads to de-gree ´ n pair-of-pants products on SH ˚ p V q , SH ˚ p V, B V q – SH ´˚ p V q , SH ˚ pB V q , and SH ˚ p W, B W q – SH ´˚ pB V q , where W “ r , s ˆ B V .More generally, one can consider open-closed TQFT operations in Floerhomology, which mix together Hamiltonian Floer homology inputs/outputsand Lagrangian Floer homology inputs/outputs. The discussion aboutweights carries over without any modification, and we refer to Appen-dix B for a discussion of gradings in this setting.4.3. Products and the mapping cone.
We consider in this sectionan algebraic setup that will help us organize the subsequent geometricarguments regarding the compatibility of the Poincar´e duality isomor-phism with product structures.Consider a chain complex p C, Bq of the form C “ C ´ ‘ C ` , B “ ˆ B ´ f B ` ˙ . Thus p C ˘ , B ˘ q are chain complexes, C ´ is a subcomplex of C , and C is the cone of the chain map f : C ` r s Ñ C ´ . We refer to [25] forconventions on cones and degree shifts (e.g., B C ` r s “ ´B ` ).Assume now that C is acyclic as a consequence of Id C being homotopicto zero, i.e., there exists K : C Ñ C r s such thatId C “ B K ` K B . Writing K with respect to the decomposition C “ C ´ ‘ C ` as K “ ˆ k ´ hg k ` ˙ , this is equivalent to the system of four equations $’’&’’% B ` g ` g B ´ “ , B ´ k ´ ` f g ` k ´ B ´ “ Id C ´ , B ` k ` ` gf ` k ` B ` “ Id C ` , B ´ h ` f k ` ` k ´ f ` h B ` “ . The first three equations amount to the fact that g : C ´ Ñ C ` r s is a chain map which is a homotopy inverse for f , the homotopiesbetween f g and gf and the corresponding identity maps being givenby k ´ : C ´ Ñ C ´ r s , k ` : C ` Ñ C ` r s . We interpret the fourth equation as giving extra information whichwe will not use in the sequel. The presence of this extra bit of in-formation comes from the fact that requiring Id C to be homotopic tozero is a stronger condition than requiring f to be a chain homotopyequivalence, which is yet a stronger condition than requiring f to be aquasi-isomorphism, which is equivalent to acyclicity of C .Let now p C, Bq be another chain complex of the form C “ C ´ ‘ C ` , B “ ˜ B ´ f B ` ¸ . The complex C need not be acylic, though an important special caseis C “ C . In the following we will adopt the following conventions fora linear map φ : C b C Ñ C : ‚ φ `´` is the part of φ mapping C ` b C ´ Ñ C ` , etc; ‚ we abbreviate φ ` : “ φ ``` and φ ´ : “ φ ´´´ ; ‚ rB , φ s : “ B ˝ φ ´ p´ q | φ | φ ˝ pB b ` b Bq .Suppose now that we are given a “product” µ : C b C Ñ C of degreezero satisfying rB , µ s “ µ ´´` “ µ ´`` “ µ `´` “ . Thus µ descends to homology and the subcomplex C ´ is a “two-sidedideal” with respect to µ . It follows that0 “ rB , µ s ´´´ “ rB ´ , µ ´´´ s and 0 “ rB , µ s ``` “ rB ` , µ ``` s , so µ ´ “ µ ´´´ : C ´ b C ´ Ñ C ´ and µ ` “ µ ``` : C ` b C ` Ñ C ` descend to products on homology. I.e., whenever one of the inputs of µ is in C ´ , the output is in C ´ . OINCAR´E DUALITY FOR LOOP SPACES 45
Remark 4.7.
The case of a product of any degree can always be re-duced to the degree zero case by a shift of the grading on C .The following lemma is a key result for this paper. Lemma 4.8.
For p C, Bq , K , p C, Bq and µ as above the following hold.(a) The negative parts ν ´ , r ν ´ : C ´ b C ´ Ñ C ´ r s of the maps ν : “ µ p K b q : C b C Ñ C r s and r ν : “ µ p b K q satisfy µ ´ “ rB ´ , ν ´ s “ rB ´ , r ν ´ s . In particular, the primary product µ ´ vanishes on the homology of C ´ .(b) The negative part σ ´ : C ´ r´ s b C ´ r´ s Ñ C ´ r´ s of the differ-ence σ : “ r ν ´ ν satisfies rB ´ , σ ´ s “ and thus descends to a secondaryproduct on the homology of C ´ r´ s .(c) The negative part η ´ “ η ´´´ : C ´ r´ s b C ´ r´ s Ñ C ´ of η : “ µ p K b K q satisfies rB ´ , η ´ s “ σ ´ ´ f µ ` p g b g q . Thus the secondary product σ ´ agrees on homology with the product µ ` transferred from C ` to C ´ r´ s via the chain map f and the homotopyinverse g of f .Proof. For part (a) we compute, using rB , µ s “ rB , K s “ rB , ν s “ B µ p K b q ` µ p K b qpB b ` b Bq“ µ rB b ` b B , K b s“ µ ` rB , K s b ˘ “ µ, and then take the C ´ part µ ´ “ rB , ν s ´´´ “ rB ´ , ν ´ s . Part (b) follows directly from part (a). For part (c) we compute, using rB , µ s “ rB , K s “ rB , η s “ B µ p K b K q ` µ p K b K qpB b ` b Bq“ µ rB b ` b B , K b K s“ µ ´ rB , K s b K ´ K b rB , K s ¯ “ µ p b K ´ K b q . Taking the C ´ part the right hand side becomes r ν ´ ´ ν ´ “ σ ´ , whilethe left hand side becomes rB , η s ´´´ “ rB ´ , η ´ s ` f η ´´` “ rB ´ , η ´ s ` f µ ` p g b g q . (cid:3) Poincar´e duality with products.
As in the previous sections V is a Liouville domain of dimension 2 n , and W “ r , s ˆ B V Ă V is thetrivial cobordism realized by a collar neighbourhood of B V in V . Recallfrom Section 4.2 that SH ˚ p W, B W q carries a product of degree ´ n de-termined by counts of rigid pairs-of-pants in combination with suitableaction truncations. We refer to this product as the primary product on SH ˚ p W, B W q . By Poincar´e duality SH ˚ p W, B W q » SH ´˚ p W q , and werefer to the corresponding degree n pair-of-pants product on SH ˚ p W q as the primary product on SH ˚ p W q “ SH ˚ pB V q . Recall also that SH ˚ pB V q carries a unital pair-of-pants product m of degree ´ n . Theorem 4.9.
Let V be a Liouville domain of dimension n . Then:(a) The primary product on SH ˚ pB V q vanishes. As a consequence, SH ˚ pB V q carries a secondary product c ˚ of degree n ´ , which is gradedcommutative and has a unit in degree ´ n .(b) The Poincar´e duality isomorphism P D : SH ˚ pB V q » ÝÑ SH ´˚ pB V q is a ring homomorphism, where SH ˚ pB V q is endowed with its degree ´ n pair-of-pants product m and SH ˚ pB V q is endowed with its degree n ´ secondary product c ˚ . It is interesting to note that our proof of (a) is inseparable from theproof of (b). In particular, we obtain the following
Corollary 4.10.
The secondary product on SH ˚ pB V q has a unit (indegree ´ n ). (cid:3) The unit is only implicitly defined within the framework of the currentpaper. We refer to [24] for an alternative, more direct description,within the framework of multiplicative structures on cones.
Proof of Theorem 4.9.
Let p V be the completion of V , with a canonicalembedding of the symplectization p ,
8q ˆ B V Ă p V . Under this em-bedding the level t u ˆ B V is canonically identified with B V , and therestriction of this embedding to p , s ˆ B V takes values in V . Denote V δ “ p V zp δ,
8q ˆ B V for δ ą τ ą ℓ ´ τ : p ,
8q Ñ R constant equal to 3 τ { p , s and linear of slope ´ τ on r , , so that ℓ ´ τ p q “
0. Fix nowparameters τ ą τ ą ă ǫ ă . Let ǫ “ ǫ p τ ´ τ q{p τ ` τ q ,so that 0 ă ǫ ă ǫ . Consider the continuous piecewise linear function h “ h ´ τ ,τ,ǫ,ǫ : p ,
8q Ñ R defined by the following conditions: ‚ h coincides with ℓ ´ τ on p , ´ ǫ s . ‚ h is linear of slope ´ τ on the interval r ´ ǫ, ´ ǫ s ; OINCAR´E DUALITY FOR LOOP SPACES 47 ‚ h is linear of slope τ on the interval r ´ ǫ , s ; ‚ h p q “ ‚ h coincides with ℓ ´ τ on the interval r , .We call h “ h ´ τ ,τ,ǫ,ǫ the p´ τ , τ, ǫ, ǫ q -dent on the function ℓ ´ τ . SeeFigure 5. I II III ´ ǫ ´ ǫ ´ τ τ r τ { ´ τ ´ τ ´ ǫ τǫτ Figure 5.
