aa r X i v : . [ m a t h . S G ] F e b Hamiltonian Floer theory on surfaces
Dustin Connery-Grigg ∗ February 23, 2021
Abstract
We develop connections between the qualitative dynamics of Hamiltonian isotopies on a surfaceΣ and their chain-level Floer theory using ideas drawn from Hofer-Wysocki-Zehnder’s theory of finiteenergy foliations. We associate to every collection of capped 1-periodic orbits which is ‘maximallyunlinked relative the Morse range’ a singular foliation on S × Σ which is positively transverse to thevector field ∂ t ⊕ X H and which is assembled in a straight-forward way from the relevant Floer modulispaces. As a consequence, we derive a purely topological and Turing-computable characterization ofthe spectral invariant c ( H ; [Σ]) for generic Hamiltonians on arbitrary closed surfaces. This completes,for generic Hamiltonians, a project initiated by Humili`ere-Le Roux-Seyfaddini, in addition to fulfillinga desideratum expressed by Gambaudo-Ghys seeking a topological characterization of the Entov-Polterovich quasi-morphism on Ham ( S ). The goal of this paper is to introduce a set of tools for tying together the qualitative dynamics of Hamil-tonian isotopies on surfaces and the chain-level structure of their Floer complexes. Since its introductionby Floer in his approach to the famed Arnol’d conjecture, Hamiltonian Floer theory has become a stan-dard tool in the modern symplectic geometer’s toolkit. However, in spite of the numerous uses to whichit has been put, the relationship between the Floer complex CF ∗ ( H, J ) of a generic Floer pair (
H, J ) andfeatures of the dynamics of the Hamiltonian isotopy φ H induced by H remains rather unclear, essentiallydue to the difficulty in understanding the relationship between solutions to Floer’s equations and thequalitative properties of the (generically non-autonomous) vector field X H used to define them.To give a better sense of what we mean by this, and to give some sense of the analogy that our workseeks to develop, consider the case when H is a C -small Morse function on a surface and J is constant;because of the tight relationship between negative gradient-flow lines and the defining Hamiltonian vectorfield — away from singular points of X H , the negative gradient trajectories are positively transverse tothe trajectories of the Hamiltonian flow — it’s easy to construct the filtered Floer complex simply fromknowledge of the qualitative dynamics of X H and the critical values of H , and contrariwise, one canreadily extract a significant amount of information about the qualitative dynamics of X H from knowl-edge of the Floer complex. The situation becomes much murkier if we allow H or J to vary with timeor even if we omit the C -smallness condition. In fact, the strongest result that exists in this directionhas been given by Humili`ere-Le Roux-Seyfaddini in [12], in which the authors give a characterization ofthe Hamiltonian spectral invariant c ( H ; [Σ]) (see Section 4.1 for the definition) in terms of the topologyof certain braids of 1-periodic orbits of X H in the case when H is any autonomous Hamiltonian andΣ = S . However, even the approach therein hints at the limitations of our collective understanding ofthe situation; in order to circumvent the difficulties posed by the relationship of more time-dependentFloer cylinders to the underlying dynamics, the authors develop an axiomatic approach to the spectralinvariant, and are forced to use techniques which are fundamentally limited to the study of autonomousHamiltonians.The view developed in this paper attacks the question of the relationship of Floer cylinders to dy-namics in low dimensions head-on, and we develop a theory which essentially reduces the study of this ∗ This work was supported by the Fonds de recherche du Qu´ebec – Nature et technologies, Grant C -small Morse functions, relatedto the underlying dynamics in a straight-forward way. This gives a transparent relationship betweenthe homologically non-trivial part of the (filtered) Floer complex and the dynamics of the Hamiltonianisotopy. As a corollary, we obtain an effective topological characterization of the spectral invariantsassociated to the point and fundamental classes on arbitrary closed symplectic surfaces. Unless otherwise noted, we always work on a closed symplectic surface (Σ , ω ). In order to treat thecase of C -small Morse functions on even footing with generic Hamiltonians, we consider the flow of thelifted vector field ˇ X H := ∂ t ⊕ X H on S × Σ. Hamiltonian Floer theory from our point of view thenbecomes a matter of studying the relationship between the dynamics of the flow of ˇ X H and cylinders ofthe form ( s, t ) ( t, u ( s, t )) ∈ S × Σ, for u a Floer cylinder running between strands of the braid givenby graphs t ( t, x ( t )) for x an element of P er ( H ), the collection of a contractible 1-periodic orbits of H .In order to treat all surfaces at once (including the case of Σ = S ), it becomes useful to introduce thenotion of a capped braid. Essentially, a capped braid ˆ X is a finite collection of capped loops { ˆ x , . . . , ˆ x k } called its strands such that the graphs of the underlying loops are all disjoint in S × Σ. We say thatˆ X is unlinked if the capping disks may be chosen to have disjoint graphs in D × Σ (see Section 3 forformal definitions). For a Hamiltonian H and a capped braid ˆ X ⊆ g P er ( H ) of capped 1-periodic orbitsof φ H , we say that ˆ X is maximally unlinked relative the Morse range if ˆ X is unlinked, each capped loopin ˆ X has Conley-Zehnder index lying in the set {− , , } , and such that ˆ X is maximal (as a subsetof g P er ( H )) with respect to these two properties. We denote the collection of such capped braids by murm ( H ). Our main result is the following Theorem A.
Let ( H, J ) be a Floer non-degenerate pair on a closed symplectic surface (Σ , ω ) , then toeach ˆ X ∈ murm ( H ) we may associate a singular foliation F ˆ X of S × Σ with singular leaves given bythe graphs of the strands of ˆ X and regular leaves parametrized by maps of the form ( s, t ) ( t, u ( s, t )) for u a Floer cylinder for ( H, J ) connecting ˆ x, ˆ y ∈ ˆ X . Moreover, the vector field ∂ t ⊕ X H is positivelytransverse to F ˆ X . Similar foliations played a crucial role in Bramham’s construction of periodic approximations forirrational pseudo-rotations of the disk in [2]. Our approach gives both the existence of such foliations forgeneric Hamiltonians, and moreover ties their behaviour directly to the structure of the Floer complexand the dynamics of the Hamiltonian isotopy.The structure of the foliations F ˆ X could be a priori rather complicated, however, with Theorem Ain hand, we can define the ˆ X -restricted action functional A ˆ X ∈ C ∞ ( S × Σ) by A ˆ X ( t, u ( s, t )) = A H (ˆ u s ),for A H : e L (Σ) → R the Hamiltonian action functional on the space of capped loops (the capping of u s is naturally induced by the cappings of the limiting orbits). A ˆ X turns out to be a Morse-Bott function,and if we define A ˆ Xt ∈ C ∞ (Σ), t ∈ S , to be its restriction to the fiber { t } × Σ, we obtain an S -family ofMorse functions, such that the negative gradient flow of A ˆ Xt provides a singular foliation which coincideswith the foliation F ˆ Xt given by intersecting F ˆ X with the fiber over t ∈ S . Sliding the fiber { } × Σalong the circles t ( t, u s ( t )) provides a loop ( ψ ˆ Xt ) t ∈ S , and we prove Theorem B.
For every t ∈ S , F ˆ Xt is a singular foliation of Morse type. Moreover, the loop ( ψ ˆ Xt ) t ∈ S is a contractible Hamiltonian loop such that the orbits of ( ψ ˆ X ) − ◦ φ H are positively transverse to thefoliation F ˆ X . We thereby reduce the study of the qualitative dynamics of the isotopy φ H to the much betterunderstood situation of dynamics which are positively transverse to a Morse-type foliation. Note thatsimilar foliations (with a weaker notion of positive transversality) have been constructed by Le Calvez in[17] for Hamiltonian homeomorphisms and play a central role in the forcing theory developed in [18]. Theabove result can be viewed as giving a Floer-theoretic construction of certain of Le Calvez’s foliations,along with additional insight into their structure in the smooth case. We also obtain as a corollary2he following somewhat surprising structural result about the topology of the braid generated by the1-periodic orbits of H (a capped braid is said to be linked if it is not unlinked): Theorem C.
Let H be non-degenerate ˆ X ∈ murm ( H ) , and let ˆ y ˆ X be a capped time- k orbit of φ H for k ∈ Z , then ˆ X ∪ ˆ y is linked. In particular every capped braid which is maximally unlinked relative theMorse range is maximally unlinked as a subset of g P er ( H ) . Finally, as the foregoing discussion suggests, our approach gives a transparent relationship betweenthe filtered Floer complex in non-trivial degrees and the dynamics of a generic Hamiltonian isotopy. Asan example of the sort of chain-level information which can be extracted by these techniques we givethe following characterization of the spectral invariants (see Section 4.1 for definitions) associated to thepoint and the fundamental class:
Theorem D.
Let H be a non-degenerate Hamiltonian on a closed symplectic surface (Σ , ω ) , then c ( H ; [Σ]) = min ˆ X ∈ murm ( H ) max ˆ x ∈ ˆ X A H (ˆ x ) ,c ( H ; [ pt ]) = max ˆ X ∈ murm ( H ) min ˆ x ∈ ˆ X A H (ˆ x ) . This seems to be the first such result giving an effectively computable formula (modulo the taskof first calculating the 1-periodic orbits) for spectral invariants of generic Hamiltonians. The closestprecedent appears to be [12], which computes these spectral invariants for autonomous Hamiltonianswhen Σ = S through similar minimax formulas over slightly different collections of orbits. There arealso characterizations of c ( H ; [Σ]) for higher dimensional manifolds with H non-autonomous but withsignificant restrictions on H (see [13] for the aspherical case and [21] for an extension to general symplecticmanifolds). Additionally, this result is easily seen to give a (non-effective) topological characterizationof the Entov-Polterovich quasi-morphism µ : Ham ( S , ω ) → R defined in [4] by µ ( φ ) := V ol ( S , ω ) · lim k →∞ c ( H ♮k ; [ S ]) k , for H any Hamiltonian with φ H = φ , in terms of the long term structure of the collections of cappedbraids murm ( H ♮k ). The question of the existence of such a topological characterization was first raisedexplicitly by Gambaudo-Ghys in [5] but little positive progress has been made since (although see [14] forsome impossibility results, which in particular imply that the ‘braid quasi-morphism averaging’ proceduredeveloped in [5] cannot produce µ ). The ideas in this paper evolve reasonably naturally from adopting the position — implied by the work ofHumili`ere-Le Roux-Seyfaddini — that the topology of the capped braids which make up g P er ( H ) oughtto be in some way related to the structure of the filtered Floer complex of H , and proceeding to studywhy this should be the case.The homological linking number , presented in Section 3, produces a homotopy invariant for pairs ofcapped braids ( ˆ X, ˆ Y ) by counting the signed number of intersections made by the strands throughout ageneric deformation which takes ˆ X to ˆ Y , subject to a certain homotopy condition. The graph of sucha deformation gives rise to a collection of cylinders in [0 , × S × Σ, which we call a braid cobordism ,whose signed intersections are precisely what is counted by the homological linking number. The proofof Theorem A exploits the interaction of this invariant with the positivity of intersections of pseudoholo-morphic curves in dimension 4 by viewing, via the Gromov trick, the Floer boundary and continuationoperators as giving rise to collections of braid cobordisms between capped braids made up of cappedorbits lying over the corresponding ends (we adopt, therefore, the field-theoretic perspective on Floertheory presented in [24]). Positivity of intersections implies that the homological linking number cannotdecrease along such cobordisms. In order to deal with cylinders which emerge from or converge to thesame orbit, we make use of the analysis of the relative asymptotic behaviour of pseudo-holomorphiccurves developed by Siefring in [29], which pairs with work of Hofer-Wysocki-Zehnder in [8] to connectthe Conley-Zehnder index of an orbit to bounds on the winding behaviour of pairs of cylinders whichemerge from or converge to that orbit. 3n the contact setting, these sorts of bounds (in the non-relative case), along with the insight thatunder appropriate index conditions on the asymptotic orbits families of pseudoholomorphic curves au-tomatically form local foliations in the symplectization of a contact manifold, go back to the pioneeringwork of Hofer-Wysocki-Zehnder in [9], [8], [10] and [11]. Siefring has also more recently put this cir-cle of ideas to use in [30] to define an intersection number for arbitrary pseudoholomorphic curves in4-dimensional symplectic cobordisms which is invariant under homotopy.In comparison with [30] and [11] which put many of these same ideas to work, the main noveltiesin this paper are our use of the homological intersection number to provide a means by which to probethe global topology of the collection of g P er ( H ), and to relate this to the behaviour of Floer modulispaces, along with the use of Floer-theoretic tools — continuation maps and the cap product — toprovide existence principles for the required families of Floer cylinders against which we may pit our apriori controls. In addition, the fact that we place ourselves in a Floer-theoretic (rather than an SFT-type setting, as in the previously mentioned works) considerably simplifies the analytic prerequisites;hopefully this work may serve to initiate non-experts to this productive circle of ideas while minimizingthe technical requirements. This circle of ideas is developed in the the form needed for our purposesin Section 4.2 (linking and asymptotic behaviour of Floer cylinders) and Section 4.3 (producing localfoliations from Floer moduli spaces).The idea of relating the global topology of the braids formed by periodic orbits of a Hamiltonianto Floer theory via positivity of intersections arguments also has some precedent on the disk in [33], inwhich the authors use Floer homology to define braid invariants and obtain a forcing theory for periodicorbits of Hamiltonians, and it is likely that the results in this paper could be combined with these toproductive ends, but we make no attempt at this here.A final novelty of our approach is that we take the topology of capped braids seriously (rather thanassigning primacy to the trivializations which they induce). In addition to allowing us to treat all surfacesin a uniform manner, the value of this perspective in the Floer-theoretic context is supported by theidentification provided by Proposition 3.23 between the area of a capping disk for a capped orbit ˆ x andthe average homological linking of ˆ x with each point of the surface (see Proposition 3.23). This providescrucial tools for relating the actions of various capped orbits to their dynamics relative other orbitsin g P er ( H ) (see Lemma 8.1 and Proposition 8.2), which prove essential in characterizing the spectralinvariants. The author would like to thank his PhD supervisor Fran¸cois Lalonde for his enthusiastic support andencouragement throughout this project. He would also like to thank Egor Shelukhin for his helpfulcomments and feedback on an earlier version of this paper, as well as Jordan Payette for many stimulatingconversations on the subject matter.