Hamiltonian profile h ´ τ ,τ,ǫ,ǫ for Poincar´e duality.Let ˜ h be a smoothing of h with the following properties: ‚ Outside the union of a small neighbourhood of r “ with a smallneighbourhood of the closed interval r ´ ǫ, s , the function ˜ h coin-cides with h . In particular, ˜ h is constant equal to 3 τ { r ď and is linear of slope ´ τ for r ě in this range. ‚ Inside the neighbourhood of the closed interval r ´ ǫ, s , and outsidesmall neighbourhoods of r “ ´ ǫ , r “ ´ ǫ , and r “
1, the function˜ h is linear of negative slope ´ τ smaller but close to ´ τ , then linearof positive slope τ larger but close to τ . ‚ Inside the neighbourhood of r “ where its derivative lies in p´ τ, q , the function ˜ h is strictly concave. ‚ Inside the neighbourhood of r “ ´ ǫ where its derivative lies in p´ τ , ´ τ q , the function ˜ h is strictly concave. ‚ In a neighbourhood of r “ ´ ǫ the function ˜ h is constant equalto ´ ǫ τ , and is strictly convex as its slope varies in p´ τ , q and p , τ q . ‚ In a neighbourhood of r “ h is constant equal to 0on some open interval, and is strictly concave as its slope varies in p , τ q and p´ τ, q .A function ˜ h as above can be interpreted as a Hamiltonian ˜ H : p V Ñ R by extending it as constant equal to 3 τ { p V zp ,
8q ˆ B V .Assume now that τ does not belong to the action spectrum of B V .Since the latter is a closed set in R , we can choose the other parameters τ , τ , τ such that τ ă τ ă τ ă τ and such that the whole interval r τ, τ s does not intersect the spectrum. The parameter ǫ is chosenarbitrarily, and the parameter ǫ is determined by ǫ , τ , and τ . The1-periodic orbits of ˜ H fall then into three groups. ‚ Group I consists of constants in V and nonconstant orbits in theconcavity region near r “ . ‚ Group II consists of orbits located in a neighbourhood of r “ ´ ǫ ,and these are themselves of three types: constants on the trivialcobordism which constitutes the minimal level (type II ), noncon-stant orbits in the convexity region near its negative boundary (type II ´ ), nonconstant orbits in the convexity region near its positiveboundary (type II ` ). ‚ Group III consists of orbits located in a neighbourhood of r “
1, andthese are again of three types: constants on the trivial cobordismwhich constitutes the maximal level (type
III ), nonconstant orbitsin the concavity region of positive slope near its negative boundary(type III ´ ), nonconstant orbits in the concavity region of negativeslope near its positive boundary (type III ` ).Let H “ H τ be a C -small perturbation of ˜ H , time-dependent and sup-ported near the nonconstant periodic orbits in the concavity or convex-ity regions, time-independent Morse in the flat regions, and such thatin each flat region the gradient of the Morse perturbation is pointingoutwards along the boundary if the latter is adjacent to a convexityregion, and is pointing inwards along the boundary if the latter is adja-cent to a concavity region. The orbits of H naturally fall into classes I , II , III as above, and their action is close to the action of correspondingorbits of ˜ H .Let p a, b q be a fixed action window with ´8 ă a ă ă b ă 8 , andassume τ is large enough so that ´ τ { ă a . Then the action of allorbits in group I lies below the action window p a, b q . Denote by p C a,b , Bq : “ F C p a,b q˚ p H q the Floer complex in the action window p a, b q . Note that, up to canon-ical chain homotopy equivalence, this complex does not depend on H “ H τ as long as τ ą max t | a |{ , b u . The generators of C a,b are OINCAR´E DUALITY FOR LOOP SPACES 49 orbits of types II and III , and we can write C a,b “ C ´ a,b ‘ C ` a,b , where C ´ a,b is the submodule generated by orbits of type III , and C ` a,b is the submodule generated by orbits of type II .It follows from [25, Lemmas 2.2, 2.3, and 2.5] that C ´ a,b is a subcomplexif the size of the perturbation H ´ ˜ H is small enough. In particular wecan think of C a,b as the cone C p f q , where f : C ` a,b Ñ C ´ a,b r´ s is thepart of the differential which maps elements of C ` a,b to elements of C ´ a,b .The Hamiltonian H is homotopic by a monotone homotopy supportedin an open neighbourhood of the region t ´ ǫ ď r ď u to the Hamil-tonian L “ L τ which coincides with H in an open neighbourhood of V and which is linear of slope ´ τ outside that neighbourhood. Thishomotopy defines a chain map F C p a,b q˚ p H q Ñ F C p a,b q˚ p L q “ ǫ ą H and L are C -close. By [25, Lemma 7.2], the reverse homotopy from L to H theninduces a chain map 0 “ F C p a,b q˚ p L q Ñ F C p a,b q˚ p H q which is a homotopyinverse for the previous map. The homotopy of homotopies betweenthe composition of these two homotopies and the constant homotopyfor H induces a chain homotopy K : C a,b Ñ C a,b r s such that Id C a,b “ rB , K s . It follows from the discussion in Section 4.3 and the definition of sym-plectic (co)homology in [25] that the map f : C ` a,b Ñ C ´ a,b r´ s aboveinduces on homology the Poincar´e duality isomorphism f ˚ : SH p a,b q˚ pB V q – ÝÑ SH ´˚p´ b, ´ a q pB V q from symplectic homology to cohomology in the action window p a, b q .Consider now the pair-of-pants product C a,b b C a,b Ñ F C p a ` b, b q˚ p H q as in Section 4.2, where we use a 1-form β satisfying dβ “ H is ofthe type H τ , C -close to the corresponding linear Hamiltonian L τ .Thus F C p a ` b, b q˚ p H q can be identified with C a ` b, b and we obtain adegree ´ n product µ : C a,b b C a,b Ñ C a ` b, b . Note that the target also splits as C a ` b, b “ C ´ a ` b, b ‘ C ` a ` b, b , with C ´ a ` b, b a subcomplex, and in the notation of § rB , µ s “ µ ´´` “ µ ´`` “ µ `´` “ , again as a consequence of [25, Lemmas 2.2, 2.3, and 2.5]. Hence weare in the situation of Section 4.3 with C “ C a,b and C “ C a ` b, b . Inthe notation of that section, it follows that µ ` “ µ ``` descends to aproduct on homology µ `˚ : SH p a,b q˚ pB V q b SH p a,b q˚ pB V q r´ n s ÝÑ SH p a ` b, b q˚ pB V q . Moreover, Lemma 4.8 implies:(a) The primary product µ ´ “ µ ´´´ vanishes on homology, i.e. µ ´˚ “ SH ´˚p´ b, ´ a q pB V q b SH ´˚p´ b, ´ a q pB V q Ñ SH ´˚p´ b, ´ a ´ b q pB V q . (b) The vanishing of the primary product in two ways gives rise to asecondary product σ ´˚ : SH ´˚p´ b, ´ a q pB V q b SH ´˚p´ b, ´ a q pB V q r´ n s ÝÑ SH ´˚p´ b, ´ a ´ b q pB V q . (c) The products µ `˚ and σ ´˚ on homology are related via the Poincar´eduality isomorphism f ˚ and its inverse g ˚ as SH p a,b q˚ pB V q b SH p a,b q˚ pB V q µ `˚ / / SH p a ` b, b q˚ pB V q f ˚ – (cid:15) (cid:15) SH ´˚p´ b, ´ a q pB V q b SH ´˚p´ b, ´ a q pB V q g ˚ b g ˚ – O O σ ´˚ / / SH ´˚p´ b, ´ a ´ b q pB V q . By construction, the maps in this diagram are compatible with theaction filtrations, so for a ă a and b ă b the diagram is related to thecorresponding diagram for a , b via the action truncation maps SH p a,b q˚ pB V q Ñ SH p a ,b q˚ pB V q , SH ´˚p´ b, ´ a q pB V q Ñ SH ´˚p´ b , ´ a q pB V q . Passing first to the inverse limit as a Ñ ´8 and then to the directlimit as b Ñ 8 , we obtain a degree ´ n product m “ µ `˚ on SH ˚ pB V q and a degree n ´ c ˚ “ σ ´˚ on SH ˚ pB V q which arerelated via the Poincar´e duality isomorphsm P D “ f ˚ as SH ˚ pB V q b SH ˚ pB V q m / / P D b P D – (cid:15) (cid:15) SH ˚ pB V q P D – (cid:15) (cid:15) SH ´˚ pB V q b SH ´˚ pB V q c ˚ / / SH ´˚ pB V q . This concludes the proof of Theorem 4.9. (cid:3)
We state the Lagrangian counterpart of Theorem 4.9. The proof isthe same, and gradings are discussed in Appendix B. Given a Liouvilledomain V as above, we consider Maslov 0 exact Lagrangians L Ă V with boundary such that B L “ L X B V and L is conical in theneighborhood of B L . OINCAR´E DUALITY FOR LOOP SPACES 51
Theorem 4.11.
Let V be a Liouville domain of dimension n and L Ă V be an exact Lagrangian as above. Then:(a) The primary product on SH ˚ pB L q vanishes. As a consequence, SH ˚ pB L q carries a secondary product c ˚ of degree ´ , which has a unitin degree .(b) The Poincar´e duality isomorphism P D : SH ˚ pB V q » ÝÑ SH n ´˚` pB V q is a ring homomorphism, where SH ˚ pB V q is endowed with its degree ´ n pair-of-pants product m and SH ˚ pB V q is endowed with its degree ´ secondary product c ˚ . (cid:3) Proof of Theorems 1.1, 1.11, and 1.13.
Now we can proveTheorems 1.1 and 1.11 from the Introduction.
Proof of Theorem 1.1.
Applied to the unit disk cotangent bundle V “ D ˚ M , Theorem 4.9 yields Theorem 1.1 from the Introduction. (cid:3) Proof of Theorem 1.11.
The existence of the double filtrations on q H ˚ Λand q H ˚ Λ and their compatibility with the products are immediateconsequences of the existence of the action filtrations SH p a,b q˚ pB V q , SH ˚p a,b q pB V q on symplectic (co)homology and their compatibility withthe pair-of-pants products, applied to the unit disc cotangent bundle V “ D ˚ M . Compatibility with Poincar´e duality follows directly fromthe proof of Theorem 4.9. Compatibility with the canonical splittingprovided by Theorem 1.4 follows from compatibility of the isomor-phisms SH ˚ p D ˚ M q – H ˚ Λ and SH ˚ p D ˚ M q – H ˚ Λ with the actionresp. length filtrations explained in the following Remark 4.12. (cid:3)
Remark 4.12 (filtrations) . In this remark we explain the relationsbetween the filtrations on the various Floer and loop homologies. Webegin with a general Liouville domain V . Let H : p V Ñ R be a Hamil-tonian on the completion p V “ V Yr , V satisfying H p r, y q “ h p r q on p ,
8q ˆ B V Ă p V . Then 1-periodic orbits t ÞÑ p r, y p t qq of X H corre-spond to (reparametrized) Reeb orbits y of period ş y α “ h p r q and withHamiltonian action rh p r q´ h p r q . Suppose that h : p ,
8q Ñ p , andits derivative h : p ,
8q Ñ p , are increasing diffeomorphisms, inparticular h p r q ą r . Then the function r ÞÑ rh p r q ´ h p r q is strictly increasing, so h determines a diffeomorphism φ h : p ,
8q Ñp , sending h p r q to rh p r q ´ h p r q . Now the definition of symplec-tic homology and a standard Floer continuation argument shows thatfiltered symplectic homology is related to filtered Floer homology (byHamiltonian action) as(26) SH p a,b q˚ p V q – F H p φ h p a q ,φ h p b qq˚ p H q . We apply this to the unit disc cotangent bundle V “ D ˚ M of a Rie-mannian manifold M and the function h p r q “ r , so H p q, p q “ | p | isthe standard quadratic Hamiltonian on T ˚ M . Its Legendre transformis the standard energy functional E p q q “ ş | q | dt . Let us also intro-duce the square root energy functional F p q q “ a E p q q “ p ş | q | dt q { .Then we have the following chain of isomorphisms SH p a,b q˚ p D ˚ M q – F H p a { ,b { q˚ p H q – M H p a { ,b { q˚ p E q– M H p a,b q˚ p F q – H p a,b q˚ Λ , (27)where the Morse homologies M H ˚ p E q , M H ˚ p F q are filtered by thevalues of the functionals E, F : Λ Ñ R , and H ˚ Λ is filtered by length ℓ p q q “ ş | q | dt . The first isomorphism follows from (26), and the secondone from [2]. The third isomorphism is obvious, and the last one followsfrom the fact that F p q q “ ℓ p q q for a critical point of F , i.e., a closedgeodesic q : r , s Ñ M parametrized proportionally to arclength.The filtered isomorphism (27) is to be understood also in view of thefiltered chain map from the Floer complex to the Morse complex con-structed in [21]. Proof of Theorem 1.13.
Applied to a fiber D ˚ q M of the unit disc cotan-gent bundle V “ D ˚ M , Theorem 4.11 yields Theorem 1.13 from theIntroduction. (cid:3) A Morse theoretic description of the Gysin sequence.
Aspreparation for the next section we discuss here the Gysin sequence interms of Morse theory. Consider an p r ´ q -sphere bundle π : S Ñ M with structure group O p r q over a closed, connected, n -dimensionalmanifold M . Suppose r ě
2. Pick a Morse function f : M Ñ R witha unique minimum p min and a unique maximum p max . Perturb thepullback π ˚ f “ f ˝ π to a Morse function φ : S Ñ R with exactly two critical points p ˘ for each critical point p of f ofindices ind p p ` q “ ind p p q , ind p p ´ q “ ind p p q ` r ´ , so p ` corresponds to the minimum and p ´ to the maximum on thefibre sphere over p (the reason for this convention will become clearin the next subsection). Write the Morse cochain complex for φ with R -coefficients as C : “ M C ˚ p φ q “ C ` ‘ C ´ , where C ˘ is the free R -module generated by the critical points p ˘ . Wedenote O R “ O b Z R , where O is the local system on M determined bythe orientations of the fibres of S . OINCAR´E DUALITY FOR LOOP SPACES 53
Lemma 4.13.
The Morse function φ : S Ñ R and the metric on S can be chosen such that the following holds (see Figure 6).(a) With respect to the splitting C “ C ` ‘ C ´ the Morse codifferentialof φ has the form B “ ˆ B ` ε B ´ ˙ . (b) The short exact sequence Ñ C ` i Ñ C p Ñ C ´ Ñ gives rise to along exact sequence which is isomorphic to the Gysin sequence of thesphere bundle π : S Ñ M via the commuting diagram ¨ ¨ ¨ H ˚ p C ` q i ˚ / / H ˚ p C q p ˚ / / H ˚ p C ´ q ε ˚ / / H ˚` p C ` q ¨ ¨ ¨¨ ¨ ¨ H ˚ p M q π ˚ / / H ˚ p S q π ˚ / / H ˚` ´ r p M ; O R q Y e p S q / / H ˚` p M q ¨ ¨ ¨ Here e p S q P H r p M ; O R q is the Euler class of the bundle S Ñ M .(c) Suppose that S “ S ˚ M . The map ε : C ´ Ñ C ` in the differential isthen trivial except for the term ε p p ´ min q “ χ ¨ p ` max , where χ “ x e p S q , r M sy is the Euler characteristic of M .(d) Suppose that S “ S ˚ M . Then each algebraic count of index zerogradient Y-graphs with two cohomological inputs from ker ε Ă C ´ andoutput in C { ker ε is zero. ε n p ´ p ` n ´ Figure 6.
Morse cochain complex for S ˚ M . Proof.
We work in the Morse-Bott limit using the Morse-Bott complexwith cascades [31, Appendix A], see also [13]. In view of the Morse-Bott Correspondence Theorem [9, Theorem 1.1], see also [14], this isequivalent to working with a perturbation φ ´ π ˚ f that is very small.We choose on S a horizontal distribution and a metric such that thehorizontal distribution is orthogonal to the canonical vertical one, andthe restriction of the metric to the horizontal distribution is the lift ofthe metric on M . The gradient of π ˚ f with respect to such a metric istangent to the horizontal distribution and projects onto the gradient of f , hence gradient trajectories of π ˚ f project onto gradient trajectoriesof f . As a consequence, the Morse complex C with generators p ˘ , p P Crit p f q is filtered by the Morse index of p . This is the setup considered in [55, § § § C ` is a subcomplex of C . Arguing by contradiction, suppose there exists a gradient line from p ` P C ` to q ´ P C ´ of index difference1 “ ind p q ´ q ´ ind p p ` q “ ind p q q ` r ´ ´ ind p p q , i.e. ind p q q “ ind p p q ` ´ r . We distinguish two cases. The case r ě . We directly obtain a contradiction using the fact thatthe Morse codifferential respects the filtration, which forbids ind p q q “ ind p p q ` ´ r ă ind p p q . The case r “ . The equality ind p q q “ ind p p q implies p “ q , sothat the only possible cascade configurations are gradient trajectoriesentirely contained in the critical fibers. In this situation there areindeed two gradient trajectories going from p ` to p ´ , but these twocancel each other algebraically.(b) To establish the isomorphism between the long exact sequences weinterpret these as special instances of the Leray-Serre spectral sequence.The filtration on the Morse-Bott complex of π ˚ f —or, equivalently,on the Morse complex of φ near the Morse-Bott limit,—determinesa spectral sequence which, for degree reasons, is equivalent to the toplong exact sequence in the diagram in (b). On the other hand, theLeray-Serre spectral sequence for the fibration S Ñ M is equivalent tothe bottom long exact sequence in the diagram in (b). This is provedin [61, § III.5] for the orientable case, and the non-orientable case isdiscussed in [66]. Finally, it is proved in [55, §
3] and [56, §
7] thatthe Leray-Serre spectral sequence is naturally isomorphic to the Morsespectral sequence for the perturbed lift φ with respect to a suitablepseudo-gradient. We infer as a particular case the identification of thetwo long exact sequences in (b).(c) Suppose now that S “ S ˚ M . For index reasons the only possiblecontribution to ε is ε p p ´ min q “ κ ¨ p ` max for some κ P R which we need toidentify. We will show that κ “ χ for an appropriate choice of metricon S as above. We fix as before the Morse function f : M Ñ R andwork in the Morse-Bott limit.Pick a section s of D ˚ M which is transverse to the zero section andwhich satisfies the following additional properties: the zero set Z “ s ´ p q does not contain any critical point of f , the section takes valuesin S ˚ M outside a small neighbourhood N of its zero set, and s p p q “ p ´ for all p P Crit p f q . We further specify the horizontal distribution on S ˚ M to be tangent to the image of s over M z N and to be integrable OINCAR´E DUALITY FOR LOOP SPACES 55 in its neighbourhood. We also fix trivializations of S over small neigh-bourhoods of the critical points of f , and assume that the section andthe horizontal distribution are constant in those trivialisations.With respect to the chosen trivializations in the neighbourhood of thecritical fibers, the gradient of π ˚ f is the trivial lift of the gradient of f .We first claim that κ is equal to the signed count of trajectories of ∇ π ˚ f running from p ´ min to p ` max . By definition of the Morse-Bott complex thevalue of κ is given by the signed count of cascade configurations from p ´ min to p ` max , and we need to show that such cascades necessarily containa single level. This is a consequence of the fact that moduli spacesof gradient trajectories of f between critical points whose differenceof index is smaller than n generically avoid Z . Thus, if a cascadeconfiguration from p ´ min to p ` max involved an intermediate critical level,the projections of the gradient trajectories of π ˚ f in the cascade wouldconnect critical points of f with difference of index smaller than n and therefore would avoid fixed neighbourhoods of the points in Z . Inparticular, the first gradient trajectory of π ˚ f in the cascade would betangent to im p s q because it starts at p ´ min P im p s q . The endpoint ofthis first gradient trajectory would also belong to im p s q and thereforewould be of the form q ´ for some q P Crit p f q . This is a contradictionbecause, by the very definition of cascade configurations, the endpointsof trajectories of π ˚ f which lie on an intermediate critical level in acascade cannot be critical points of the perturbing function. This showsthat cascades from p ´ min to p ` max contain a single level, hence κ equalsthe signed count of trajectories of ∇ π ˚ f from p ´ min to p ` max .Each trajectory of ∇ π ˚ f running from p ´ min to p ` max projects onto atrajectory of ∇ f running from p min to p max , and moreover each suchprojected trajectory must necessarily intersect N . The proof of thisfact is a variant of the previous argument: the trajectories issued from p ´ min whose projections do not intersect N must be tangent to im p s q and they are therefore asymptotic at the other endpoint to s p p max q “ p ´ max ‰ p ` max .Without loss of generality we can take N to be a disjoint union of flowboxes around the points in Z , i.e. N “ \ x P Z N x with N x “ D x ˆr´ , s ,where D x ” D x ˆ t´ u is a p n ´ q -disc transverse to the flow of f ,and each segment y ˆ r´ , s for y P D x is parametrized by the flow of f . See Figure 7.Consider now M x p p ´ min , S p max q , the moduli space of gradient lines of π ˚ f running from p ´ min to S p max and whose projection to M intersects N x . This is a smooth mani-fold of dimension n ´ M x p p ´ min , S p max q Ñ S p max . We have a canonical identification M x p p ´ min , S p max q ” D x which associates to a flow line its intersection ∇ f p min p max p ` max p ´ min D x N x “ D x ˆ r´ , s x S p max p ´ max SM π ∇ π ˚ f Figure 7.
Proof of Lemma 4.13(c).with D x , and, with respect to this identification, we have ev | B D x ” p ´ max .Denoting κ x “ deg p ev : D x {B D x Ñ S p max q and taking into account that p ` max P S p max is a regular value, we obtain κ “ ÿ x P Z κ x . Similarly, we can express the Euler characteristic as a sum over zeroesof s , i.e. χ “ ÿ x P Z ε x , where ε x “ ˘ ds vert p x q : T x M Ñ T ˚ x M preserves or not the local orientation. Givena trivialization S | N x » N x ˆ S n ´ , we can alternatively express ε x “ deg p s : B N x Ñ S n ´ q . We finally claim the equality(28) deg p ev : D x {B D x Ñ S p max q “ deg p s : B N x Ñ S n ´ q for each x P Z , which implies κ “ χ .To prove this, let us assume without loss of generality that the trivi-alization of S over N x “ D x ˆ r´ , s is such that s is constant over D x ˆ t˘ u . Consider the map τ : N x “ D x ˆ r´ , s Ñ S n ´ , p y, t q ÞÑ ϕ t ` ∇ π ˚ f p s p y qq . OINCAR´E DUALITY FOR LOOP SPACES 57
Then deg τ | B N x “ τ | B D x ˆt u is constant,this translates intodeg τ | D x YB D x ˆr´ , s{B D x ˆt u ` deg τ | D x ˆt u{B D x ˆt u “ . The flow line segments of π ˚ f contained in S | B D x ˆr´ , s and contained inflow lines issued from p ´ min are exactly the sectional lifts of the segments y ˆ r´ , s for y P B D x , hencedeg τ | D x YB D x ˆr´ , s{B D x ˆt u “ deg s | B N x . On the other hand, by definition of the evaluation map ev : D x {B D x Ñ S p max we have that deg τ | D x ˆt u{B D x ˆt u “ ´ deg ev , where the minus sign comes from the fact that D x “ D x ˆ t´ u and D x ˆt u inherit opposite boundary orientations from D x ˆr´ , s . Thisproves (28), and hence the equality κ “ χ .(d) It is convenient to revert for the proof of (d) to the Morse complexof φ near the Morse-Bott limit. Suppose that there exists an index 0gradient Y-graph with inputs a, b P ker ε and output c P C { ker ε . Sinceall indices in ker ε Ă C ´ are ě n ´ C { ker ε are ď n ,it follows that0 “ ind p c q ´ ind p a q ´ ind p b q ď n ´ p n ´ q “ ´ n, so this is only possible for n ď
2. The case n “ C ` , C ´ lie in different connected components of S ˚ S , so it remainsto consider the case n “
2. Here for index reasons the only possibilityis a “ αp ´ min , b “ βp ´ min (of index 1) and c “ γp ` max (of index 2), with α, β, γ P R such that αχ “ βχ “ S can be computed using Y-graphs for which one incoming half-edge(say the one with input a “ αp ´ min ) and the outgoing half-edge fittogether into a (parametrized) gradient line of a given Morse function,and for which the perturbation on the second incoming half-edge ischosen generically. Secondly, by part (b) the signed count of gradienttrajectories of φ running from αp ´ min to p ` max is equal to αχ and thereforevanishes in R . Thirdly, we claim that we can choose an s -dependentperturbation on the second incoming half-edge of the Y-graph suchthat, for each trajectory p ´ min Ñ p ` max , there is a unique Y-graph whichcontains that trajectory as the concatenation of its first incoming half-edge with its outgoing half-edge. As a consequence, up to a global signindeterminacy the count of such Y-graphs is the same as the count ofrigid gradient trajectories αp ´ min Ñ p ` max and therefore vanishes in R .The s -dependent perturbation of ∇ φ is constructed as follows. The un-stable manifold W u p p ´ min , ∇ φ q is 2-dimensional and locally a 2-disc. Wechoose an s -dependent perturbation ˜ ∇ φ of ∇ φ such that W u p p ´ min , ˜ ∇ φ q is locally a graph over W u p p ´ min , ∇ φ q in the transverse direction, ob-tained by spreading in an S -family—where S is identified to the ori-ented projectivisation of T p ´ min W u p p ´ min , ∇ φ q —the graph of a smoothfunction h : r ,
8q Ñ R such that h p q “ h has vanishing derivativesat 0, h p r q ă r ą h has a single transverse zero on p , q ,and h vanishes on r , . See Figure 8. We parametrize the corre-sponding flow so that the transverse zero of h corresponds to s “ p ´ min .0 h W u p p ´ min , ∇ φ q W u p p ´ min , ˜ ∇ φ q p ´ min Figure 8.