It will be convenient in what follows to have some explicit language with which to speak about singularfoliations. To that end, we will make use of some elementary notions from the theory of
Stefan-Sussmannfoliations (cf. [3] and the references therein, in particular [31]) which will be suitable to our purposes.Let G k ( M ) → M denote the k -Grassmannian of M , having fiber Gr ( k, T x M ) over x ∈ M , and let G ∗ ( M ) := ⊔ nk =0 G k ( M ), where n = dim M denote the total Grassmannian of M . Definition 2.1. A (generalized) distribution on a manifold M is a section D : M → G ∗ ( M ). A localsection X : M → T M is said to belong to D if X ( x ) ∈ D ( x ) for all x ∈ dom ( X ). The set of all smoothlocal sections X ∈ X loc ( M ) belonging to D is denoted by ∆ D . Definition 2.2.
A generalized distribution D is said to be smooth if, D ( x ) = span h X ( x ) i X ∈ ∆ D , forevery x ∈ M . Definition 2.3. A (smooth) k -leaf of M is a subset L ⊆ M equipped with a differentiable structure σ such that ( L, σ ) is a connected k -dimensional immersed submanifold of M and such that for any4ontinuous map f : N → M with f ( N ) ⊆ L and N a locally connected topological space, we have that f : N → ( L, σ ) is continuous.
Definition 2.4.
A ( C ∞ ) -singular (Stefan-Sussmann) foliation of M is a partition F of M intosmooth leaves of M such that for every x ∈ M , there exists a local smooth chart ϕ : U → O ( x ) ⊆ M ,from U ⊆ R n an open neighbourhood of 0 ∈ R n , with ϕ (0) = x and1. U = V × W for V an open neighbourhood of 0 in R k and W an open neighbourhood of 0 in R n − k ,where k is the dimension of the smooth leaf L x ∈ F containing x .2. L ∩ ϕ ( U × W ) = ϕ ( U × l ), for every leaf L ∈ F , where l := { w ∈ W : ϕ (0 , w ) ∈ L } . Definition 2.5.
A smooth generalized distribution D is said to be integrable if for every x ∈ M thereexists an immersed submanifold L ⊆ M , such that x ∈ L , and T y L ⊆ D ( y ) for all y ∈ L . Such animmersed submanifold is called an integral submanifold of D . Theorem 2.6 ([31]) . If D is a smooth integrable generalized distribution and F D is the partition of M formed by taking the collection maximal connected integral submanifolds of D , then F D is a smoothsingular Stefan-Sussmann foliation. For a singular foliation F , we let d ( − , F ) denote the function which sends x ∈ M to the dimensionof the leaf of F passing through x ∈ M . Definition 2.7.
A smooth singular foliation is said to have codimension k if n − k = max x ∈ M d ( x, F ).For such a foliation, we define the domain to be dom ( F ) := { x ∈ M : d ( x, F ) = n − k } , while we definethe singular set of F to be sing ( F ) := M \ dom ( F ).A leaf of F is said to be regular if it is of maximal dimension, otherwise it is said to be singular . F is said to be oriented if every regular leaf of F is in addition equipped with an orientation, and thelocal charts about points on the regular leaves may be taken to be orientation-preserving. Throughout this section and the rest of the paper, Σ will always denote a smooth symplectic surface(Σ , ω ), L ( M ) the (smooth) loop space of the manifold M , L ( M ) its space of contractible loops and e L ( M ) its Novikov covering space (see [19], Section 12.1). For x ∈ L ( M ), we write ˇ x ( t ) := ( t, x ( t ))for its graph. For u : I × S → M , where I ⊆ R , we write ˜ u ( s, t ) := ( s, t, u ( s, t )) for its graph andˇ u ( s, t ) := ( t, u ( s, t )) for the projection of its graph onto S × M . Definition 3.1.
For any k ∈ N , we define the k -configuration space C k (Σ) := { ( z , . . . , z k ) ∈ Σ k : ( i = j ) ⇒ z i = z j } Definition 3.2. An (ordered) k -braid is a an element X = ( x , . . . , x k ) ∈ L ( C k (Σ)). Denote by B k (Σ) the space of ordered k -braids. The loop x i is called the i -th strand of X , for i = 1 , . . . , k . Definition 3.3. An unordered k -braid is an element [ X ] ∈ L ( C k (Σ)) /S k , where S k acts by permu-tation of coordinates. Such unordered braids may be identified with certain finite subsets of L (Σ). Remark.
We raise the distinction between ordered and unordered braids here mainly to flag for thereader that we will make no real effort outside of this section to maintain the distinction between thesetwo concepts. In particular, we will routinely treat ordered braids as finite subsets of L (Σ) and performset-wise operations on them, when properly speaking we should be speaking of the unordered braidswhich they represent. We will moreover speak simply of ‘braids’ relying on the context to make clearwhether these braids are ordered or unordered. For the remainder of this section, we will make a cleardistinction between ordered and unordered braids, mainly to convince the suspicious reader that nothingessential is lost in making this elision. Definition 3.4.
The graph ˇ X of an (ordered) k -braid is the set-valued map ˇ X ( t ) = ⊔ ki =1 ˇ x ( t ) ⊆ S × Σ, t ∈ S . The graph of an unordered braid [ X ] is the graph of some (hence every) representative X of [ X ].5 efinition 3.5. An ordered k ′ -braid Y ∈ B k ′ (Σ) is an ordered sub-braid of X ∈ B k (Σ) if Y ⊆ X , asan ordered set. An unordered braid [ Y ] is a sub-braid of [ X ] if [ Y ] ⊆ [ X ] as sets. There is an obviouspartial ordering on the collection of sub-braids of X (resp. of [ X ]). Definition 3.6. X ∈ B k (Σ) is contractible if each strand of X is a contractible loop. We write B k (Σ)for the space of contractible ordered k -braids. [ X ] ∈ B k (Σ) /S k is contractible if some (hence every)representative X of [ X ] is contractible. Definition 3.7.
A continuous map h : [0 , → B k (Σ) with h (0) = X , h (1) = Y is a braid homotopyfrom X to Y . When such a map exists, we shall say that X and Y are braid homotopic , denoted X ≃ Y . The map ( s, t ) h i ( s, t ) is called the i -th strand of h . To any braid homotopy, we associateits graph ˜ h ( s, t ) = ⊔ ki =1 ˜ h i ( s, t ) ⊆ [0 , × S × Σ, ( s, t ) ∈ [0 , × S . Definition 3.8.
An ordered braid X will be said to be trivial if all of its strands are constant maps. Wewill sometimes write 0 ∈ B k (Σ) to stand for some fixed but arbitrary trivial braid, when the particularchoices of the constant maps are unimportant. An unordered braid [ X ] is trivial if some (hence every)ordered representative is trivial. Definition 3.9. X ∈ B k (Σ) is unlinked if X ≃
0. A ordered braid is linked if it is not unlinked. Anunordered braid is unlinked (resp. linked) if some (hence every) ordered representative is unlinked (resp.linked).
Definition 3.10.
Two unordered braids [ X ] , [ Y ] are said to be unlinked if [ X ] S [ Y ] is unlinked, andare said to be linked otherwise. Two ordered braids X, Y ∈ B k (Σ) are said to be unlinked (resp. linked)if [ X ] and [ Y ] are unlinked (resp. linked). Definition 3.11.
A continuous map h : [0 , → L (Σ) k with h (0) = X ∈ B k (Σ), h (1) = Y ∈ B k (Σ) willbe called a braid cobordism if there exists some δ > h ( s ) ∈ B k (Σ) , ∀ s ∈ (0 , δ ) ∪ (1 − δ, Remark.
We will frequently find ourselves concerned with maps h : I → L (Σ) k , where I = R or I = [ a, b ]for some a, b ∈ R , and in the case that I = R , it will always be the case that h extends continuouslyto a map ¯ R → L (Σ) k such that on some neighbourhood of ±∞ , the graphs of the strands of h do notintersect. In such a case, we will speak freely of ‘the’ braid cobordism induced by h , which is simplyany braid cobordism h ◦ ϕ , where ϕ : ¯ I → [0 ,
1] is any orientation-preserving diffeomorphism.
Definition 3.12.
An (ordered) capped k -braid ˆ X is an equivalence class [ X, ~w ] where X ∈ B k (Σ) and ~w = ( w , · · · , w k ) with w i : D → Σ a capping disk for the i -the strand of X , subject to the equivalencerelation [ X, ~w ] ∼ [ X ′ , ~w ′ ] if and only if X = X ′ and [ w i ] − [ w ′ i ]) = 0 ∈ π (Σ) for each i = 1 , · · · , k . Thespace of ordered capped k -braids is denoted by e B k (Σ). The capped loop ˆ x i = [ x i , w i ] ∈ e L (Σ) is calledthe i -th strand of ˆ X . The notion of capped sub-braids ˆ Y ⊆ ˆ X is defined in the obvious way.The distinction between ordered capped braids and unordered capped braids obtains here as well,and we adopt parallel conventions as those discussed in the case of braids in Remark 3. π (Σ) k acts on e B k (Σ) by the obvious ‘gluing of spheres’:( A , . . . , A k ) · ([ x , w ] , . . . , [ x k , w k ]) = ([ x , A w ] , . . . , [ x k , A k w k ]) , where here we abuse notation slightly by thinking of A i ∈ π (Σ , x i (0)) as being both a homotopy classof maps, as well as a particular choice of a representative from that class. This action does not descendto an action on e B k (Σ) /S k . However, if we denote by F ix S k ( π (Σ) k ) ≃ π (Σ) the set of fixed points ofthe action of the symmetric group on π (Σ) k by permutation of coordinates, we obtain a well-definedinduced action on unordered braids given by ( A, . . . , A ) · [ ˆ X ] = [( A, . . . , A ) · ˆ X ], for A ∈ π (Σ). Definition 3.13.
A trivial braid 0 ∈ B k (Σ) has a naturally associated capping ˆ0 ∈ e B k (Σ) given bycapping each strand of 0 with the constant capping. We call any such braid a trivial capped braid .When the particular components of a trivial capped braid are unimportant, we we denote some fixedbut arbitrary capped braid by the symbol ˆ0. An unordered capped braid is said to be trivial if some(hence every) ordered representative is trivial. 6 efinition 3.14. For A = ( A , · · · , A k ) ∈ π (Σ) k , an ordered braid cobordism h from X to Y will becalled an A -cobordism from [ X, ~w ] to [ Y, ~v ] if [ w i ] h i ] − [ v i ]) = A i , for all i = 1 , · · · , k . Wheneversuch a map exists, [ X, ~w ] and [
Y, ~v ] will be said to be A -cobordant . This notion descends to unorderedcapped braids provided that A ∈ F ix S k ( π (Σ) k ). Definition 3.15. If u : [0 , → L (Σ) is homotopy from x to y , then for any choice of cappingsˆ x = [ x, w x ] and ˆ y = [ y, w y ], u is an A -cobordism from ˆ x to ˆ y , for A = [ w x ] u ] − [ w y ]) ∈ π (Σ).Moreover, for any s ∈ [0 , u s ∈ L (Σ). Namely,if we write α s ( τ ) := u ( s · τ, t ) and β ( τ ) := u (1 − (1 − s ) · τ, t ) for s ∈ [0 , u s either of the cappings [ u s , w x α s ] or [ u s , w y β s ], and these two cappings are obviously related by A · [ u s , w y β s ] = [ u s , w x α s ]. Consequently, if u is a 0-homotopy between ˆ x and ˆ y , these two cappingsagree and we may associate a unique cappingˆ u s := [ u s , w x α s ] = [ u s , w y β s ]to each u s in this case. We will call such a capping the natural capping of u s whenever u is such a0-homotopy. Definition 3.16. An A -cobordism h from ˆ X = [ X, ~w ] to ˆ Y = [ Y, ~v ] is called an A -homotopy if h isin addition a braid homotopy from X to Y . In such a situation, we will say that ˆ X and ˆ Y are A -homotopic , and we will denote the relation by ˆ X ≃ A ˆ Y . This notion descends to unordered cappedbraids, provided that A ∈ F ix S k ( π (Σ)). Definition 3.17. ˆ X ∈ e B k (Σ) is unlinked if ˆ X ≃ ˆ0. An unordered capped braid [ ˆ X ] is unlinked ifsome (hence every) ordered representative is unlinked. The notion of linkedness for a capped braid orfor a pair of capped braids is defined as in the case of braids. To any x ∈ L (Σ), we may associate the set S x := { y ∈ L (Σ) : ∃ t ∈ S such that x ( t ) = y ( t ) } , with the property that L (Σ) \ S x consists of precisely those loops y such that ( x, y ) ∈ L (Σ) is a braid.We fix some family J = ( J t ) t ∈ S of ω -compatible almost complex structures, and let g J = ( g J t ) t ∈ S denote the associated family of compatible metrics. This data provides us with an exponential neigh-bourhood O ⊆ L ( M ) of x , along with a diffeomorphism Exp : U → O , defined by ( Exp ( ξ ))( t ) := exp J t x ( t ) ( ξ ( t )), for U ⊆ Γ ∞ ( x ∗ T Σ) a neighbourhood of the zero section.Remark that any choice of a lift ˆ x = [ x, α ] ∈ e L (Σ) of x gives rise to a lift ˜ O α of O , and T ˆ x e L (Σ) ≃ ( x ∗ T Σ , J, ω ) comes equipped with a homotopically unique unitary trivialization T ˆ x : S × ( R , J , ω ) → ( x ∗ T Σ , J, ω ) , provided by any trivialization which extends over the capping. For any y ∈ O \ S x and any cappingˆ x α := [ x, α ] of x , let ˆ y α denote the unique lift of y lying in e O α . We define the winding number of ˆ x α andˆ y α by ℓ (ˆ x α , ˆ y α ) := wind (( T − x ◦ g Exp − )(ˆ y α )) , where wind ( ξ ) denotes the classical winding number of a non-vanishing family of vectors t ξ ( t ) for t ∈ S in R . Note that for A ∈ π (Σ), we clearly have ℓ ( A · ˆ x α , A · ˆ y α ) = ℓ (ˆ x α , ˆ y α ) + c ( A ), and ℓ issymmetric in its arguments. In order to extend this definition to arbitrary cappings of the orbits x and y , let A, B ∈ π (Σ) and we define ℓ ( A · ˆ x α , B · ˆ y α ) := ℓ (ˆ x α , ˆ y α ) + 12 ( c ( A ) + c ( B )) . It is easy to check that this definition does not depend on the choice of α , nor the choice of compatiblealmost complex structure J = ( J t ) t ∈ S , and agrees with the previous definition in the case that ˆ x and ˆ y are close in e L (Σ). A more geometric view of this formula will be provided in the following subsection.7 .2 The homological linking number for capped braids Definition 3.18.
Let ˆ X, ˆ Y ∈ e B k (Σ) and A ∈ π (Σ) k . We define the homological ( A )-linkingnumber of ˆ Y relative to ˆ X L A ( ˆ X ; ˆ Y ) := X ≤ i The above definition may be generalized straightforwardly by replacing the cylinder [0 , × S with a surface S g,k − ,k + of genus g , having k − negatively oriented boundary components and k + positivelyoriented boundary components. This provides a family of homotopy invariants for collections of k − ‘input’and k + ‘output’ capped braids in the obvious way. These invariants likely provide interesting insightinto the structure of the field-theoretic operations in Floer theory described in [24] at the chain level,but we will not pursue this here.The following proposition summarizes the main properties of the homological linking number whichwe will need in our investigations. Proposition 3.19. For any ˆ X, ˆ Y , ˆ Z ∈ e B k (Σ) and A, B ∈ π (Σ) k we have that:1. L A ( ˆ X, ˆ Y ) is well-defined.2. For any σ ∈ S k , L σ · A ( σ · ˆ X ; σ · ˆ Y ) = L A ( ˆ X ; ˆ Y ) .3. L A ( ˆ X, ˆ Y ) + L B ( ˆ Y , ˆ Z ) = L A + B ( ˆ X, ˆ Z ) .4. If ˆ X and ˆ Y are A -homotopic, then L A ( ˆ X, ˆ Y ) = 0 .5. L A ( ˆ X, ˆ Y ) = − L − A ( ˆ Y , ˆ X ) .6. L A ( ˆ X, B · ˆ Y ) = L A + B ( ˆ X, ˆ Y ) .7. L ( ˆ X, A · ˆ X ) = ( k − P ki =1 c ( A i )2 .Proof. Items 1-6 are straightforward consequences of the definition of L A ( ˆ X, ˆ Y ). To prove item 7, wenote first that item 3 implies that L (ˆ0 , ˆ X ) + L ( ˆ X, A · ˆ X ) + L ( A · ˆ X, A · ˆ0) = L (ˆ0 , A · ˆ0) . Next, items 5 and 6 imply that L ( A · ˆ X, A · ˆ0) = L A ( A · ˆ X, ˆ0) = − L − A (ˆ0 , A · ˆ X ) = − L (ˆ0 , ˆ X ) , whence we need only show that the desired formula holds when ˆ X = ˆ0. To reduce to an even simplercase, let us write A as( A , · · · , A k ) = ( A , , · · · , 0) + (0 , A , , · · · , 0) + · · · + (0 , , · · · , A k ) =: A ′ + · · · + A ′ k . By items 3 and 6, demonstrating the desired equality is therefore equivalent to showing that L (ˆ0 , A ′ i · ˆ0) =( k − · c ( A i )2 for any i = 1 , · · · , k . In what follows, let (ˆ p , · · · , ˆ p k ) = ˆ0 represent the trivial capped braid.Since the statement is trivial when Σ = S , as then π (Σ) = 0 and every capped braid is 0-homotopicto itself, we now suppose Σ = S . For m ∈ Z , if u i : ( S , ∗ ) → (Σ , p i ) represents A i = m [ S ] ∈ π (Σ , p i ),we may pull u i back along the quotient [0 , × S → S , given by collapsing the boundary circles topoints, to a map which we will denote h i : [0 , × S → Σ. If we take h to be the 0-cobordism from ˆ0to A ′ i · ˆ0 given by h i as the i -th strand and the constant strand h j ( s, t ) ≡ p j for all other strands j = i ,then the important point is that P D ( c ) = 2[ S ] and hence the intersection of the graph of h i with theconstant cylinder h j ( s, t ) ≡ p j for j = i contributes precisely ( u i ∗ [ S ]) ∩ [ p j ] = c ( A i )2 to the sum defining L (ˆ0 , A ′ i · ˆ0), and such intersections are the only ones that occur, since all other strands are constant anddisjoint. The desired equality follows. 8 roposition 3.20. For A ∈ π (Σ) k , and [ ˆ X ] , [ ˆ Y ] ∈ e B k (Σ) /S k , the function L A ([ ˆ X ]; [ ˆ Y ]) := L A ( ˆ X ; ˆ Y ) ,is well-defined.Proof. The previous proposition implies that for any σ, τ ∈ S k , we have L A ( σ · ˆ X ; τ · ˆ Y ) = L A (ˆ0; ˆ0) − L (ˆ0; σ · ˆ X ) + L (ˆ0; τ · ˆ Y ) , so it suffices to show that the expression L (ˆ0; σ · ˆ X ) is independent of σ ∈ S k . To see this, notethat item 2 of the previous proposition, together with the fact that 0 ∈ F ix S k ( π (Σ) k ) implies that L (ˆ0; σ · ˆ X ) = L ( σ − · ˆ0; ˆ X ). Moreover, it is easy to see that ˆ0 = (ˆ p , . . . , ˆ p k ) is 0-homotopic to σ · ˆ0 forany σ ∈ S k , and consequently, L ( σ − · ˆ0; ˆ X ) = L (ˆ0; ˆ X ), which is independent of σ ∈ S k . Proposition 3.21. Let ˆ X = (ˆ x , ˆ x ) ∈ e B (Σ) with x lying in some exponential of neighbourhood of x in L (Σ) , then L (ˆ0; ˆ X ) = ℓ (ˆ x , ˆ x ) .Proof. As L (ˆ0; ˆ X ) depends on ˆ X only up to a 0-homotopy, we may assume without loss of generalitythat x is a constant loop. Moreover, noting that if A, B ∈ π (Σ), then L (ˆ0; ( A, B ) · ˆ X ) − L (ˆ0; ˆ X ) = 12 ( c ( A ) + c ( B )) = ℓ ( A · ˆ x , B · ˆ x ) − ℓ (ˆ x , ˆ x ) , and so it suffices to prove the statement in the case in which ˆ x is a trivially capped constant loop andˆ x lies inside an exponential neighbourhood of ˜ x in e L (Σ).As discussed in Section 3.1, T ˆ x e L (Σ) is naturally identified (up to a homotopy of trivializations)with Γ ∞ ( S × R ) and so (in the notation of that section) we may write ˜ x in local coordinates as v ( t ) := ( T − x ◦ g Exp − )(˜ x ), and we have that ℓ (ˆ x , ˆ x ) = wind ( v ) by definition. By the capped braidhomotopy-invariance of the homological linking number, and the homotopy invariance of the windingnumber in R \ { } , we may assume that v ( t ) = r e πilt ∈ C , t ∈ [0 , r > l = ℓ (ˆ x ; ˆ x ). The claim then immediately follows from computing the transverse intersections of thestrands of the braid homotopy given by h ( s, t ) = (0 , (1 − s ) r + s v ( t )2 ).The previous proposition justifies the following extension of the winding of capped loops with closestrands in Section 3.1. Definition 3.22. For ˆ x, ˆ y ∈ e L (Σ) such that ˆ X = (ˆ x, ˆ y ) ∈ e B (Σ), we define the linking of ˆ x and ˆ y as ℓ (ˆ x, ˆ y ) := L (ˆ0; ˆ X ). As a matter of convention, we also define ℓ (ˆ x, A · , ˆ x ) := c ( A )2 for A ∈ π (Σ), andfor A = 0 ∈ π (Σ). We will additionally declare, simply as a matter of convention, that ˆ x and A · ˆ x arelinked.The following proposition hints at the role that linking plays