The function h and the perturbation ˜ ∇ φ . (cid:3) Remark 4.14.
We gather here a few observations about the proof ofLemma 4.13.(1) It is possible to give a direct proof of the isomorphism between thetwo long exact sequences in (b) without referring to the Leray-Serrespectral sequence. Indeed, the proof of (c) constitutes the main stepfor S “ S ˚ M , and it can be adapted in a quite straightforward wayto arbitrary sphere bundles with structure group O p r q . We do notpursue this discussion since it is outside of the main focus of the paper.Statement (b) is also proved by Furuta in [34], and there are manysimilarities between our proof of (c) and [34].(2) Statement (a) holds for a much larger class of perturbations if r ě f : M Ñ R such thatind p q q ą ind p p q implies f p q q ą f p p q for all critical points p, q (e.g. f could be self-indexing), and the perturbation φ ´ f ˝ π to be smallerthan the minimal difference of values of f between critical points ofindex difference 1. Under these assumptions C ` is a subcomplex of C .The proof goes by contradiction. Assuming the existence of a gradientline from p ` P C ` to q ´ P C ´ of index difference 1 “ ind p q ´ q ´ ind p p ` q “ ind p q q ` r ´ ´ ind p p q , we get ind p q q “ ind p p q ` ´ r andtherefore ind p q q ă ind p p q if r ě
3. By the assumption on φ this implies f p q q ă f p p q and φ p q ´ q ă φ p p ` q , in contradiction with the fact that thevalue of φ increases along a gradient trajectory. OINCAR´E DUALITY FOR LOOP SPACES 59 (3) Statement (c) admits a much simpler proof if M is orientable. Weconsider the relevant portion of the commuting diagram of exact se-quences in (b), where the local system O coincides with the orientationlocal system on M and is therefore trivial: H n ´ p C ´ q ε ˚ / / H n p C ` q H p M ; R q Y e p S q / / H n p M ; R q . The composition R “ H p M ; R q Y e p S q ÝÑ H n p M ; R q – H p M ; R q “ R is given by multiplication with the Euler characteristic χ “ x e p S q , r M sy .On the other hand ε ˚ is induced by multiplication with κ , hence κ “ χ .Unfortunately, this method of proof does not apply if M is not ori-entable: the last composition is a map R Ñ R { R , where R is the2-torsion subgroup of R . If R has no 2-torsion, we obtain no informa-tion on κ .(4) We find it useful to stress that, while it is easy to achieve thatgradient trajectories of π ˚ f project onto gradient trajectories of f , thisis not true for the gradient trajectories of φ regardless of how small theperturbation φ ´ π ˚ f is. This shortcoming of the Morse complex wascircumvented in [55, 56] by using a suitable pseudo-gradient. However,implanting the pseudo-gradient Morse complex into Floer theoretic ar-guments is a subtle matter, see [56]. Our approach to the proof ofLemma 4.13 is to infer the filtration property for the Morse complex of φ from the filtration property for the Morse-Bott complex of π ˚ f viathe Morse-Bott Correspondence Theorem [9, Theorem 1.1].(5) Our final remark concerns the equality χ “ x e p S q , r M sy for S “ S ˚ M . While this is discussed in many places for orientable M ,it is not so easy to find a reference for the non-orientable case. Ourreference is Steenrod [63, Theorem 39.7], which relies on Alexandroffand Hopf [8, § XIV.4]. Since the Euler class is the primary obstructionclass to the existence of a nowhere vanishing section, it follows thata closed manifold M , possibly non-orientable, admits a nonvanishingvector field if and only if its Euler characteristic is zero.4.7. Canonical splitting of the duality sequence and proof ofTheorem 1.4.
Let now V be a Liouville domain of dimension 2 n andrecall its duality sequence(29) ¨ ¨ ¨ SH ´˚ p V q ε / / SH ˚ p V q ι / / SH ˚ pB V q π / / SH ´˚ p V q ¨ ¨ ¨ where ι is a ring map with respect to the pair-of-pants products andthe map ε factors as ε : SH ´˚ p V q ÝÑ H n ´˚ p V, B V q ÝÑ H n ´˚ p V q ÝÑ SH ˚ p V q , with the middle map being the restriction map. As in the Introductionwe define the “reduced” (co)homology groups(30) SH ˚ p V q : “ coker ε, SH ˚ p V q : “ ker ε, so the long exact sequence induces a short exact sequence(31) 0 / / SH ˚ p V q ι / / SH ˚ pB V q π / / SH ´˚ p V q / / . The following theorem, combined with the main result in [21] iden-tifying the secondary pair-of-pants product on SH ˚ p D ˚ M q with thecohomology product on H ˚ Λ, yields Theorem 1.4 from the Introduc-tion.Given a ring R and an integer m , we denote m R “ t a P R : ma “ u . Theorem 4.15 (canonical splitting) . Suppose that V “ D ˚ M is theunit disk cotangent bundle of a closed connected n -dimensional mani-fold M .If M is orientable, the short exact sequence (31) has a canonical split-ting (32) 0 / / SH ˚ p V q ι / / SH ˚ pB V q p q q π / / SH ´˚ p V q i q q / / with arbitrary coefficients, where i is a ring map with respect to the sec-ondary product on SH ´˚ p V q and the pair-of-pants product on SH ˚ pB V q .If M is non-orientable, the short exact sequence (31) admits such asplitting if the following two conditions hold:— the map R Ñ R { χ R induced by the inclusion R ã Ñ R is surjective.— the map R { χ R Ñ R { R induced by multiplication by χ is injective.In particular, such a splitting exists in each of the following situations:(i) χ “ in R , (ii) R is -torsion, e.g. R “ Z { . Remark 4.16.
One may wonder whether the conditions from Theo-rem 4.15 in the non-orientable case are also necessary for the existenceof a splitting compatible with products. It seems to us that the appro-priate framework to address this question is the one from [24]. Whatis certain is that these conditions are also necessary for our proof ofTheorem 4.15 to work.
Proof.
Pick numbers µ, τ ą B V and δ, ε P p , { q . Let H : p V Ñ R be a Hamiltonian on thecompletion p V “ V Y r ,
8q ˆ B V which is a smoothing and time-dependent perturbation of the following map, see also Figure 9. OINCAR´E DUALITY FOR LOOP SPACES 61 ‚ H ” p ´ ε ´ δ q µ on V δ : “ V zp δ, s ˆ B V ; ‚ H is linear with slope ´ µ on r δ, ´ ε s ˆ B V ; ‚ H is linear with slope τ on r ` ε,
8q ˆ B V . ‚ H p r, y q “ p r ´ q ´ ε ` ψ p r, y q on r ´ ε, ` ε s for a C -smallfunction ψ which vanishes near r “ ˘ ε and satisfies ψ p r, y q “ φ p y q near r “
0, with a function φ : B V Ñ R as in Section 4.6 (here weuse that B V Ñ M is a sphere bundle).1 ´ ε ´ δ ´ µ rτ δ ´ ε ` ε Figure 9.
Hamiltonian profile for the canonical splitting.The free R -module underlying the Floer chain complex of H splits as C : “ F C ˚ p H q “ C F ‘ C I , where C F is generated by the 1-periodic orbits near V δ , and C I by theorbits near r ´ ε, ` ε sˆB V . The action of orbits in C F is ď ´p ´ ε ´ δ q µ and the action of orbits in C I is ě ´p ´ ε qp µ ´ η µ q , where η µ ą µ to the action spectrum of B V . Thus all actions in C F are smaller than all actions in C I provided that δµ ă p ´ ε q η µ , which we will always arrange for given µ, τ, ε by making δ small. Itfollows that C F is a subcomplex of C . The R -module C I splits furtheras C I “ C ´ I ‘ C ` I , where C ˘ I is generated by the nonconstant 1-periodic orbits near t ˘ ε u ˆ B V and the critical points p ˘ of φ : B V Ñ R on t u ˆ B V . Here weuse the separation of critical points on the sphere bundle B V Ñ M intotwo groups p ˘ from Section 4.6. (Recall that the Floer chain complexrestricts to the Morse cochain complex on the constant orbits.) Notethat all nonconstant orbits in C ´ I (resp. C ` I ) have negative (resp. posi-tive) Hamiltonian action. This observation together with Lemma 4.13 implies that with respect to the splitting C I “ C ´ I ‘ C ` I the Floerdifferential on C I (viewed as a quotient complex of C ) has the form B I “ ˆ B ´ I ˚ ε I B ` I ˙ , where ε I : C ´ I Ñ C ` I is trivial except for ε p p ´ min q “ χ ¨ p ` max , with χ “ x e pB V q , r M sy the Euler characteristic of M . Define C ´ I : “ ker ε I , C ` I : “ C I { C ´ I . Then C ´ I is a subcomplex of p C I , B I q with quotient complex C ` I , so weget a commuting diagram of short exact sequences0 (cid:15) (cid:15) (cid:15) (cid:15) / / C F ε / / C F ‘ C ´ I ι / / i (cid:15) (cid:15) C ´ I / / i (cid:15) (cid:15) / / C F ε / / C ι / / p (cid:15) (cid:15) C I / / p (cid:15) (cid:15) C ` I (cid:15) (cid:15) C ` I (cid:15) (cid:15) ¨ ¨ ¨ (cid:15) (cid:15) ¨ ¨ ¨ (cid:15) (cid:15) H p C ` I qr s (cid:15) (cid:15) H p C ` I qr s (cid:15) (cid:15) ¨ ¨ ¨ H p C F q ε ˚ / / H p C F ‘ C ´ I q ι ˚ / / i ˚ (cid:15) (cid:15) H p C ´ I q π ˚ / / i ˚ (cid:15) (cid:15) H p C F qr´ s ¨ ¨ ¨¨ ¨ ¨ H p C F q ε ˚ / / H p C q ι ˚ / / p ˚ (cid:15) (cid:15) H p C I q π ˚ / / p ˚ (cid:15) (cid:15) H p C F qr´ s ¨ ¨ ¨ H p C ` I q H p C ` I q Let us compute A : “ H p C F ‘ C ´ I q . By the usual Floer continuationargument, it can be computed for a Hamiltonian as above whose slope ´ µ is so close to zero that the Floer complex reduces to the Morsecochain complex. Now each critical point p of the Morse function OINCAR´E DUALITY FOR LOOP SPACES 63 f : M Ñ R underlying φ : B V Ñ R (see Section 4.6) gives rise to threegenerators of C : p ˘ P C ˘ I located on t u ˆ B V , and p F P C F locatedon the zero section M Ă V . See Figure 10. n ´ rτp F n n n p ´ p ` Figure 10.
Profile with small slope µ .Their indices areind p p ` q “ ind p p q , ind p p ´ q “ ind p p q ` n ´ , ind p p F q “ ind p p q ` n, and the Morse coboundary operator B M satisfies B M p ´ min “ p F min ` “ B Mf p p min q ‰ ´ ` χ ¨ p ` max , B M p ´ “ p F ` “ B Mf p p q ‰ ´ for all p ‰ p min , B M p ` “ “ B Mf p p q ‰ ` , B M p F “ “ B Mf p p q ‰ F . Here “ B Mf p p q ‰ ‹ is a notation for ř n q q ‹ , where B Mf p p q “ ř n q q and ‹ “ ´ , ` , F . See Figure 11. nn ´ ε “ ¨ χC F C ´ I C ` I n Figure 11.
Morse cochain complex for a profile withsmall slope µ .We claim that A “ H p C F ‘ C ´ I q “ p R { χ R, q where χ R “ t a P R : χa “ u and p R { χ R, q means that A is isomor-phic to R { χ R living in degree 0. For the computation we consider theshort exact sequence0 Ñ C F, ą min ‘ C ´ I, ą min Ñ C F ‘ C ´ I Ñ ¨˝ Rp F min Ò χ Rp ´ min ˛‚ Ñ , where C F, ą min ‘ C ´ I, ą min represents the subcomplex generated by criticalpoints p F , p ´ other than p F min , p ´ min . This complex is identified up toa shift to the cone of the identity on M C ˚ą min p f ; O q and is thereforeacyclic. Here O is the orientation local system on M and M C ˚ą min represents the subcomplex generated by critical points p other than p min . The claim follows since A is generated by the class r p F min s ofMorse index n , hence of Conley–Zehnder index 0.The group A persists as we pass to the inverse limit as µ Ñ 8 and thento the direct limit as τ Ñ 8 . In the inverse-direct limit H p C F q , H p C q and H p C I q become SH ´˚ p V q , SH ˚ p V q and SH ˚ pB V q , respectively. Letus denote the inverse-direct limits of H p C ` I q and H p C ´ I q by SH ˚ p V q and SH ´˚ p V q , respectively. Then the diagram in the inverse-directlimit reads ¨ ¨ ¨ (cid:15) (cid:15) ¨ ¨ ¨ (cid:15) (cid:15) SH ˚` p V q (cid:15) (cid:15) SH ˚` p V q (cid:15) (cid:15) ¨ ¨ ¨ SH ´˚ p V q ε ˚ / / A ι ˚ / / i ˚ (cid:15) (cid:15) SH ´˚ p V q π ˚ / / i ˚ (cid:15) (cid:15) SH ´˚ p V q ¨ ¨ ¨¨ ¨ ¨ SH ´˚ p V q ε ˚ / / SH ˚ p V q ι ˚ / / p ˚ (cid:15) (cid:15) SH ˚ pB V q π ˚ / / p ˚ (cid:15) (cid:15) SH ´˚ p V q ¨ ¨ ¨ SH ˚ p V q SH ˚ p V q Let us focus on the commutative square(33) SH ´˚ p V q ε ˚ / / A i ˚ (cid:15) (cid:15) SH ´˚ p V q ε ˚ / / SH ˚ p V q . OINCAR´E DUALITY FOR LOOP SPACES 65
At chain level and for finite action this is induced by the diagram C F incl / / C F ‘ C ´ I (cid:15) (cid:15) C F incl / / C F ‘ C ´ I ‘ C ` I C ` I . „ incl o o The top-right maps induce in degree 0 and at zero action(34) H p M ; O q incl ˚ / / A ¨ χ (cid:15) (cid:15) H p M ; O q . That the vertical map is multiplication by χ follows from the fact that p F min is homologous in the total complex to χp ` max . The commutativesquare (33) factors therefore in degree 0 as SH p V q ” H Λ / / / / ε ˚ H p M ; O q / / A ¨ χ (cid:15) (cid:15) i ˚ ~ ~ H p M ; O q (cid:15) (cid:15) (cid:15) (cid:15) SH p V q ” H Λ ε ˚ / / SH p V q ” H Λ . (Recall that we are using twisted coefficients on Λ.)We are seeking conditions under which ε ˚ is surjective and i ˚ is in-jective, which is equivalent to the surjectivity of the horizontal mapand injectivity of the vertical map in diagram (34). At this point theorientable and the non-orientable case need to be discussed separately.(a) If M is orientable, diagram (34) becomes R / / R { χ R ¨ χ (cid:15) (cid:15) R. The horizontal map is induced by the identity R Id ÝÑ R and is thereforesurjective, the vertical map is induced by the multiplication R ¨ χ ÝÑ R and is therefore injective when the kernel χ R is quotiented out. Thus, inthe orientable case, the map ε ˚ is surjective and the map i ˚ is injectivewith arbitrary coefficients. (b) If M is non-orientable, diagram (34) becomes R / / R { χ R ¨ χ (cid:15) (cid:15) R { R. The horizontal map is induced by the inclusion R Ñ R , the verticalmap is induced by the multiplication R ¨ χ ÝÑ R . Thus, surjectivity ofthe horizontal map and injectivity of the vertical map are preciselythe conditions from the statement of the theorem. When χ “ R we have R { χ R “ R is 2-torsion we have R “ R and R { R ” R , and the conditions aresatisfied just like in the orientable case. Thus, under the conditionsfrom the statement of the theorem, the map ε ˚ is surjective and themap i ˚ is injective.We can now replace A by im ε ˚ Ă SH ˚ p V q and the diagram becomes0 (cid:15) (cid:15) (cid:15) (cid:15) ¨ ¨ ¨ SH ´˚ p V q ε ˚ / / im ε ˚ / / i ˚ (cid:15) (cid:15) SH ´˚ p V q / / i ˚ (cid:15) (cid:15) SH ´˚ p V q ¨ ¨ ¨¨ ¨ ¨ SH ´˚ p V q ε ˚ / / SH ˚ p V q ι ˚ / / p ˚ (cid:15) (cid:15) SH ˚ pB V q π ˚ / / p ˚ (cid:15) (cid:15) SH ´˚ p V q ¨ ¨ ¨ SH ˚ p V q (cid:15) (cid:15) SH ˚ p V q (cid:15) (cid:15) SH ´˚ p V q – ker ε ˚ , SH ˚ p V q – SH ˚ p V q{ im ε ˚ in agreement with our earlier definition (30). It follows that ι ˚ in-duces a map ι ˚ : SH ˚ p V q Ñ SH ˚ pB V q and the map π ˚ : SH ˚ pB V q Ñ SH ´˚ p V q lands in SH ´˚ p V q “ ker ε ˚ , so we have derived the splitshort exact sequence (32).Finally, observe that the subcomplex C ´ I Ă C I is a subring with respectto the pair-of-pants product. To see this, consider pairs-of-pants withtwo inputs a, b P C ´ I and output c P C ` I . Since all actions of generatorsof C ´ I are ď C ` I are ě
0, such pairs-of-pants canonly exist if a, b, c are constant orbits. But in this case they degenerateto gradient Y-graphs, whose algebraic count is zero by Lemma 4.13.
OINCAR´E DUALITY FOR LOOP SPACES 67
It follows that the inclusion i : C ´ I ã Ñ C I induces a ring map i ˚ : H p C ´ I q Ñ H p C I q , which in the inverse-direct limit gives a ring map i ˚ : SH ´˚ p V q Ñ SH ˚ pB V q . Here SH ˚ pB V q is equipped with its pair-of-pants product and SH ´˚ p V q is equipped with the degree ´ n productinduced by the pair-of pants product on H p C ´ I q . By construction, thelatter product equals the restriction to SH ´˚ p V q of the secondaryproduct c ˚ on SH ´˚ pB V q in Theorem 4.9. (cid:3) Remark 4.17.
In [24] we generalize the class of Weinstein domainsfor which Theorem 4.15 holds. The generalization uses mapping cones.We end the section with a discussion of the Lagrangian setting.
Proof of Theorem 1.15. (a) The proof is similar to that of Theorem 4.15,with the important simplification that the role of the Gysin sequencefor the bundle S ˚ M Ñ M is now played by the trivial Gysin sequenceof the (trivial) bundle S ˚ q M Ñ q . The interested reader can trace allthe arguments within the proof of Theorem 4.15.(b) This is the same as the proof of Theorem 1.11. (cid:3) Open-closed TQFT structures
In this section we discuss in more detail the open-closed noncompactTQFT structures from Section 5, and prove Theorem 1.18.It will be convenient to describe the oriented 2-dimensional surfaceswith boundary and corners which are the morphisms in 2 ´ Cob ` inthe following way. Any such surface ˜Σ is identified with the closedoriented 2-dimensional surface with boundary and punctures Σ ob-tained by collapsing and then discarding the boundary circles and theboundary intervals of ˜Σ which correspond to objects in 2 ´ Cob ` . Thepunctures of Σ can be either interior punctures or boundary punctures,and they are labeled by ` if they are incoming, respectively by ´ ifthey are outgoing. Conversely, the surface ˜Σ can be reconstructed from Σ by performing a real oriented blow-up at the punctures.5.1.
Primary open-closed TQFT structure in homology.
Proof of Theorem 1.18(1).
For a Liouville domain V the symplectic ho-mology SH ˚ p V q is a closed noncompact TQFT, and for any exact con-ical Lagrangian L Ă V the Lagrangian symplectic homology SH ˚ p L q isan open noncompact TQFT (Ritter [57]). Ritter’s arguments general-ize directly to show that the pair p SH ˚ p L q , SH ˚ p V qq is an open-closednoncompact TQFT. The case V “ D ˚ Q and L “ D ˚ q Q for a base-point q P Q yields p SH ˚ p L q , SH ˚ p W qq » p H ˚ Ω , H ˚ Λ q and shows in particular that there are infinite-dimensional examples of open-closednoncompact TQFTs.The arguments in [25, §
10] further show that p SH ˚ pB L q , SH ˚ pB W qq is an open-closed noncompact TQFT. Specializing to W “ D ˚ Q and L “ D ˚ q Q , q P Q yields that p q H ˚ Ω , q H ˚ Λ q is an open-closed noncompactTQFT. (cid:3) Canonical operations.
Closed-open and open-closed maps.
Definition 5.1 (Lauda-Pfeiffer [50]) . The zipper is the disc with oneincoming interior puncture and one outgoing boundary puncture. The cozipper is the disc with one incoming boundary puncture and one out-going interior puncture. See Figure 3.
The following definition describes the simplest instances of closed-openand open-closed maps in Floer theory.
Definition 5.2.
Let i : Ω Ñ Λ be the inclusion of constant loops. The homological “shriek” map i ! : q H ˚ Λ Ñ q H ˚´ n Ω is the operation defined by the zipper. The homological pushforward map i ˚ : q H ˚ Ω Ñ q H ˚ Λ is the operation defined by the cozipper.The cohomological “shriek” map i ! : q H ˚ Ω Ñ q H ˚` n Λ is the operation defined by the zipper. The cohomological pullback map i ˚ : q H ˚ Λ Ñ q H ˚ Ω is the operation defined by the cozipper. Proposition 5.3.
Poincar´e duality exchanges the operations definedby the zipper and the cozipper.Proof.
This is a straightforward consequence of the existence of theopen-closed or closed-open map at the level at the acyclic complexused in the proof of Theorem 1.1. (cid:3)
OINCAR´E DUALITY FOR LOOP SPACES 69
Module and comodule structures.