in understanding the relation of theaction functional to dynamics and will prove crucial in Section 8. Proposition 3.23. Suppose that ˆ γ = [ γ, w ] is a capped loop such that γ is smooth, then Z D w ∗ ω = Z Σ \ im γ ℓ (ˆ γ, ˆ x ) ω, where for x ∈ Σ , ˆ x = [ x, x ] denotes the trivially capped constant loop based at x .Proof. Suppose without loss of generality that w : D → Σ is a smooth map with w (0) = γ (0). Themain point is to notice that for any x ∈ Σ \ im γ such that w is transversal to x , we have that ℓ (ˆ γ, ˆ x ) = deg ( w ) x , where deg ( w ) x denotes the local degree of w at x . Indeed, transversality of w to x implies the transversality of the maps into [0 , × S × Σ defined by ˜ w ( s, t ) = ( s, t, w ( se πit )) and˜ x ( s, t ) = ( s, t, x ), for ( s, t ) ∈ [0 , × S , and obviously the algebraic count of the intersection numberbetween these two graphs is identical with deg ( w ) x . One immediate consequence is that ℓ (ˆ γ, ˆ x ) = 0 forall x im w , so it suffices to show that Z D w ∗ ω = Z im w \ im γ ℓ (ˆ γ, ˆ x ) ω. 9o this end, denote by S ( w ) ⊆ D the set of points such that Dw is not of full rank and define G ( w ) ⊆ D to be G ( w ) = w − (im γ ) \ S ( w ). Note that G ( w ) is a set of measure 0, since we may realize G ( w ) as theprojection onto D \ S ( w ) of ( w | D \ S ( w ) × γ ) − ( △ ) which is a submanifold of D \ S ( w ) of codimension2. Next note that we must have R S ( w ) w ∗ ω = 0, since w ∗ ω vanishes on S ( w ). Consequently, writing N := S ( w ) ∪ G ( w ), we note that im γ ⊆ w ( N ) and so it suffices to establish that Z D \ N w ∗ ω = Z w ( D \ N ) ℓ (ˆ γ, ˆ x ) ω = Z w ( D \ N ) deg ( w ) x ω. The local degree is a locally constant function of x , and so we obtain Z w ( D \ N ) deg ( w ) x ω = X C ∈ π ( w ( D \ N )) deg ( w ) | C Z C ω = X C ∈ π ( w ( D \ N )) Z w − ( C ) w ∗ ω = Z D \ N w ∗ ω as claimed. In this section, we give a rapid overview of the elements of Floer theory of which we will have need, mainlyto fix notation and conventions. For a more detailed treatment, see [1], [25] for standard accounts ofHamiltonian Floer theory (see also [7] for its adaptation to the weakly monotone case) and [27], [24] or [15]for a more detailed treatment of how Floer theory fits into a field theory over surfaces. Throughout, weassume that ( M, ω ) is a weakly monotone compact symplectic manifold of dimension 2 n , while J ( M, ω )denotes the space of all smooth ω -compatible almost complex structures. For convenience, we work onlywith Z -coefficients, but this restriction is inessential. A smooth Hamiltonian function H : S × M → R induces a time-dependent vector field ( X tH ) t ∈ [0 , on M defined by the relation ω ( X tH , − ) = − dH t . The Hamiltonian isotopy obtained as the flow by thisvector field is denoted φ H := ( φ tH ) t ∈ [0 , .The Hamiltonian H defines a corresponding action functional on the space of capped loops e L ( M ), A H ([ γ, v ]) := R H t ( γ ( t )) dt − R D v ∗ ω . We write g P er ( H ) := Crit A H , and P er ( H ) := π ( g P er ( H )) ⊆L ( M ) which consists precisely of the contractible 1-periodic orbits of φ H . H is said to be non-degenerate if for all x ∈ P er ( H ), ( Dφ H ) x (0) has no eigenvalues equal to1. When H is non-degenerate, there exists a well-defined Conley-Zehnder index µ ([ x, v ]) ∈ Z , for[ x, v ] ∈ g P er ( H ). We shall normalize the Conley-Zehnder index by insisting that if H is a C -smallMorse function and x a critical point of H , then µ (ˆ x ) = µ Morse ( x ) − n , where µ Morse is the Morse indexof x , and ˆ x denotes the trivial capping of the constant orbit x . For k ∈ Z , and any ˆ P ⊆ g P er ( H ) wedefine ˆ P ( k ) to be the collection of capped orbits in ˆ P with Conley-Zehnder index k .Given ˆ x ± = [ x ± , w ± ] ∈ g P er ( H ), we write C ∞ ˆ x − , ˆ x + ( R × S ; M ) for the subspace of C ∞ ( R × S ; M )consisting of cylinders which induce a 0-homotopy from ˆ x − to ˆ x + . Letting E → C ∞ ( R × S ; M ) ˆ x − , ˆ x + be the infinite dimensional vector bundle with fiber E u = Γ ∞ ( u ∗ T M ) at u , any smooth S -family J =( J t ) t ∈ S ⊆ J ( M, ω ), permits the definition of the Floer operator F H,J ( u ) := ∂ s u + J ( ∂ t u − X H ) ∈ E u ,for u ∈ C ∞ ˆ x − , ˆ x + ( R × S ; M ). After passing to appropriate Banach space completions, F H,J defines aFredholm operator with index µ (ˆ x − ) − µ (ˆ x + ). The intersection of F H,J with the 0-section gives rise to Floer’s equation ∂ s u + J t ( ∂ t u − X tH ) = 0 (1)for smooth maps u : R × S → M . If we define the energy of u ∈ C ∞ ( R × S ; M ) by E ( u ) := R R × S k ∂ s u k J t dtds , then the finite energy solutions of Floer’s equation may be thought of as the pro-jections to M of negative gradient flow lines of A H with respect to the L -metric on e L ( M ) induced10y J . It follows easily from this that if u ∈ C ∞ ˆ x − , ˆ x + ( R × S ; M ) is such a finite energy solution, then E ( u ) = A H (ˆ x − ) − A H (ˆ x + ).For any ˆ x ± ∈ g P er ( H ), we define define f M (ˆ x − , ˆ x + ; H, J ) to be the zero set of F H,J on C ∞ ˆ x − , ˆ x + ( R × S ; M ). It carries an obvious R -action given by translation in the s -coordinate. The reduced modulispace is defined by M (ˆ x, ˆ y ; H, J ) := f M (ˆ x, ˆ y ; H, J ) / R . Definition 4.1. A pair ( H, J ) with H and J as above, H non-degenerate, are said to be Floer non-degenerate if the intersection of the Floer operator with the 0-section is transverse (after taking ap-propriate Banach space completions) for all ˆ x, ˆ y ∈ g P er ( H ) with µ (ˆ x ) − µ (ˆ y ) ≤ H non-degenerate, let J nd ( H ) ⊆ C ∞ ( S ; J ( M, ω )) denote the space of S -families of complexstructures such that ( H, J ) is Floer non-degenerate. J nd ( H ) is residual in C ∞ ( S ; J ( M, ω )).If ( H, J ) is Floer non-degenerate, then M (ˆ x, ˆ y ; H, J ) is a compact manifold of dimension 0 whenever µ (ˆ x ) − µ (ˆ y ) = 1, and in this case we may define the Floer chain complex CF ∗ ( H, J ) := Z h ˆ x i ˆ x ∈ g P er ( H ) ,which is graded by µ and which has differential defined on generators by ∂ H,J ˆ x := P µ (ˆ x ) − µ (ˆ y )=1 n (ˆ x, ˆ y )ˆ y ,with n (ˆ x, ˆ y ) being the mod 2 count of elements in M (ˆ x, ˆ y ; H, J ). The homology of this complex F H ∗ ( H )is the Floer homology of H and is independent of the choice of J .The Floer complex has the structure of a filtered complex , with the filtration coming from the actionfunctional. Explicitly, for σ = P ˆ x ∈ g P er ( H ) a ˆ x ˆ x ∈ CF ∗ ( H, J ), we define supp σ := { ˆ x ∈ g P er ( H ) : a ˆ x =0 } , and we define the level of σ to be λ H ( σ ) := sup ˆ x ∈ supp σ A H (ˆ x ) . For a Floer homology class α ∈ HF ∗ ( H ) the spectral invariant associated to α is defined by c ( H ; α ) := inf { λ H ( σ ) : [ σ ] = α } . A cycle σ such that [ σ ] = α and λ H ( σ ) = c ( H ; α ) is called tight (for α ). It is a non-trivial fact thatsuch cycles always exist (see [32] or [22]).Let H S ( M ) denote the image in H ( M ; Z ) of the Hurewicz morphism, and let Γ ω := H S ( M ) / ker c ∩ ker[ ω ]. We define the Novikov ring Λ ω := { X A ∈ Γ ω λ A e A : λ A ∈ Z , { λ A = 0 , ω ( A ) ≤ c } < ∞ , for all c ∈ R } . This is a graded commutative ring with grading given by declaring deg ( A ) := 2 c ( A ). CF ∗ ( H, J ) is a Λ ω -module where the action of e A ∈ Λ ω is defined on generators ˆ x = [ x, v ] of CF ∗ ( H, J ) by e A · ˆ x := [ x, A v ],and extended linearly. Note that we have the relations µ ( e A · ˆ x ) = µ (ˆ x ) − c ( A ) , A H ( e A · ˆ x ) = A H (ˆ x ) − ω ( A ) . It is a standard fact in Floer theory that if f ∈ C ∞ ( M ) is a sufficiently C -small Morse function and J ∈ J ( M, ω ) is such that ( f, g J ) is Morse-Smale, then CF ∗ ( f, J ) = C Morse ∗ + n ( f, g J ) ⊗ Λ ω . Taking homologythen gives a natural identification with the quantum homology of ( M, ω ): HF n ( f ) = H ∗ + n ( M ; Λ ω ) = QH ∗ + n ( M, ω ). For X a smooth manifold, a function F ∈ C ∞ ( R × X ) is said to be T -adapted for T ∈ (0 , ∞ )if ( ∂ s F ) s ≡ | s | ≥ T . F is said to be adapted if it is T -adapted for some T . For X = C ∞ ( S × M ), and H ± ∈ C ∞ ( S × M ), we denote by H ( H − , H + ) the space of adapted ho-motopies H having lim s →±∞ H ( s ) ≡ H ± . We make a similar definition for J ( J − , J + ) in the case where11 = C ∞ ( S ; J ( M, ω )).A pair ( H , J ) is an adapted homotopy of Floer data from ( H − , J − ) to ( H + , J + ) if H ∈ H ( H − , H + ) and J ∈ J ( J − , J + ). We will write HJ ( H − , J − ; H + , J + ) for the collection of all suchadapted homotopies, often omitting the dependence on ( H ± , J ± ) if it is clear from context. Just as in the s -independent case, for any adapted homotopy of Floer data ( H , J ), we obtain a corresponding Floer op-erator F H ,J . For any pair ˆ x ± ∈ g P er ( H ± ), consideration of the zeros of F H ,J along C ∞ ( R × S ; M ) ˆ x − , ˆ x + gives rise to the s -dependent Floer equation ∂ s u + J st ( ∂ t u − X H st ) = 0 , (2)and everything proceeds as before, with the proviso that now, if u ∈ C ∞ ˆ x − , ˆ x + ( R × S ; M )) solves Equation2, then its energy is given by E ( u ) = A H − (ˆ x − ) − A + H (ˆ x + ) + R ∞−∞ R ( ∂ s H )( s, t, u ( s, t )) dtds . The modulispace M (ˆ x − , ˆ x + ; H , J ) is defined to be the zero set of F H ,J on C ∞ ˆ x − , ˆ x + ( R × S ; M ). Remark. When ( H − , J − ) = ( H + , J + ), then the s -independent homotopy ( H , J ) = ( H − , J − ) = ( H + , J + )is a special case of an adapted homotopy. In this case, M (ˆ x, ˆ y ; H , J ) = f M (ˆ x, ˆ y ; H ± , J ± ). In the sequel,when we speak of adapted homotopies of Floer data, this case is included. Definition 4.2. Given ( H ± , J ± ) Floer non-degenerate, ˆ x ± ∈ g P er ( H ± ), and ( H , J ) ∈ HJ , we will saythat ( H , J ) is (ˆ x − , ˆ x + ) -regular if F H ,J is transverse to the zero section along C ∞ ˆ x − , ˆ x + ( R × S ; M ). We de-note the collection of all such adapted homotopies by HJ reg ˆ x − , ˆ x + . ( H , J ) will be said to be Floer-regular if it is (ˆ x − , ˆ x + )-regular whenever µ (ˆ x − ) − µ (ˆ x + ) ≤ 1. We denote the space of Floer-regular adaptedhomotopies from ( H − , J − ) to ( H + , J + ) by HJ reg ( H − , J − ; H + , J + ), suppressing the dependence on( H ± , J ± ) when no confusion will arise.For any fixed J ∈ J ( J − , J + ), the set H reg ( J ; H − , H + ) ⊆ H ( H − , H + ) of adapted homotopies H such that ( H , J ) is Floer regular is residual.For ( H , J ) ∈ HJ reg , the spaces M (ˆ x − , ˆ x + ; H , J ) are all compact manifolds of dimension 0 whenever µ H − (ˆ x − ) = µ H + (ˆ x + ), and so we may define the continuation morphism h H ,J : CF ∗ ( H − , J − ) → CF ∗ ( H + , J + )on generators by setting h H ,J ( x − ) := P µ (ˆ x − ) − µ (ˆ x + )=0 n (ˆ x − , ˆ x + )ˆ x + , where n (ˆ x − , ˆ x + ) is the mod 2 countof elements in the moduli space M (ˆ x − , ˆ x + ; H , J ). The continuation morphism is a morphism of com-plexes, which descends to an isomorphism at the level of homology. Moreover any two continuationmaps between ( H − , J − ) and ( H + , J + ) define the same map at the level of homology, and further theseisomorphism satisfy the obvious composition law h ◦ h = h , where h ji : HF ( H i ) → HF ( H j ).Consequently, for any quantum homology class α ∈ QH ∗ + n ( M ) ≃ H Morse ∗ + n ( f ; Λ ω ) ≃ HF ∗ ( f ), with f some C -small Morse function, letting h : HF ∗ ( f ) → HF ∗ ( H ) be such a continuation morphism, h ( α ) ∈ HF ∗ ( H ) is a well-defined homology class, independent of the Morse function f and the continu-ation morphism h . Definition 4.3. For H Floer non-degenerate and α ∈ QH ∗ + n ( M ) the spectral invariant of H associ-ated to α is defined by c ( H ; α ) := c ( H ; h ( α )).We conclude this by recalling the so-called ‘Gromov trick’, which forms the basis of much of thispaper by establishing that we may use pseudo-holomorphic techniques to analyze the graphs of Floer-type cylinders. Theorem 4.4 (1.4.C’. in [6]) . Let ( H , J ) be an adapted homotopy of Floer data, then there exists aunique almost complex structure ˜ J on R × S × M with the property that a section ˜ u : R × S → R × S × M ( s, t ) ( s, t, u ( s, t )) is ( j , ˜ J ) -holomorphic if and only if u satisfies Equation 2, where j denotes the standard complex struc-ture on the cylinder. .2 Asymptotic analysis for pseudoholomorphic cylinders The main analytic fact that gives us control over the asymptotic winding behaviour of Floer cylinders,as well as that of vector fields lying in the the kernel of the Floer differential, is the following theoremwhich describes the asymptotic behaviour of solutions to an appropriately perturbed Cauchy-Riemannequation. This result is originally due to [20], although the version we reproduce here for the convenienceof the reader is from the appendix of [29] Theorem 4.5. Let w : [0 , ∞ ) × S → R n satisfy the equation ∂ s w + J ∂ t w + ( S ( t ) − ∆( s, t )) w = 0 , (3) where S : S → End ( R n ) is a smooth family of symmetric matrices and ∆ : [0 , ∞ ) × S → End ( R n ) is smooth. Suppose that for β ∈ N , there exist constants M β , d > such that | ( ∂ β ∆)( s, t ) | ≤ M β e − ds ,and | ( ∂ β w )( s, t ) | ≤ M β e − ds . Then either w ≡ or w ( s, t ) = e λs ( ξ ( t ) + r ( s, t )) , where λ is a negativeeigenvalue of the self-adjoint operator A : H ( S ; R n ) ⊆ L ( S ; R n ) → L ( S ; R n ) h 7→ − J ( ∂ t − J S ) h,ξ : S → R n is an eigenvector of A with eigenvalue λ , and r satisfies the decay estimates | ( ∂ β r )( s, t ) | ≤ e − d ′ s M ′ β for d ′ , M ′ β > , for all β ∈ N . This theorem is useful in the following setting. Let ( H, J ) be Floer non-degenerate. To any x ∈ P er ( H ), we may assign the asymptotic operator A x,J : Γ( x ∗ T M ) → Γ( x ∗ T M )as follows. Viewing ξ ∈ Γ( x ∗ T M ) as a section of the vertical tangent bundle V| ˇ x ≤ T ( S × M ) | ˇ x alongthe graph ˇ x of x , we let ˇ X H := ∂ t ⊕ X H ∈ X ( S × M ), and we view J = ( J t ) t ∈ S as an endomorphism ofthe vertical tangent bundle by setting ˇ J t,x := J t ( x ). A x,J is then defined by setting A x,J ( ξ ) := − ˇ J L ˇ X H ξ ,where L X Y denotes the Lie derivative of Y along X . A x,J extends to an unbounded self-adjoint operatorwith discrete spectrum (still denoted A x,J ) from W , ( x ∗ T M ) to L ( x ∗ T M ).By taking an exponential chart as in Section 3.1 on a neighbourhood ˜ O of ˆ x ∈ g P er ( H ), Floer’sequation may be written in the local coordinates provided by this chart in the form of Equation 3, with A x,J being sent via these coordinates to A . Following [29], we define Definition 4.6. Let x ∈ L ( M ) and suppose that lim s →∞ u s ≡ x for a map u : R × S → M . For any R > 0, a map U + : [ R, ∞ ) → Γ( x ∗ T M ) will be said to be a positive asymptotic representative of u if u ( s, t ) = Exp ( U + ( s ))( t ) for all ( s, t ) ∈ [ R, ∞ ) × S , where Exp is as in Section 3.1. The notion ofa negative asymptotic representative of u , U − : ( −∞ , − R ] → Γ( x ∗ T M ) is defined in the obviousanalogous manner.Every Floer-type cylinder considered in this paper admits, due to exponential convergence at theends, essentially unique positive and negative asymptotic representatives, determined up to a restrictionof the domains of U ± to larger values of | R | .The main result that we will need from [29] (paraphrased for our setting) is the following Theorem 4.7. Let ( H, J ) be Floer non-degenerate, x ∈ P er ( H ) and let u, v solve Equation 2 for s >> (resp. for s << ), where the adapted homotopy used in defining Equation 2 satisfies ( H + , J + ) = ( H, J ) (resp. ( H − , J − ) = ( H, J ) ). Suppose moreover that u s and v s both converge to x as s → ∞ (resp. s → −∞ ). Let U and V be positive (resp. negative) asymptotic representatives of u and v respectively.Then either U ≡ V or there exists a strictly negative (resp. strictly positive) eigenvalue λ ∈ σ ( A x,J ) andan eigenvector ξ with eigenvalue λ such that ( U − V )( s, t ) = e λs ( ξ ( t ) + r ( s, t )) , where the remainder term satisfies the decay estimates |∇ is ∇ it r ( s, t ) | ≤ M ij e − ds for all ( i, j ) ∈ N and M i,j , d > (resp. d < ). u, v and x are as above, we will write ξ + u,v (resp. ξ − u,v ) for the eigenvectors of A x,J whoseexistence is guaranteed by the above theorem. We will call ξ ± u,v the positive (resp. negative) asymp-totic eigenvector of v relative u . Note that the above result only requires that u and v solve Equation2 on some neighbourhood of s = ∞ (resp. s = −∞ ), and that, for ( H, J ) ∈ HJ , the trivial cylinder v ( s, t ) = x ( t ) is always a solution to Equation 2 outside some compact set. We will write ξ ± u := ξ ± u,x andcall these the (positive and negative) asymptotic eigenvectors of u .This asymptotic information becomes especially useful when combined with the following fact (see [8] p.285 or [29] p.1637). Proposition 4.8. If ξ ∈ Γ( x ∗ T M ) is an eigenvector of A x,J , then ξ ( t ) = 0 for all t ∈ S . Corollary 4.9. Let u, v : R × S → M be distinct finite energy solutions of Equation 2, then there is acompact subset K ⊆ R × S such that u ( s, t ) = v ( s, t ) only if ( s, t ) ∈ K . These results become even stronger in the case when dim M = 2, as in this case Proposition 4.8implies that eigenvectors of the asymptotic operator have a well-defined winding number, once we fix atrivialization of x ∗ T M via a choice of capping disk. More precisely, when M = Σ, if ˆ x ∈ e L (Σ), and T ˆ x : S × ( R , ω ) → ( x ∗ T Σ , ω ) is a symplectic trivialization as in Section 3, then for any ξ ∈ A x,J ,the map t T ˆ x ( t ) − ξ ( t ) has a well-defined winding number wind ( ξ ; ˆ x ), by Proposition 4.8. Proposition3.21 then implies Corollary 4.10. Let u, v be distinct finite energy solutions of Equation 2 with lim s →−∞ u s = lim s →−∞ v s = x . Then there exists R > such that for all s < − R and any capping ˆ x = [ x, α ] , we have ℓ (ˆ v αs , ˆ u αs ) = wind ( ξ − u,v ; ˆ x ) , where ˆ u αs (resp. ˆ v αs ) denotes the capping of u s (resp. v s ) such that [ x, α ] and ˆ u s (resp. ˆ v s )are -homotopic. The analogous statement when lim s →∞ u s = lim s →∞ v s = x also holds. If we combine the positivity of intersection of holomorphic curves in dimension 4 with the foregoingdiscussion, we arrive at the principal point of this section Lemma 4.11. Let ( H ± , J ± ) be Floer non-degenerate, ( H , J ) ∈ HJ and let u, v ∈ C ∞ ( R × S ; Σ) bedistinct finite energy solutions to Equation 2 for ( H , J ) . Then for any lifts ˆ u, ˆ v of u, v : R → L (Σ) , thefunction ℓ ˆ u, ˆ v ( s ) := ℓ (ˆ u s , ˆ v s ) is non-decreasing, locally constant, and well-defined for all but finitely manyvalues s ∈ R . Moreover, for s, s ′ ∈ dom ( ℓ ˆ u, ˆ v ) , with s < s ′ , ℓ ˆ u, ˆ v ( s ) = ℓ ˆ u, ˆ v ( s ′ ) if and only if there exists s ∈ ( s, s ′ ) and some t ∈ S such that u ( s , t ) = v ( s , t ) .Proof. That ℓ ˆ u, ˆ v has only finitely many points at which it is ill-defined follows the fact that, by definition, ℓ (ˆ u s , ˆ v s ) is undefined only when there exists t ∈ S such that u ( s , t ) = v ( s , t ). By Corollary 4.9,the set of all such ( s , t ) ∈ R × S must lie inside some compact set, and we may then apply Theorem 4.4to choose an almost complex structure on R × S × Σ such that the graphs ˜ u and ˜ v are pseudoholomorphic,whence all such intersections must be isolated, and so finite in number. That ℓ ˆ u, ˆ v is non-decreasing followsby applying the positivity of intersections for holomorphic curves in dimension 4 at these intersections,which implies moreover that any such intersection contributes strictly positively to the change in ℓ ˆ u, ˆ v ( s )as s increases and passes from one connected component of dom ( ℓ ˆ u, ˆ v ) to another. Definition 4.12. For ( H ± , J ± ) Floer non-degenerate, ( H , J ) ∈ HJ and u, v ∈ M (ˆ x, ˆ y ; H , J ), u = v ,we define ℓ ±∞ ( u, v ) := lim s →±∞ ℓ (ˆ u s , ˆ v s ), where ˆ u s and ˆ v s are the natural cappings of u s and v s (cf.Definition 3.15).Note that the previous lemma implies that these quantities exist and are finite. Indeed, if ˆ u s and ˆ v s tend to ˆ x as s → ±∞ , then ℓ ±∞ ( u, v ) = wind ( ξ ± u,v ; ˆ x ) by Corollary 4.10, while if lim s →±∞ u s = ˆ x andlim s →±∞ v s = ˆ y with x = y , then ℓ ±∞ ( u, v ) = ℓ (ˆ x, ˆ y ). A x,J We summarize here some necessary facts from [8] on the winding numbers of eigenvectors of A x,J whichappeared in the previous subsection (while [8] works in the aspherical case, our previous discussion makesclear how this winding number depends on the choice of cappings of x and y and this is all that is neededto extend the results to capped orbits). For a loop x ∈ L (Σ), let π (Σ; x ) denote the set of homotopyclasses of capping disks for x . 