Let Σ be the Riemann surfacegiven by the closed disc with three punctures as follows: one positiveinterior puncture, one positive boundary puncture, and one negativeboundary puncture. See Figure 2. We fix a conformal structure on Σ.This defines a degree ´ n module structure on q H ˚ Ω over q H ˚ Λ via thediagram SH i p S ˚ M q b SH j ` n p S ˚ q M q / / SH i ` j p S ˚ q M q q H i Λ b q H j Ω / / q H i ` j ´ n Ω . (The map i ! : q H i Λ Ñ q H i ´ n Ω can be alternatively described in terms ofthis module structure as multiplication with 1 P H Ω.) This modulestructure combines with the algebra structure of q H ˚ Ω in order to make q H ˚ Ω into an algebra over q H ˚ Λ.In cohomology, the Riemann surface Σ realizes q H ˚ Ω as a comodule ofdegree n over q H ˚ Λ. This is expressed by the diagram SH k ` n p S ˚ q M q / / À i ` j “ k ` n SH i p S ˚ M q b SH j ` n p S ˚ q M q q H k Ω / / À i ` j “ k ` n q H i Λ b q H j Ω . The comodule structure combines with the coalgebra structure of q H ˚ Ωin order to make q H ˚ Ω into a coalgebra over q H ˚ Λ.Let Σ _ be the Riemann surface given by the closed disc with threepunctures with opposite signs as those of Σ: one negative interiorpuncture, one negative boundary puncture and one positive bound-ary puncture. We again fix a conformal structure on Σ. This defines adegree ´ n primary comodule structure on q H ˚ Ω over q H ˚ Λ which van-ishes, and a degree ´ n ` SH k ` n p S ˚ q M q / / À i ` j “ k ´ n ` SH i p S ˚ M q b SH j ` n p S ˚ q M q q H k Ω / / À i ` j “ k ´ n ` q H i Λ b q H j Ω . This combines with the secondary coalgebra structure on q H ˚ Ω in orderto make q H ˚ Ω into a coalgebra over q H ˚ Λ. In cohomology, the same Riemann surface Σ _ defines a degree n ´ q H ˚ Ω over q H ˚ Λ expressed by the dia-gram SH i p S ˚ M q b SH j ` n p S ˚ q M q / / SH i ` j ` n ´ p S ˚ q M q q H i Λ b q H j Ω / / q H i ` j ` n ´ Ω . (From this perspective, the map i ˚ : q H i Λ Ñ q H i Ω is obtained by mul-tiplication with the unit 1 P q H ´ n Ω.) This module structure combineswith the R -algebra structure on q H ˚ Ω in order to make q H ˚ Ω into analgebra over q H ˚ Λ. Proposition 5.4.
The Poincar´e duality isomorphisms q H i Λ » q H ´ i ` Λ , q H i Ω » q H ´ i ` ´ n Ω interchange the above module and comodule structures.Proof. The proof is mutatis mutandis the same as the proof that thePoincar´e duality isomorphisms interchange the algebra and coalgebrastructures (Theorem 1.1). (cid:3)
Remark 5.5.
Consider the following structures:(i) q H ˚ Ω is canonically a coalgebra over q H ˚ Λ. The comultiplication hasdegree ´ n ` q H ˚ Ω is canonically a coalgebra over q H ˚ Λ. The comultiplicationhas degree n .The compatibility of the Poincar´e duality isomorphisms for free andbased loops with these coalgebra structures is expressed by commuta-tive diagrams q H k Ω P D Ω » (cid:15) (cid:15) / / À i ` j “ k ´ n ` q H i Λ b q H j Ω P D Λ b P D Ω » (cid:15) (cid:15) q H ´ k ` ´ n Ω / / À i ` j “ k ´ n ` q H ´ i ` Λ b q H ´ j ` ´ n Ω5.3.
Duality of open-closed TQFTs.
We are now ready to proveTheorem 1.18 and Theorem 1.9.
Proof of Theorem 1.18(2), (3).
We start by recalling the structure the-orem of Lauda-Pfeiffer [50] which singles out generators for the cobor-dism category that defines open-closed noncompact TQFT’s. Moreprecisely, the generators are given by the two discs with three punc-tures not all of the same sign, by the two spheres with three punctures
OINCAR´E DUALITY FOR LOOP SPACES 71 not all of the same sign, by the disc with one negative puncture and thesphere with one negative puncture, by the zipper and by the cozipper.To prove (2), note that the existence of the Poincar´e duality isomor-phism implies the existence of an open-closed noncompact TQFT struc-ture on p q H ˚ Ω , q H ˚ Λ q induced from the open-closed noncompact TQFTstructure on the pair p q H ˚ Ω , q H ˚ Λ q . With this definition, point (3) inthe statement of the Theorem is tautological.However, the point is to show that the induced open-closed noncom-pact TQFT is generated by the canonical products on q H ˚ Λ and q H ˚ Ω,together with the canonical operations described in § (cid:3) The following is a straightforward consequence of the arguments in § Corollary 5.6.
The pairs p H ˚ Λ , H ˚ Ω q and p H ˚ Λ , H ˚ Ω q define open-closed noncompact TQFT structures. (cid:3) Proof of Theorem 1.9.
Apart from the vanishing of the primary co-product on q H ˚ Λ and q H ˚ Ω, the statement of Theorem 1.9 is subsumedby that of Theorem 1.18.We only prove the vanishing of the primary coproduct for q H ˚ Λ sincethe proof for q H ˚ Ω is mutatis mutandis the same. More generally, weprove the vanishing of the primary coproduct for SH ˚ pB V q , the sym-plectic homology of the boundary of an arbitrary Liouville domain V .This fact is implicit in the description of the secondary product on thecohomology SH ˚ pB V q from [24, § λ ă ă µ consider the Hamiltonian H λ,µ : p V Ñ R which is asmoothing of the Hamiltonian which is constant equal to | λ |{ V ,linear with respect to r of slope λ on r , sˆB V , zero at B V ” t uˆB V ,and finally linear with respect to r of slope µ on r ,
8q ˆ B V . (Up to adifferent choice of parameters this is the shape of Hamiltonian depictedin Figure 9.) Consider also the Hamiltonian L λ : p V Ñ R which is asmoothing of the Hamiltonian which is constant equal to | λ |{ V and which is linear with respect to r of slope λ on r ,
8q ˆ B V . Theprimary coproduct(35) c primary : SH ˚ pB V q Ñ SH ˚ pB V q b SH ˚ pB V q is induced in the inverse-direct limit in homology by the family ofaction-truncated coproducts c p a,b q primary : F C p a,b q˚ p H λ,µ q Ñ F C p a ,b ´ a q˚ p H λ ,µ ´ λ q b for ´8 ă a ă ă b ă 8 , which are in turn induced by c ě a primary : F C ě a ˚ p H λ,µ q Ñ F C ě a ˚ p H λ ,µ ´ λ q b . The point now is that, for a suitablechoice of defining data, the coproduct c p a,b q primary factors at chain level as F C p a,b q˚ p H λ,µ q / / c p a,b q primary * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ F C p a ,b ´ a q˚ p L λ q b F C p a ,b ´ a q˚ p H λ ,µ ´ λ q (cid:15) (cid:15) F C p a ,b ´ a q˚ p H λ ,µ ´ λ q b . If 2 λ ă a then all orbits of L λ fall below the fixed action window p a , b ´ a q , so that the Floer complex F C p a ,b ´ a q˚ p L λ q is null. As a consequencethe action-truncated coproduct c p a,b q primary , ˚ : F H p a,b q˚ p H λ,µ q Ñ F H p a ,b ´ a q˚ p H λ ,µ ´ λ q b vanishes for λ negative enough, and so does c primary in (35). (cid:3) Topological interpretation of the open-closed and closed-open maps.
In this section we identify the open-closed and closed-open maps described above with their topological counterparts, thetopological shriek or pullback/pushforward maps, denoted i top ! , i ˚ top etc. Note that the latter are also well-defined at the level of reduced(co)homology groups. Lemma 5.7.
With respect to the canonical splittings for q H ˚ Λ and q H ˚ Ω in Theorems 1.4 and 1.15, the following hold:(1) The shriek map i ! : q H ˚ Λ Ñ q H ˚´ n Ω restricts to maps H ˚ Λ Ñ H ˚´ n Ω and H ´˚` p Λ q Ñ H ´˚` Ω which coincide respectivelywith the topological shriek map i top ! and the cohomological pull-back map i ˚ top .(2) The pushforward map i ˚ : q H ˚ Ω Ñ q H ˚ Λ restricts to maps H ˚ Ω Ñ H ˚ Λ and H ´˚` ´ n Ω Ñ H ´˚` Λ which coincide respectivelywith the topological pushforward map i top ˚ and the cohomolog-ical shriek map i ! top .(3) The shriek map i ! : q H ˚ Ω Ñ q H ˚` n Λ restricts to maps H ˚ Ω Ñ H ˚` n Λ and H ´˚` ´ n Ω Ñ H ´˚´ n ` Λ which coincide respectivelywith the topological shriek map i ! top and the topological pushfor-ward map i top ˚ . OINCAR´E DUALITY FOR LOOP SPACES 73 (4) The pullback map i ˚ : q H ˚ Λ Ñ q H ˚ Ω restricts to maps H ˚ Λ Ñ H ˚ Ω and H ´˚` Λ Ñ H ´˚` ´ n Ω which coincide respectivelywith the topological pullback map i ˚ top and the topological shriekmap i top ! .Proof of Lemma 5.7. We prove only assertion (1), the proofs of theother ones being similar. The topological shriek map i top ! : H ˚ p Λ q Ñ H ˚´ n p Ω q descends to i top ! : H ˚ p Λ q Ñ H ˚´ n p Ω q because the class ofa point lies in its kernel for degree reasons. It admits the followingdescription in Morse theory (see [4, 21]). 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 asmooth action 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.View Ω Ă Λ as a codimension n Hilbert submanifold and choose thebase point q generic so that all critical points of S L lie outside Ω. Given a P Crit p S L q and b P Crit p S L | Ω q set M p a ; b q : “ W ´ p a q X W ` p b q , where W ´ p a q is the unstable manifold of a with respect to the flow of ´ ∇ S L on Λ, and W ` p b q is the stable manifold of b with respect to theflow of ´ ∇ p S L | Ω q on Ω. Generically, this is a manifold of dimensiondim M p a, b q “ ind p a q ´ ind p b q ´ n , where ind p a q and ind p b q denote theMorse indices of a and b with respect to S L and S L | Ω , respectively. Ifits dimension is zero this manifold is compact and defines the map i top ! .The shriek map i ! : H ˚ Λ Ñ H ˚´ n Ω can alternatively be described usingthe usual Hamiltonian profile for the symplectic homology SH ˚ p T ˚ M q ,i.e. a Hamiltonian that is constant equal to zero on D ˚ M and linearof positive slope with respect to the radial coordinate outside D ˚ M .We use the isomorphism Ψ Λ : SH ˚ p D ˚ M q » ÝÑ H ˚ Λ from [4, 7]. Itis given by a count of mixed configurations consisting of a disc withboundary on the zero section and one positive interior puncture, solvinga perturbed Cauchy-Riemann equation, together with a semi-infinitedescending gradient line in Λ starting at the loop determined by therestriction of the disc to its boundary. Recall that in H ˚ Λ we usetwisted coefficients as described in Remark 1.3.We also use the isomorphism Ψ Ω : SH ˚ p D ˚ q M q » ÝÑ H ˚´ n Ω with Ω “ Ω q M , i.e., the Lagrangian counterpart of Ψ Λ defined in [4, 7]. It isgiven by a count of mixed configurations consisting of a disc with threeboundary punctures, with one boundary component constrained to M , the other two boundary components constrained to T ˚ q M , a positivepuncture at the end bordered by the two T ˚ q M -components, solvinga perturbed Cauchy-Riemann equation, together with a semi-infinitedescending gradient line in Ω starting at the loop determined by therestriction of the strip to the boundary component which is constrainedto M . Note that the strip must be asymptotic to q at each of thepunctures bordered by M and T ˚ q M , since q is the unique intersectionpoint of M and T ˚ q M .We need to show that the following diagram commutes SH ˚ p D ˚ M q i ! / / Ψ Λ » (cid:15) (cid:15) SH ˚ p D ˚ q M q Ψ Ω » (cid:15) (cid:15) H ˚ Λ i top ! / / H ˚´ n Ω . (Recall that in H ˚ Λ and H ˚´ n Ω we use twisted coefficients as describedin Remark 1.3.) We prove that each of the compositions is equal to themap Γ : SH ˚ p D ˚ M q Ñ H ˚´ n Ω induced by the count of moduli spacesconsisting of a disc with boundary on the zero section, one positiveinterior puncture, solving a perturbed Cauchy-Riemann equation, suchthat the origin of the loop on the boundary is constrained to be equalto q , together with a semi-infinite descending gradient line in Ω startingat the loop determined by the restriction of the disc to its boundary.We first discuss the composition i top ! ˝ Ψ Λ . After gluing, the compo-sition is described by an obvious moduli space involving (starting atthe component which contains the positive puncture) a disc, a finitelength descending gradient line in Λ whose lowest energy point is aloop based at q , and a semi-infinite descending gradient line in Ω. Webring the length of the intermediate cylinder to zero in a 1-parameterfamily. This produces a homotopic operation and we observe that atthe 0-length end of the homotopy we recover the map Γ.We now discuss the composition Ψ Ω ˝ i ! . After gluing, it is described bythe count of mixed configurations consisting of a disc with one interiorpositive puncture, two boundary punctures, one boundary arc con-strained to M , the other boundary arc constrained to T ˚ q M , solvinga perturbed Cauchy-Riemann equation, together with a semi-infinitedescending gradient line in Ω starting at the loop determined by therestriction of the disc to the boundary arc which is constrained to M .Note that the disc must be asymptotic to q at each boundary puncture,since q is the unique intersection point of M and T ˚ q M . We vary theconformal type of the disc in a 1-parameter family within its modulispace by bringing the two boundary punctures together and shrinkingthe boundary arc labeled T ˚ q M , see Figure 12. This produces a homo-topic operation. We find at the other end of the moduli space a nodal OINCAR´E DUALITY FOR LOOP SPACES 75 disc with two irreducible components D and D , together with a semi-infinite descending gradient line in Ω. The irreducible component D of the nodal disc contains the interior puncture, it has the node on itsboundary, and the boundary is labeled M . The irreducible component D contains the two boundary punctures and the node, the boundaryarcs adjacent to the node are labeled M and the third boundary arc islabeled T ˚ q M . On both components D and D the resulting curves u and u solve a Cauchy-Riemann equation perturbed by a non-negative1-form and a Hamiltonian H as in the definition of symplectic homol-ogy which vanishes near M . The maximum principle forces the curve u to be contained in D ˚ M , where the Hamiltonian vanishes, so thatthe curve solves a genuine, unperturbed Cauchy-Riemann equation.Since both M and T ˚ q M are exact it follows that u is constant, neces-sarily equal to q . (This fact is akin to [7, Ch. 13, Exercise 5.3].) As aconsequence, the node on the boundary of u is sent to the point q andwe find that the count of such moduli spaces defines the map Γ. (cid:3) T ˚ q MqqMT ˚ q M T ˚ q M Mqq qq qqT ˚ q M qq MM MMM T ˚ q MT ˚ q M T ˚ q M Figure 12.
The 1-dimensional moduli space of discswith 1 interior puncture and 2 boundary punctures.6.
BV structures and Poincar´e duality
For dim M “ n we denote q H ˚ Λ “ q H ˚` n Λ , H ˚ Λ “ H ˚` n Λ , H ˚ M “ H ˚` n M. We discuss in this section the BV algebra structure on these three R -modules and the extension of Poincar´e duality in this setting. Thisstructure plays a crucial role in [22], where we give applications to thegeometry and dynamics of compact rank one symmetric spaces. Sinceall results in this section are straightforward adaptations of results inM. Abouzaid’s article [7], we will only sketch the proofs.6.1. Twisted BV-structures.
Consider an R -module C with a Z -grading deg p a q and an additional Z { w p a q , a P C . Definition 6.1 ([7]) . A BV structure on C twisted by w consists of “BV” stands for “Batalin–Vilkovisky”. ‚ a degree bilinear product ‚ : C ˚ b C ˚ Ñ C ˚ , a b b ÞÑ a ‚ b not. “ ab which is associative, satisfies w p ab q “ w p a q ` w p b q , has a two-sidedunit e P C , and is twisted commutative in the sense that ab “ p´ q deg p a q deg p b q` w p a q w p b q ba, ‚ and a degree linear map ∆ : C ˚ Ñ C ˚` satisfying ∆ “ , ∆ p e q “ , w p ∆ a q “ w p a q , and the twisted 7-term relation∆ p abc q ` ∆ p a q bc ` p´ q deg a a ∆ p b q c ` p´ q deg a ` deg b ab ∆ p c q“ ∆ p ab q c ` p´ q deg a a ∆ p bc q ` p´ q p ` deg a q deg b ` w p a q w p b q b ∆ p ac q . A BV structure for w “ untwisted . A twisted BV structurecan be turned into an untwisted one at the expense of losing the Z -grading as follows. Combine the Z -grading and the Z { Z { Ą deg p a q : “ deg p a q ` w p a q mod 2 and define newoperations r ‚p a b b q : “ p´ q deg p a q w p b q ab, r ∆ p a q : “ p´ q w p a q ∆ p a q . A straightforward calculation shows that the new operations define anuntwisted BV structure. Note that the new product r ‚ has degree 0 withrespect to the new Z -grading Ą deg, but this grading will in general notlift to an integer grading for which r ‚ has degree 0.6.2. BV structure on loop homology.
Let now M be a closed man-ifold. The S -action ρ : S ˆ Λ Ñ Λ given by reparametrization at thesource gives rise to the BV operator∆ : H ˚ Λ Ñ H ˚` Λ , ∆ p a q “ ρ ˚ pr S s ˆ a q . It was already observed by Chas and Sullivan [15] that, in the orientablecase, the homology H ˚ Λ endowed with the loop product and the oper-ator ∆ is a BV algebra (untwisted). This extends to the nonorientablecase and local coefficients as
Theorem 6.2 (Abouzaid [7]) . Let r O and σ be the orientation andspin local systems from Remark 1.3. The loop homologies H ˚ p Λ; r O q and H ˚ p Λ; σ b r O q both carry BV structures, twisted by the Z { -grading (36) ,consisting of the loop product and the canonical BV operator. (cid:3) BV structure on extended loop homology.
In [7] Abouzaidconstructs a BV structure on the Floer homology of asymptoticallylinear Hamiltonians on T ˚ M , twisted by the Z { w p γ q “ " γ preserves the orientation , γ reverses the orientationfor a loop γ . Following Seidel [60, § OINCAR´E DUALITY FOR LOOP SPACES 77 moduli space can be naturally identified with the circle S . Takingdirect limits, this gives rise to a twisted BV structure on symplectichomology of D ˚ M . Taking instead direct-inverse limits over V-shapedHamiltonians, we obtain part (a) of Theorem 6.3. (a) The extended loop homology q H ˚ Λ carries a BVstructure, twisted by the Z { -grading (36) , consisting of the primarypair-of-pants product and the canonical BV operator.(b) The extended loop cohomology q H ´ n ´˚ Λ carries a twisted BV struc-ture, twisted by the Z { -grading (36) , consisting of the secondary pair-of-pants product and the canonical BV operator.(c) The Poincar´e duality isomorphism P D : q H ˚ Λ » ÝÑ q H ´ n ´˚ Λ is an isomorphism of BV algebras.Proof. (b) The BV operator together with the primary pair-of-pantscoproduct defines a twisted co-BV structure at chain level in homol-ogy. The twisted co-BV relations then automatically follow for the sec-ondary pair-of-pants coproduct. Equivalently, we obtain a twisted BVstructure on cohomology with respect to the secondary pair-of-pantsproduct.(c) The Poincar´e duality isomorphism is given at chain level by a chainmap f whose cone is naturally a Floer complex. The compatibilitybetween the Floer differential and the BV operator on the cone impliesthat the map f commutes with the BV operators on the factors. (Thiscompatibility between the Floer differential and the BV operator is ageneral feature of Hamiltonian Floer homology.) (cid:3) The BV operator decreases the action, and therefore induces BV op-erators on all level homology groups q H p a,b q˚ Λ. In applications, a veryuseful feature of the BV structure is the following.
Proposition 6.4.
The induced BV operator on the -level homology q H “ ˚ Λ vanishes.Proof. Recall that q H “ ˚ Λ “ q H p´ ǫ,ǫ q˚ Λ for ǫ ą q H ˚ Λ near its zerolevel. When the size of the perturbation is small enough, all Floertrajectories involved in the definition of the induced BV operator atlevel 0 are contained in a collar neighbourhood of the unit cotangentbundle, where the Hamiltonian is a C -small Morse function. One canapply the argument of Salamon-Zehnder [59] to an S -family of almostcomplex structures which is time independent and constant in orderto show that these Floer trajectories are actually independent of time and therefore the corresponding moduli spaces are empty. This showsthat the induced BV operator is zero. (cid:3) Theorem 6.5.
The BV structures on q H ˚ Λ and q H ´ n ´˚ Λ induce BVstructures on the reduced (co)homology groups H ˚ Λ and H ´ n ´˚ Λ . Theinduced BV structure on H ˚ Λ agrees with the one on H ˚ Λ from Theo-rem 6.2 descended to the quotient.Proof. The factors H ˚ Λ and H ´ n ´˚ Λ are stable under products, asseen in § (cid:3) Appendix A. Poincar´e duality product on SH ă ˚ p V, B V q viahomotopies The definition of the secondary product c ˚ of degree n ´ SH ˚ pB V q is based on the algebraic Lemma 4.8. In this Appendix we spell outthe moduli spaces which underlie that Lemma and give an explicit de-scription of c ˚ in terms of pairs-of-pants with suitable Hamiltonian ho-motopies at the positive punctures. We rephrase the proof of Poincar´eduality with products using those moduli spaces, as summarized inFigure 14.For simplicity, we focus on the restriction of c ˚ to SH ˚ą pB V q . For aneasier connection to Lemma 4.8, we shall describe the equivalent degree ´ n ` SH ă ˚ p V, B V q » SH ´˚ą pB V q , denoted(37) σ P D : SH ă ˚ p V, B V q b SH ă ˚ p V, B V q Ñ SH ă ˚´ n ` p V, B V q . In Lemma 4.8 and the surrounding sections 4.3 and 4.4 we used twoHamiltonian profiles as in Figure 5: the Hamiltonian L is linear of slope ´ τ outside V { , and the Hamiltonian H is a “dented” perturbation of L . We restrict to negative range of the action: for F C p a,b q˚ p H q “ C “ C ´ ‘ C ` we have ´8 ă a ă b ă b close to 0, so that C ´ ,which is a subcomplex, is generated by orbits of type III ` , and C ` isgenerated by orbits of type II ´ . The chain homotopy K . The key object in Lemma 4.8 is the chainhomotopy K : C Ñ C r s such that Id C “ rB , K s . The map K isinduced by a homotopy of homotopies between the constant homotopyon H and the deformation H Ñ L Ñ H . (The latter induces at chainlevel the zero map because F C p a,b q˚ p L q “ K .Given R ą χ R : R Ñ r , s supportedon r , R s , constant equal to 1 on r , R ´ s , increasing on r , s anddecreasing on r R ´ , R s . See Figure 13. OINCAR´E DUALITY FOR LOOP SPACES 79 Rχ R R ´ Figure 13.