14 roposition 4.13. Let x ∈ P er ( H ) with H non-degenerate and J : S → J (Σ , ω ) arbitrary. There isa well-defined function W = W x,J : π (Σ; x ) × σ ( A x,J ) → Z ( α, λ ) wind ( T − x,α ] ◦ ξ ) , where ξ ∈ Γ( x ∗ T Σ) is any eigenvector with eigenvalue λ . Moreover, W satisfies the following properties1. For any α ∈ π (Σ; x ) , λ < λ ′ ⇒ W ( α, λ ) ≤ W ( α, λ ′ ) .2. For any α ∈ π (Σ; x ) , and any k ∈ Z , P λ ∈ W − α ( k ) dim E λ = 2 ,where W α ( λ ) = W ( α, λ ) , and E λ is the eigenspace associated to the eigenvalue λ .3. For any A ∈ π (Σ) , W ( A · α, λ ) = W ( α, λ ) + c ( A ) . In view of the control over the sign of the eigenvalue provided by Theorem 4 . 7, combined with themonotonicity of the winding number provided by item (1) of the above proposition, we make the following Definition 4.14. For ( H, J ) and ˆ x = [ x, α ] ∈ P er ( H ) as above, define a (ˆ x ) = a (ˆ x ; H ) := sup λ ∈ σ ( A x,J ) ∩ ( −∞ , W ( α, λ ) b (ˆ x ) = b (ˆ x ; H ) := inf λ ∈ σ ( A x,J ) ∩ (0 , ∞ ) W ( α, λ ) . Remark that we have, by the monotonicity of W and by Theorem 4 . 7, that Corollary 4.15. Let ( H ± , J ± ) be Floer non-degenerate, ( H , J ) ∈ HJ , x ± i ∈ g P er ( H ± ) for i = 0 , .If u i ∈ M (ˆ x − , ˆ x + i ; H , J ) , i = 0 , , then b (ˆ x − ) ≤ ℓ −∞ ( u , u ) . If v i ∈ M (ˆ x − i , ˆ x +0 ; H , J ) , i = 0 , , then ℓ ∞ ( v , v ) ≤ a (ˆ x +0 ) . The result which relates this discussion to the behaviour of the Floer complex is the following Theorem 4.16 ([8] Theorem 3.10) . − µ (ˆ x ) = a (ˆ x ) + b (ˆ x ) Remark. Strictly speaking, it is shown in [8] that − µ ( x ) = 2 a ( x ) + p ( x ), where p ( x ) denotes the parity of the index, but it is straight-forward to show, using Proposition 4.13, that b ( x ) = a ( x ) + p ( x ). Notethat our sign convention for the Conley-Zehnder index is the negative of that used in [8].Recall from Section 4.1 that to any ( H , J ) ∈ HJ ( H − , J − ; H + , J + ), we associate the operator F H ,J : C ∞ ( R × S ; Σ) → E . Whenever F H ,J ( u ) = 0, for u ∈ C ∞ ( R × S ; Σ) ˆ x − , ˆ x + , T u E splits canonicallyas T u C ∞ ( R × S ; Σ) ˆ x − , ˆ x + ⊕ E u . In such a case, we denote by D F H ,J the projection of the differential of F H ,J onto E u , and we call D F H ,J the linearized Floer operator . The transversality of F H ,J to the0-section of E at u is equivalent to the surjectivity of ( D F H ,J ) u , which is in turn related to the behaviourof its kernel by the Fredholm property. The following result is essentially proved in [8] as Proposition5 . Proposition 4.17. Let ( H ± , J ± ) be Floer non-degenerate, let ( H , J ) ∈ HJ , u ∈ f M (ˆ x − , ˆ x + ; H , J ) , andlet ξ ∈ ker( D F H ,J ) u . Suppose that ξ and denote by Z ( ξ ) the algebraic count of the number of zerosof ξ , then Z ( ξ ) is finite and satisfies the inequality ≤ Z ( ξ ) ≤ a (ˆ x + ) − b (ˆ x − ) .Proof. It is a standard result in Floer theory (see for instance [25], Section 2 . 2) that for any u ∈ f M (ˆ x − , ˆ x + ; H , J ), any element ξ ∈ ker( D F H ,J ) u may be expressed (with respect to the unitary trivial-ization Φ : R × S × ( R , J ) → u ∗ ( T Σ , J ) along u induced by the cappings of ˆ x − and ˆ x + ) as solving anequation of the form ∂ s ξ + J ∂ t ξ + Sξ = 0 , S on the positive and negative ends as S ± ( s, t ) = Φ − A x ± ,J − ∆ ± ( s, t ), with ∆ ± satisfying the decay estimates of Theorem 4.5. Consequently, any ξ ∈ ker( D F ) u must be non-vanishingoutside of some compact neighbourhood of R × S , and the Carlemann similarity principle, combinedwith positivity of intersections of holomorphic curves in dimension 4, implies that Z ( ξ ) is finite andnon-negative.To see that Z ( ξ ) ≤ a (ˆ x + ) − b (ˆ x − ), we take R > ξ is non-vanishing outsideof ( − R, R ) × S and consider the homotopy of 2-braids in R induced by h ( s ) = (0 , Φ − ξ s ) ∈ L ( R ) , s ∈ [ − R, R ]. Theorem 4.5 implies that for R > ℓ (0 , ξ − R ) = wind (Φ − ξ − R ) ≥ b (ˆ x − ) , andℓ (0 , ξ R ) = wind (Φ − ξ R ) ≤ a (ˆ x + ) , (since R is aspherical, we omit any mention of cappings), and the algebraic count zeros of ξ correspondto the algebraic count of the intersections of the graphs of the strands of h from which the propositionfollows. We proceed to an investigation of the relationship between the topology of the capped braid (ˆ x, ˆ y )for ˆ x, ˆ y ∈ g P er ( H ), and collective behaviour in S × Σ of maps ( s, t ) ( t, u ( s, t )), as u varies in f M (ˆ x, ˆ y ; H, J ). Lemma 4.18. Let ( H , J ) be an adapted homotopy of Floer data with ( H ± , J ± ) Floer non-degenerate.Suppose that ˆ x ± ∈ g P er ( H − ) ∩ g P er ( H + ) , with x − = x + , and that M (ˆ x − , ˆ x + ; H , J ) , M (ˆ x − , ˆ x − ; H , J ) , and M (ˆ x + , ˆ x + ; H , J ) are all non-empty, then b (ˆ x − ; H − ) ≤ ℓ (ˆ x − , ˆ x + ) ≤ a (ˆ x + ; H + ) .Proof. That b (ˆ x − ; H − ) ≤ ℓ (ˆ x − , ˆ x + ) follows from applying Corollary 4.15 with u ∈ M (ˆ x − , ˆ x + ; H , J )and u ∈ M (ˆ x − , ˆ x − ; H , J ). The second inequality uses v ∈ M (ˆ x + , ˆ x + ; H , J ).Applying the preceding lemma in the case where ( H , J ) is s -independent yields Corollary 4.19. Let ( H, J ) be Floer non-degenerate and suppose that f M (ˆ x, ˆ y ; H, J ) = ∅ , x = y , then b (ˆ x ) ≤ ℓ (ˆ x, ˆ y ) ≤ a (ˆ y ) . Applying Lemma 4.11 to the constant cylinder u ∈ f M (ˆ x ′ , ˆ x ′ ; H, J ) and v ∈ f M (ˆ x, ˆ y ; H, J ) shows Proposition 4.20. Let ( H, J ) be Floer non-degenerate and let ˆ x, ˆ y ∈ g P er ( H ) with ˆ y ∈ supp ∂ H,J ˆ x ,then for all ˆ x ′ ∈ g P er ( H ) , x ′ 6∈ { x, y } , we have ℓ (ˆ x, ˆ x ′ ) ≤ ℓ (ˆ y, ˆ x ′ ) . A straight-forward computation using Theorem 4.16 gives Lemma 4.21. Suppose that µ HCZ (ˆ x ) ∈ { k − , k } for some k ∈ Z , then a (ˆ x ; H ) = − k . If µ HCZ (ˆ x ) ∈{ k, k + 1 } for some k ∈ Z , then b (ˆ x ; H ) = − k And the preceding lemma combines with Corollary 4.19 to give Corollary 4.22. If µ (ˆ x ) , µ (ˆ y ) ∈ { k − , k, k + 1 } for some k ∈ Z and f M (ˆ x, ˆ y ; H, J ) = ∅ , then ℓ (ˆ x, ˆ y ) = − k . The main geometric input for this section is the following (cf. Theorem 5 . Proposition 4.23. Let ( H, J ) be Floer non-degenerate, ˆ x, ˆ y ∈ g P er ( H ) , k − ≤ µ (ˆ x ) , µ (ˆ y ) ≤ k + 1 ,for some k ∈ Z . Then the map f Ev : R × S × f M (ˆ x, ˆ y ; H, J ) → R × S × Σ defined by f Ev ( s, t, u ) :=˜ u ( s, t ) = ( s, t, u ( s, t )) is a diffeomorphism onto its image. roof. We may suppose that f M (ˆ x, ˆ y ; H, J ) = ∅ , or else the lemma is vacuously true. Moreover, if ˆ x = ˆ y ,then the lemma is obvious. Thus, we may suppose that µ (ˆ x ) − µ (ˆ y ) ∈ { , } . We will show that f Ev isa proper injective immersion. That f Ev is one-to-one follows from the fact that by Lemma 4.18, we havethat for any u, v ∈ f M (ˆ x, ˆ y ; H, J ), u = v , and all s ∈ R , we have0 = b (ˆ x ) ≤ ℓ (ˆ u s , ˆ v s ) ≤ a (ˆ y ) = 0 , and so by Lemma 4.11, the graphs ˜ u and ˜ v cannot intersect for any u = v . That f Ev is proper essentiallyfollows from compactness results in Floer theory; if { ( s n , t n , u n ) } n ∈ N ⊆ R × S × f M (ˆ x, ˆ y ; H, J ) is somesequence which eventually leaves any compact set, then either s n → ±∞ and ( s n , t n , u n ) converges toa point on either the graph ˜ x ( s, t ) = ( s, t, x ( t )) or on the graph ˜ y ( s, t ) = ( s, t, y ( t )), or ( s n ) n ∈ N remainsbounded, in which case ( s n , t n , u n ) must converge to a point on the graph of some broken Floer cylinderbetween x and y . In either case, the sequence ( s n , t n , u n ) eventually leaves every compact subset ofim f Ev .It remains to show that f Ev is an immersion when µ (ˆ x ) − µ (ˆ y ) ∈ { , } . We note that T ( R × S × f M (ˆ x, ˆ y ; H, J )) = T ( R × S ) ⊕ (ker( d F )) | f M (ˆ x, ˆ y ; H,J ) , and since d f Ev is obviously non-vanishing on T ( R × S ), the problem reduces to showing that ξ ∈ ker( d F ) u is a nowhere-vanishing vector field along u , which follows by combining Proposition 4.17 with Corollary4.22 to deduce that Z ( ξ ) = 0 whenever ξ is not identically zero.Note that as a consequence of the previous lemma, whenever µ (ˆ x ) , µ (ˆ y ) ∈ { k − , k, k + 1 } for k ∈ Z , then f Ev ( R × S × f M (ˆ x, ˆ y ; H, J )) carries a smooth 2-dimensional foliation e F ˆ x, ˆ y , the leaves ofwhich are nothing but the graphs ˜ u of u ∈ f M (ˆ x, ˆ y ; H, J ). Definition 4.24. For ˆ x, ˆ y ∈ g P er ( H ), the connecting subspace of ˆ x and ˆ y will denote the subspace W (ˆ x, ˆ y ) := { ( t, u ( s, t )) ∈ S × Σ : s ∈ R , u ∈ f M (ˆ x, ˆ y ; H, J ) } .Remark that if we write f W (ˆ x, ˆ y ) := f Ev ( R × S × f M (ˆ x, ˆ y ; H, J )), then the map ˇ π : R × S × Σ → S × Σrestricts to a projection ˇ π : f W (ˆ x, ˆ y ) → W (ˆ x, ˆ y ), with fiber ˇ π − ( t, p ) = { f Ev ( s, t, u ) : u ( s, t ) = p } , whichunder the hypotheses of Proposition 4.23 may be identified via f Ev with the orbit of any ( s , t, u ) ∈ R × S × f M (ˆ x, ˆ y ; H, J ) such that u ( s , t ) = p under the R -action τ · ( s, t, u ) = ( s − t, t, u τ ), for τ ∈ R ,where u τ ( s, t ) = u ( s + τ, t ). Consequently, f Ev descends to a well-defined map ˇ Ev ([ s, t, u ]) = ( t, u ( s, t )) ∈ W (ˆ x, ˆ y ), [ s, t, u ] ∈ ( R × S × f M (ˆ x, ˆ y ; H, J )) / R .Hence, under the hypotheses of Proposition 4.23, ˇ π restricts to a submersion on f W (ˆ x, ˆ y ) with fiberdiffeomorphic to R . Moreover, if we choose a section σ : M (ˆ x, ˆ y ; H, J ) → f M (ˆ x, ˆ y ; H, J ), then we maythereby (non-canonically) identify φ σ : R × S × M (ˆ x, ˆ y ; H, J ) ≃ −→ ( R × S × f M (ˆ x, ˆ y ; H, J )) / R ( s, t, [ u ]) [ s, t, σ ([ u ])] . Finally, to understand the behaviour of the foliation e F ˆ x, ˆ y under this projection, note that ker d ˇ π = h ∂ s i ,and that since the tangent space of any leaf of e F is given by h ∂ s + ( ∂ s u ) u ( s,t ) , ∂ t + ( ∂ t u ) u ( s,t ) i , where ∂ s u ∈ ker( d F ) u , and is therefore nowhere-vanishing whenever u is not an orbit cylinder by our indexconstraint. So the leaves of the foliation are nowhere tangent to the fibers of the projection map wheneverˆ x = ˆ y . As a consequence, we deduce Corollary 4.25. Let ˆ x, ˆ y ∈ g P er ( H ) satisfy µ (ˆ x ) , µ (ˆ y ) ∈ { k − , k, k + 1 } , for some k ∈ Z then forany section σ : M (ˆ x, ˆ y ; H, J ) → f M (ˆ x, ˆ y ; H, J ) , as above, ˇ Ev ◦ φ σ : R × S × M (ˆ x, ˆ y ; H, J ) W (ˆ x, ˆ y )( s, t, [ u ]) ( t, σ ([ u ])( s, t )) is a smooth embedding. Moreover, writing σ ([ u ]) = u σ , the partition F ˆ x, ˆ y := { im ˇ u σ } [ u ] ∈M (ˆ x, ˆ y ; H,J ) is asmooth -dimensional foliation of W (ˆ x, ˆ y ) whenever ˆ x = ˆ y . Contracting Seidel morphisms In order to build the foliation described by Theorem A, our goal is to find appropriate conditions oncapped braids ˆ X ⊆ g P er ( H ) such that the foliated sectors W (ˆ x, ˆ y ) from the preceding section consideredfor pairs ˆ x, ˆ y ∈ ˆ X will glue together to produce the desired foliation. Minimally, any such braid will haveto be unlinked. In this section, we introduce a useful tool for reducing the study of unlinked cappedbraids to that of trivial capped braids without changing the qualitative dynamics or the Floer-theoreticproperties of the situation.We briefly recall the definition of the Seidel morphism on Floer chain complexes introduced in [28].Let G = C ∞ (( S , Ham ( M, ω ) , id )) be the space of Hamiltonian loops, based at the identity. Any g ∈ G induces a diffeomorphism g ∗ : L ( M ) → L ( M ), defined by ( g ∗ x )( t ) := g t ( x ( t )), t ∈ S . We denoteby ˜ G the covering of G given by pairs ( g, ˜ g ) such that g ∈ G and ˜ g : e L ( M ) → e L ( M ) is a Γ ω -equivariantlift of the map g ∗ . For any Floer non-degenerate pair ( H, J ), an element ( g, ˜ g ) ∈ ˜ G with g = φ G , where G is a 1-periodic Hamiltonian, gives rise to an isomorphism of Floer complexes (modulo a grading shift) S ( g, ˜ g ) : CF ∗ ( H, J ) CF ∗ +2 I ( G H, g ∗ J )ˆ x ˜ g (ˆ x ) , where I = I ( g, ˜ g ) ∈ Z is the Maslov index of the loop of symplectic linear maps T ˜ g (ˆ x ) ( t ) ◦ Dg t ( x ( t )) ◦ T − x ( t ). In addition, S ( g, ˜ g ) shifts the action upwards by a constant a ( g, ˜ g ) which, when ( M, ω ) ismonotone, is a scalar multiple of I ( g, ˜ g ). S ( g, ˜ g ) is called the Seidel morphism associated to ( g, ˜ g ) ∈ ˜ G .The next proposition gives a topological interpretation of I ( g, ˜ g ) in the case that g is a contractibleHamiltonian loop. Proposition 5.1. Let g ∈ G be contractible and ˜ g ∈ ˜ G any lift of g , then I ( g, ˜ g ) = k if and only if forsome (hence any) ˆ x ∈ e L ( M ) , there exists some homotopy s g s ∈ G , s ∈ [0 , , from g ≡ id to g = g ,such that the induced homotopy of loops given by h ( s ) := ( g s ) ∗ x ∈ L ( M ) , s ∈ [0 , , is an A -homotopyfrom ˆ x to ˜ g (ˆ x ) for A ∈ π ( M ) with c ( A ) = k .Proof. Let ( g, ˜ g ) be as above and fix an arbitrary ˆ x = [ x, α ] ∈ e L ( M ) and an arbitrary homotopy( g s ) s ∈ [0 , from id to g . Let us write c ( s, t ) := ( g s ∗ x )( t ), ( s, t ) ∈ [0 , × S be the cylinder in M traced outby g s ∗ x throughout the homotopy. Note first that h is obviously a 0-homotopy between ˆ x and [ g ∗ x, α c ],and the map ( s, t ) T − g s ) ∗ x,α c | [0 ,s ] ] ( t ) ◦ ( Dg s ( t )) x ( t ) ◦ T ˆ x ( t ) , ( s, t ) ∈ [0 , × S provides a homotopy from the constant loop to the loop T − g ∗ x,α c ] ( t ) ◦ ( Dg ( t )) x ( t ) ◦ T ˆ x . So, in the eventthat ˜ g (ˆ x ) = [ g ∗ x, α ]¸ (and so h is a 0-homotopy), we have that I ( g, ˜ g ) = 0. Next note that we mayalways write ˜ g (ˆ x ) = [ g ∗ x, A α c ] for some A ∈ π ( M ), and in such a case, we know both that h is a( − A )-homotopy from ˆ x to ˜ g (ˆ x ) and also that T [ g ∗ x,A α c ] ( t ) = T [ g ∗ x,α c ] ( t ) ◦ L t for ( L t ) t ∈ S a loop ofsymplectic matrices with Maslov index equal to I ( g, ˜ g c ) − c ( A ), where ˜ g c is the lift of g sending ˆ x to[ g ∗ x, α c ]. The statement follows readily. Corollary 5.2. Let M = Σ be a surface, and let ( g, ˜ g ) ∈ ˜ G be such that I ( g, ˜ g ) = 0 . Then, for anycapped k -braid ˆ X = (ˆ x , . . . , ˆ x k ) , L (ˆ0 , ˆ X ) = L (ˆ0 , ˜ g ( ˆ X )) , where ˜ g ( ˆ X ) = (˜ g (ˆ x ) , . . . , ˜ g (ˆ x k )) .Proof. This follows immediately from Proposition 3.19 and consideration of the braid homotopy h ( s ) =( g s ∗ x , . . . , g s ∗ x k ) for g s a path of contractible loops from id to g .The following maybe be viewed as an analogue of Lemma 9 . Lemma 5.3. For any ˆ x ∈ e L ( M ) , there exists a contractible loop g ∈ G and a lift ˜ g ∈ ˜ G such that I ( g, ˜ g ) = 0 and ˜ g (ˆ x ) = ˆ x , where ˆ x = [ x , x ] denotes the trivially capped constant loop based at x = x (0) .Proof. Consider the map Ev x : Ham ( M, ω ) → M . It is easy to see that this a locally trivial fibrationwith model fiber S x = { φ ∈ Ham ( M, ω ) : φ ( x ) = x } over x . Let w : ( D , − → ( M, x ) represent18he capping of ˆ x (here we view D as the unit disk in C ), because ( D , − 1) is homeomorphic as a pairto ([0 , , [0 , × { } ∪ { , } × [0 , x to a map ˜ w : [0 , → Ham ( M, ω ) which covers w via Ev x and which moreover satisfies w ( s, 0) = w (0 , t ) = w (1 , t ) = id , for all ( s, t ) ∈ [0 , . Let us write k s ( t ) := ˜ w ( s, t ) ∈ Ham ( M, ω ), and notice that s k s ∈ G gives a homotopy from the constant loop to k , with k ∗ x = x . Moreover, since ˜ w covers w , it is immediate that h ( s ) := k s ∗ x ∈ L ( M ) provides a0-homotopy from ˆ x to ˆ x , and so the lift of k defined by insisting that ˜ k (ˆ x ) = ˆ x satisfies I ( k , ˜ k ) = 0by Proposition 5.1. Setting ( g, ˜ g ) = (( k ) − , (˜ k ) − ) proves the statement.If ˆ X = (ˆ x , . . . , ˆ x k ) is an unlinked capped braid, then reasoning in an analogous manner to theprevious subsection — with the fibration Ev x : Ham ( M, ω ) → M replaced with the fibration Ev ~x : Ham ( M, ω ) → C k ( M ) given by Ev ~x ( φ ) := ( φ ( x (0)) , . . . , φ ( x k (0)) — allows us to prove the following Lemma 5.4. For any unlinked capped braid ˆ X , there exists a contractible loop g ∈ G and a lift ˜ g ∈ ˜ G such that I ( g, ˜ g ) = 0 and ˜ g (ˆ x ) = [ x (0) , x (0)] for every ˆ x ∈ ˆ X . We will call any Seidel morphism S ( g, ˜ g ) satisfying the conclusions of the above lemma for a given ˆ X a contracting Seidel morphism associated to ˆ X . Since the precise choice of ( g, ˜ g ) will be irrelevantfor our purposes, we will write S ( ˆ X ) := S ( g, ˜ g ) for any contracting Seidel morphism associated to thecapped orbit ˆ X . Let ( H, J ) be a non-degenerate Floer pair. To any capped braid ˆ X ⊆ g P er ( H ), we may associate thesubmodule C ∗ ( ˆ X ) := Λ ω h ˆ x i ˆ x ∈ ˆ X , which comes with the projection π ˆ X : CF ∗ ( H, J ) → C ∗ ( ˆ X ) associatedto the splitting CF ∗ ( H, J ) = C ∗ ( ˆ X ) ⊕ C ∗ ( ˆ Y ), for ˆ Y ⊆ g P er ( H ) any capped braid such that P er ( H ) = X ⊔ Y . C ∗ ( ˆ X ) is not generally a subcomplex of CF ∗ ( H, J ), since there is no reason that Floer cylindersshould only run between strands of ˆ X . However, we will see that if we define the restricted differential ∂ ˆ X := π ˆ X ◦ ∂ H,J , then under suitable conditions on ˆ X , CF ∗ ( ˆ X ; H, J ) := ( C ∗ ( ˆ X ) , ∂ ˆ X )is a chain complex whose homology is isomorphic to the homology of the full complex CF ∗ ( H, J ). Definition 6.1. For any capped braid ˆ X ⊆ g P er ( H ), we define P os ( ˆ X ) := { σ ∈ CF ∗ ( H, J ) : ∀ ˆ γ ∈ supp σ, ∀ ˆ x ∈ ˆ X, ℓ (ˆ x, ˆ γ ) ≥ } ,P os ∗ ( ˆ X ) := { σ ∈ P os ( ˆ X ) : ∀ ˆ γ ∈ supp σ, ∃ ˆ x ∈ ˆ X, such that ℓ (ˆ x, ˆ γ ) > } . We define N eg ( ˆ X ) and N eg ∗ ( ˆ X ) in the obvious manner simply by reversing the inequalities in the above. Definition 6.2. Let ˆ X ⊆ g P er ( H ) be a capped braid for some Hamiltonian H . ˆ X will be said to be maximally unlinked if it is unlinked and if for any ˆ y ∈ g P er ( H ) either ˆ y ∈ ˆ X or ˆ y and ˆ X are linked.We write mu ( H ) for the collection of all such capped braids. Definition 6.3. Let ˆ X ⊆ g P er ( H ) be a capped braid for some Hamiltonian H . ˆ X of X will be said tobe maximally unlinked relative the Morse range ˆ X is unlinked, µ (ˆ x ) ∈ {− , , } for all ˆ x ∈ ˆ X ,and moreover if for any ˆ y ∈ g P er ( H ) such that µ (ˆ y ) ∈ {− , , } , either ˆ y ∈ ˆ X or ˆ y and ˆ X are linked.We write murm ( H ) for the collection of all capped braids ˆ X ⊆ g P er ( H ) which are maximally unlinkedrelative the Morse range.The next lemma is a direct consequence of the definitions. Lemma 6.4. Let ˆ X ⊆ g P er ( H ) be an unlinked braid, then P os ∗ ( ˆ X ) , N eg ∗ ( ˆ X ) ⊆ ker π ˆ X . The following situation will occur frequently enough that it will be useful to isolate it as a19 emma 6.5. Let ( H ± , J ± ) be Floer non-degenerate, ( H , J ) ∈ HJ , and ˆ x i ∈ g P er ( H − ) ∩ g P er ( H + ) , i = 1 , . . . k . Suppose that µ (ˆ x ) ∈ { , } , ˆ X = (ˆ x , . . . , ˆ x k ) is