The cut-off function χ R .Define the deformation H Ñ L Ñ H by the s -dependent Hamiltonian H R p s q : “ H ` χ R p s qp L ´ H q , s P R , and consider the 1-parameter family H λ : “ p ´ λ q H R ` λH “ H ` p ´ λ q χ R p s qp L ´ H q , λ P r , s . The map K is determined by the count of elements of 0-dimensionalmoduli spaces of solutions to the 1-parametric Floer problem M p x ; y q : “ p λ, u q ˇˇ λ P r , s , u : R ˆ S Ñ p V , B s u ` J p u qpB t u ´ X H λ p s, u qq “ , lim s Ñ`8 u p s, t q “ x p t q , lim s Ñ´8 u p s, t q “ y p t q ( . The p C ´ , C ` q -component of K is given by the count of elements of 0-dimensional moduli spaces M “ p x ; y q with x P C ´ and y P C ` andis denoted g : C ´ Ñ C ` r s . This is a chain homotopy inverse of the p C ` , C ´ q -component of the Floer differential on C , denoted f : C ` Ñ C ´ r´ s , which induces the Poincar´e duality isomorphism. The chain level secondary product.
We describe the moduli spaceswhich give rise to the chain level product σ P D : C ´ b C ´ Ñ C ´ r´ n ` s , with C “ F C p a ` b, b q˚ p H q . The latter induces in homology and in thelimit a Ñ ´8 the product (37).Let Σ be a genus 0 Riemann surface with two positive punctures z ` , z ` and one negative puncture z ´ , endowed with cylindrical ends Z i ` “r ,
8q ˆ S at the positive punctures z i ` and Z ´ “ p´8 , s ˆ S at thenegative puncture z ´ . We define a 1-parameter family of Σ-dependentHamiltonians H λ Σ : Σ Ñ R , λ P r , s , by H λ Σ p z q : “ $’&’% H λ p s q z “ p s, t q P Z ` , λ P r , { s ,H ´ λ p s q z “ p s, t q P Z ` , λ P r { , s ,H otherwise . As usual we pick a 1-form β on Σ satisfying dβ “ structure compatible with the cylindrical ends on Σ and on p V . For x , x P P III ` p H q and y P P III ` p H q , define the moduli space N p x , x ; y q : “ p λ, u q ˇˇ λ P r , s , u : Σ Ñ p V , p du ´ X H λ Σ b β q , “ , lim s Ñ`8 z “p s,t qÑ z i ` u p z q “ x i p t q , i “ , , lim s Ñ´8 z “p s,t qÑ z ´ u p z q “ y p t q ( . All the elements of this moduli space have image contained in a fixedcompact set and the dimension of the moduli space isdim N p x , x ; y q “ CZ p x q ` CZ p x q ´ CZ p y q ´ n ` . The count of elements of 0-dimensional moduli spaces N “ p x , x ; y q with signs determined by a choice of coherent orientations defines σ P D : C ´ b C ´ Ñ C ´ r´ n ` s . If the parameter R ą H R islarge enough, the moduli spaces N λ “ p x , x ; y q and N λ “ p x , x ; y q areempty. This implies that σ P D is a chain map.
Isomorphism with the primary product on SH ă ˚ pB V qr´ n s . Sofar we have spelled out the geometric content of parts (a) and (b) ofLemma 4.8. We now spell out Lemma 4.8(c).With R ą H λ , Σ as above, consider the familyof Σ-dependent Hamiltonians H λ ,λ Σ parametrized by p λ , λ q P r , s defined by H λ ,λ Σ p z q : “ $’&’% H λ p s q , z “ p s, t q P Z ` ,H λ p s q , z “ p s, t q P Z ` ,H, otherwise . This is an extension to the square r , s of the previous family H λ Σ ,which is read on the top and right side of the square, parametrizedby p λ, q for λ P r , { s and by p , ´ λ q for λ P r { , s . SeeFigure 14. As before, we pick a 1-form β on Σ satisfying dβ “ J “ J λ ,λ compatiblewith the cylindrical ends on Σ and on p V . Given x , x P P III ` p H q and OINCAR´E DUALITY FOR LOOP SPACES 81 y P P III ` p H q we consider the moduli space N p x , x ; y q : “ p λ , λ , u q ˇˇ p λ , λ q P r , s , u : Σ Ñ p V , p du ´ X H λ ,λ b β q , “ , lim s Ñ`8 z “p s,t qÑ z i ` u p z q “ x i p t q , i “ , , lim s Ñ´8 z “p s,t qÑ z ´ u p z q “ y p t q ( . All the elements of this moduli space have image contained in a fixedcompact set and the dimension of the moduli space isdim N p x , x ; y q “ CZ p x q ` CZ p x q ´ CZ p y q ´ n ` . The count of elements of 0-dimensional moduli spaces N “ p x , x ; y q defines an operation η ´ : C ´ b C ´ Ñ C ´ r´ n ` s . When it has dimension 1 the moduli space admits a Floer compactifi-cation into a manifold with boundary B N “ p x , x ; y q “ ž x P III ` p H q CZ p x q“ CZ p x q´ M p x ; x q ˆ N “ p x , x ; y q> ž x P III ` p H q CZ p x q“ CZ p x q´ M p x ; x q ˆ N “ p x , x ; y q> ž y P III ` p H q CZ p y q“ CZ p y q` N “ p x , x ; y q ˆ M p y ; y q> ž y P II ´ p H q CZ p y q“ CZ p y q` N “ p x , x ; y q ˆ M p y ; y q> N λ “ , λ “ , p x , x ; y q . Here N λ “ , λ “ , p x , x ; y q is the set of elements of N p x , x ; y q lying above points p λ , λ q P Br , s . One argues directly that themoduli spaces N λ “ p x , x ; y q and N λ “ p x , x ; y q are empty, so thatwe have a canonical identification N λ “ , λ “ , p x , x ; y q ” N “ p x , x ; y q , where N “ p x , x ; y q is the moduli space that was discussed previ-ously. The first three lines in the above description of B N “ p x , x ; y q give the commutator of η with the respective boundary operators. Letus analyze the fourth line. The moduli space M p y ; y q gives the chainmap f “ f : C ` Ñ C ´ r´ s . When the length R cylinders in thepositive ends where H λ ,λ Σ is supported are close to `8 , the elements of the moduli space N “ p x , x ; y q are canonically identified withbroken configurations consisting of one rigid H λ -Floer cylinder withinput x P C ´ and output y P C , one other rigid H λ -Floer cylinderwith input x P C ´ and output y P C , and one rigid p H, β q -Floer pair-of-pants with inputs y , y and output y P C ` . Since C ´ is an idealfor the pair-of-pants product, we must have y , y P C ` , so the twocylinders correspond to the map g defined above and the pair-of-pantscorresponds to the primary product µ “ µ ` on C ` . Putting everythingtogether, the above description of B N “ p x , x ; y q yields the relation rB ´ , η ´ s “ σ P D ´ f µ ` p g b g q from Lemma 4.8(c), where the product σ P D is denoted σ ´ . λ λ Figure 14.
The secondary product σ P D is read on thetop and right sides of the parametrizing square.
Appendix B. Grading conventions
Our standing convention is to grade Hamiltonian Floer homology bythe Conley-Zehnder index of orbits, and Lagrangian Floer homologyby the Conley-Zehnder index of chords.Given a preferred trivialization, the Conley-Zehnder index CZ p γ q ofa Hamiltonian orbit γ is such that the Fredholm indices of the cap-ping operators o ˘ p γ q defined on the sphere with one positive/negativepuncture, and with asymptotic behavior at the puncture given by thelinearized Hamiltonian flow along the periodic orbit γ , are given by(see [30, § p o ´ p γ qq “ n ´ CZ p γ q , index p o ` p γ qq “ n ` CZ p γ q . Given a preferred trivialization, the Conley-Zehnder index CZ p c q of aHamiltonian chord c is such that the Fredholm indices of the cappingoperators o ˘ p c q defined on the disc with one boundary puncture whichis positive/negative, and with asymptotic behavior at the boundarypuncture given by the linearized Hamiltonian flow along the Hamilton-ian chord c with endpoints on the Lagrangian, are given by (see [30, § p o ´ p c qq “ n ´ CZ p c q , index p o ` p c qq “ CZ p c q . OINCAR´E DUALITY FOR LOOP SPACES 83
The case in point is the cotangent fiber L “ D ˚ q M . This is a Maslov0 exact Lagrangian. Moreover, there is a preferred trivialization of T T ˚ M along Hamiltonian chords given by the complexification of theLagrangian vertical distribution, and this determines a canonical grad-ing on the symplectic homology and cohomology groups.The degree of an operation determined by a punctured Riemann surfaceis calculated by gluing negative/positive capping operators at the posi-tive/negative punctures. For example the half-pair-of-pants product on SH ˚ p L q is defined by the disc with 2 positive boundary punctures andone negative boundary puncture, so that it has degree ´ n . The half-pair-of-pants primary coproduct on SH ˚ p L q is defined by the disc withone positive boundary puncture and 2 negative boundary punctures,so that it has degree 0. The half-pair-of-pants secondary coproducton SH ą ˚ p L q has degree one higher, i.e. `
1, because it is defined bya parametrized Floer problem with one-dimensional parameter space.Its (algebraic) dual half-pair-of-pants product on SH ˚ą p L q has oppositedegree, i.e. ´ §
5. Recall the zipper , which is the closed disc with one inte-rior positive puncture and one boundary negative puncture. It definesmaps i ! : SH ˚ p D ˚ M q Ñ SH ˚ p L q and i ! : SH ˚ p L q Ñ SH ˚ p D ˚ M q andwe now compute their degree to be 0. Let M p γ ; c q be the moduli spaceof such discs asymptotic to an orbit γ at the positive puncture and toa chord c at the negative puncture. To compute the Fredholm index I of the linearized problem we glue to the linearized operator the cap-ping operators o ´ p γ q at the positive puncture and o ` p c q at the negativepuncture. Using additivity of the index under gluing and the fact thatthe index of the Cauchy-Riemann problem on a disc with boundary ona Maslov 0 Lagrangian is equal to n , we obtain I ` p n ´ CZ p γ qq ` CZ p c q “ n, so that I “ CZ p γ q ´ CZ p c q and therefore CZ p c q “ CZ p γ q for index 0moduli spaces. This shows that i ! and i ! have degree 0.By Lemma 5.7, the maps i ! and i ! correspond to the topological shriekmaps via the commuting diagrams which involve the isomorphismsfrom Theorems 1.2 and 1.14: SH ˚ p D ˚ M q i ! / / – (cid:15) (cid:15) SH ˚ p L q – (cid:15) (cid:15) H ˚ Λ i ! / / H ˚´ n Ω , SH ˚ p L q i ! / / – (cid:15) (cid:15) SH ˚ p D ˚ M q – (cid:15) (cid:15) H n ´˚ Ω i ! / / H ˚ Λ . In these diagrams, if the coefficients for symplectic homology are un-twisted then the coefficients for loop homologies must be twisted.
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