unlinked, and u i ∈ M (ˆ x i , ˆ x i ; H , J ) for i = 1 , . . . , k . Let v ∈ M (ˆ x , ˆ x + ; H , J ) , with ˆ x + ∈ g P er ( H + ) such that ˆ X ∪ { ˆ x + } is linked, then ˆ x + ∈ P os ∗ ( ˆ X ) .Proof. Note that we have 0 = ℓ (ˆ x , ˆ x i ) ≤ ℓ (ˆ x + , ˆ x i ) for i = 2 , . . . , k by Lemma 4.11 and 0 = b (ˆ x ) ≤ ℓ (ˆ x , ˆ x + ) by Lemma 4.18, so we need only show that ℓ (ˆ x + , ˆ x i ) > i = 1 , . . . , k . We write h ( s ) = ( u s , . . . , u ks , v s ), s ∈ R . h does not induce a braid cobordism, because u and v degenerate tothe same orbit as s → −∞ , however Lemma 4.11 implies that for R > h | ( − R, ∞ ) induces a braid cobordism from ˆ X ∪ { ˆ v − R } to ˆ X ∪ { ˆ y } . Since b (ˆ x ) ≥ 0, there are two possibilities:either 0 < ℓ −∞ ( u , v ), or ℓ −∞ ( u , v ) = 0. In the former case, Lemma 4.11 immediately implies that0 < ℓ ∞ ( u , v ) = ℓ (ˆ x , ˆ x + ), and we are done.We may therefore assume that ℓ −∞ ( u , v ) = 0. In this case, ˆ X ∪ { ˆ v − R } is unlinked. Indeed byCorollary 4.9, R > v has no intersections with ˜ u i , i = 1 , . . . , k for s < − R ,and the property of being unlinked is invariant under 0-homotopies, so ˆ X ∪ { ˆ v − R } is unlinked only if { ˆ x , ˆ v − R } is unlinked. But we may take R > v − R ∈ e L (Σ) lies in anexponential neighbourhood of ˆ x , and in this neighbourhood the homological linking number reduces tothe classical winding number by Proposition 3.21, and so that { ˆ x , ˆ v − R } is unlinked follows directly fromthe fact that the winding number classifies homotopy classes of loops into R \ X ∪ { ˆ v − R } is unlinked, while ˆ X ∪ { ˆ x + } is linked, whence the graphs of some of the strands of h must intersect. Since ℓ −∞ ( u i , u j ) = ℓ ∞ ( u i , u j ) = ℓ (ˆ x i , ˆ x j ) = 0 for i = j , it follows from Lemma 4.11that the graphs of u i and u j are disjoint for i = j . Thus, there exists some i = 1 , . . . , k such that thegraphs of u i and v intersect, so 0 < ℓ ∞ ( u i , v ) = ℓ (ˆ x i , ˆ x + ), as claimed. Lemma 6.6. Let ˆ X ∈ murm ( H ) , then for all ˆ x ∈ ˆ X , ∂ H,J ˆ x ∈ Z h ˆ x i ˆ x ∈ ˆ X ⊕ P os ∗ ( ˆ X ) .Proof. Let ˆ x ∈ ˆ X , and ˆ y ∈ supp ∂ H,J ˆ x . Either µ (ˆ x ) = − µ (ˆ x ) ∈ { , } . If µ (ˆ x ) = − 1, then b (ˆ x ) = 1and so Corollary 4.19 implies that 1 ≤ ℓ (ˆ x, ˆ y ) while Proposition 4.20 implies that 0 ≤ ℓ (ˆ x ′ , ˆ y ) for allˆ x ′ ∈ ˆ X , ˆ x ′ = ˆ x , and so ˆ y ∈ P os ∗ ( ˆ X ). If µ (ˆ x ) ∈ { , } , then Corollary 4.19 and Proposition 4.20 implythat 0 ≤ ℓ (ˆ x ′ , ˆ y ) for all ˆ x ′ ∈ ˆ X . To see that ˆ y ∈ ˆ X ∪ P os ∗ ( ˆ X ), note that either ˆ X ∪ { ˆ y } is unlinked,in which case ˆ y ∈ ˆ X by the maximality of ˆ X , or ˆ X ∪ { ˆ y } is linked, in which case Lemma 6.5 directlyimplies that ˆ y ∈ P os ∗ ( ˆ X ).Proposition 4.20 immediately implies Lemma 6.7. Let ˆ X ⊆ g P er ( H ) be any capped braid, then ∂ H,J P os ∗ ( ˆ X ) ⊆ P os ∗ ( ˆ X ) . Theorem 6.8. Let ˆ X ∈ murm ( H ) , then CF ∗ ( ˆ X ; H, J ) is a chain complex. That is, ∂ ˆ X ◦ ∂ ˆ X = 0 .Proof. First, consider that Σ is either an aspherical surface, in which case Λ ω = Z , or else a sphere, inwhich case Σ has minimal Chern number 2, and so in the case that Σ = S , CF ∗ ( ˆ X ; H, J ) vanishes inany degree congruent to 2 mod 4, whence by the Λ ω -equivariance of the Floer boundary map, it sufficesin all cases to prove that ( ∂ ˆ X ) vanishes in the Morse range. To wit, by the previous two lemmas, wesee that for any ˆ x ∈ ˆ X , ∂ H,J ˆ x = ∂ ˆ X ˆ x + ( π P os ∗ ( ˆ X ) ◦ ∂ H,J )(ˆ x ) , where π P os ∗ ( ˆ X ) denotes projection onto P os ∗ ( ˆ X ). Thus, since ∂ H,J = 0, ( ∂ ˆ X ) ˆ x + σ = 0, where σ ∈ P os ∗ ( ˆ X ). It follows that ( ∂ ˆ X ) ˆ x = 0.We will write HF ∗ ( ˆ X ; H ) for the homology of the complex CF ∗ ( ˆ X ; H, J ) when ˆ X ∈ murm ( H ). We obtain results giving us a measure of control over the moduli spaces which contribute to continuationmorphisms. In addition, we introduce the notion of pseudolinear adapted homotopies which enable usto avoid regularity issues which arise in considering linear homotopies while retaining sufficient controlover the moduli spaces for our linking arguments to go through.20 roposition 6.9. Let ( H ± , J ± ) be Floer non-degenerate and ˆ x ± ∈ g P er ( H ± ) . Suppose that µ (ˆ x ± ) =2 k + 1 , k ∈ Z , then any ( H , J ) ∈ HJ is (ˆ x − , ˆ x + ) -regular.Proof. Regularity of ( H , J ) for (ˆ x − , ˆ x + ) is claim about the surjectivity of the linearized Floer operator D F H ,J associated ( H , J ) at any u ∈ M (ˆ x − , ˆ x + ; H , J ). Proposition 4.17 implies that for any non-zero ξ ∈ ker( DF H ,J ) u , the number Z ( ξ ) of zeros of ξ satisfies the bounds 0 ≤ Z ( ξ ) ≤ a (ˆ x + ) − b (ˆ x − ). Theindex constraint µ (ˆ x ± ) = 2 k + 1 implies that a (ˆ x + ) < b (ˆ x − ) by Lemma 4.21, rendering this absurd. Weconclude that (ker DF H ,J ) u = 0, and since index ( DF H ,J ) u = 0, this implies the desired surjectivity. Proposition 6.10. Let ( H ± , J ± ) be Floer non-degenerate, then for any ˆ x ± ∈ g P er ( H ± ) such that µ (ˆ x ± ) = 2 k + 1 , k ∈ Z , and any ( H , J ) ∈ HJ , we have |M (ˆ x − , ˆ x + ; H , J ) | ∈ { , } and this quantity isindependent of the pair ( H , J ) .Proof. The fact that |M (ˆ x − , ˆ x + ; H , J ) | is independent of ( H , J ) ∈ HJ follows from the fact thatthat every such pair is regular. Indeed, in this event, any path ( H λ , J λ ), λ ∈ [0 , 1] produces a cobor-dism ∪ λ ∈ [0 , M (ˆ x − , ˆ x + ; H λ , J λ ) between M (ˆ x − , ˆ x + ; H , J ) and M (ˆ x − , ˆ x + ; H , J ) such that the pro-jection map π : ∪ λ ∈ [0 , M (ˆ x − , ˆ x + ; H λ , J λ ) → [0 , 1] is a submersion with compact fibers. To see that |M (ˆ x − , ˆ x + ; H , J ) | ≤ 1, suppose that there exist distinct u, v ∈ M (ˆ x − , ˆ x + ; H , J ), then Corollary 4.15and Lemma 4.11 together imply that b (ˆ x − ) ≤ ℓ −∞ ( u, v ) ≤ ℓ ∞ ( u, v ) ≤ a (ˆ x + ), which is absurd since µ (ˆ x ± ) = 2 k + 1 implies that a (ˆ x + ) < b (ˆ x − ) by Lemma 4.21.We fix, once and for all, some smooth, surjective non-decreasing function β : R → [0 , 1] whichis constant outside some compact set. Given two Hamiltonians H ± ∈ C ∞ ( S × Σ), we will call theadapted homotopy H lin ( s, t ) := β ( s ) H − ( t ) + (1 − β ( s )) H + ( t ) the linear homotopy from H − to H + .If x ∈ P er ( H − ) ∩ P er ( H + ), then the trivial cylinder u ( s, t ) = x ( t ) solves Equation 2 for ( H lin , J ) with J ∈ J ( J − , J + ) arbitrary. Corollary 6.11. Let ( H ± , J ± ) be Floer non-degenerate, then for any ˆ x ∈ g P er ( H − ) ∩ g P er ( H + ) suchthat µ (ˆ x ; H ± ) = 2 k + 1 , k ∈ Z , we have |M (ˆ x, ˆ x ; H , J ) | = 1 for every ( H , J ) ∈ HJ . Corollary 6.12. Let ( H ± , J ± ) be Floer non-degenerate, ( H , J ) ∈ HJ and ˆ x ∈ g P er ( H − ) ∩ g P er ( H + ) such that µ (ˆ x ; H ± ) = 2 k + 1 for some k ∈ Z . Let ˆ x ± ∈ g P er ( H ± ) , ˆ x ± = ˆ x , also satisfy µ (ˆ x ± ) = 2 k + 1 .Then M (ˆ x − , ˆ x ; H , J ) = ∅ only if ℓ (ˆ x, ˆ x − ) ≤ a (ˆ x ) , and M (ˆ x, ˆ x + ; H , J ) = ∅ only if b (ˆ x ) ≤ ℓ (ˆ x, ˆ x + ) .Proof. Let u ∈ M (ˆ x − , ˆ x ; H , J ). By the previous corollary, there exists v ∈ M (ˆ x, ˆ x ; H , J ) and so byCorollary 4.15 and Lemma 4.11, this implies that ℓ (ˆ x, ˆ x − ) ≤ ℓ ∞ ( u, v ) ≤ a (ˆ x ). The statement involvingˆ x + is proved similarly.We denote by H cst ( H lin ) the collection of adapted homotopies H ∈ H ( H − , H + ) which agree with H lin up to second order along the graphs ( s, t, x ( t )) ∈ R × S × M for all x ∈ P er ( H − ) ∩ P er ( H + ) suchthat µ (ˆ x ; H − ) = µ (ˆ x ; H + ) for some (hence every) capping ˆ x of x (thus we admit arbitrary perturbationsalong the graphs of constant cylinders which are 0-homotopies between capped orbits with index differingon each end). We may use the fact that any finite energy solution to Equation 2 has only finitely manyintersections with any given constant cylinder solving the same equation to adapt the usual proof (see theproof of Lemma 11.1.9 in [1], for example) that generic perturbations of H suffice to achieve regularityof the Floer operator to establish the following Proposition 6.13. Let ( H ± , J ± ) be Floer non-degenerate and J ∈ J ( J − , J + ) . There exists a residualset H regcst ( H lin ; J ) inside H cst ( H lin ) such every ( H , J ) ∈ H regcst ( H lin ; J ) is (ˆ x − , ˆ x + ) -regular for all ˆ x ± ∈ g P er ( H ± ) with either ˆ x − = ˆ x + and µ (ˆ x − ) − µ (ˆ x + ) ≤ or with ˆ x − = ˆ x + and µ (ˆ x − ; H − ) = µ (ˆ x + ; H + ) . The essential point is that if u is a non-constant cylinder, then we may always find some ( s , t ) ∈ R × S and a neighbourhood U of ( s , t , u ( s , t )) ∈ R × S × Σ which is disjoint from the graph ofevery constant solution and construct a perturbation with support contained in U .If P er ( H − ) ∩ P er ( H + ) = ∅ , we cannot expect ( H , J ) ∈ HJ with H ∈ H regcst ( H lin ; J ) to be Floer-regular, owing to the fact that constant solutions are not in general regular points of the Floer operator.The following lemma provides the existence of regular pairs ( H ′ , J ) ∈ HJ reg such that, for the purposesof controlling the linking behaviour of orbits contributing to the continuation map h H ′ ,J , we may reason‘as if’ these constant solutions exist. 21 emma 6.14. Let ( H ± , J ± ) be Floer non-degenerate and J ∈ J ( J − , J + ) . There exists an open neigh-bourhood U of H cst ( H lin ) ⊆ H ( H − , H + ) such that for every H ∈ H reg ( J ) ∩ U there exists some H ′ ∈ H regcst ( H lin ; J ) with the property that for all ˆ x ± ∈ g P er ( H ± ) , ˆ x − = ˆ x + , with µ (ˆ x − ) = µ (ˆ x + ) , M (ˆ x − , ˆ x + ; H , J ) = ∅ ⇒ M (ˆ x − , ˆ x + ; H ′ , J ) = ∅ . (4) Proof. It will suffice to show that if we are given a pair ˆ x ± ∈ g P er ( H ± ) such that µ (ˆ x − ) = µ (ˆ x + ), thenaround each H ′ ∈ H regcst ( H lin ; J ), there exists an open neighbourhood V = V ˆ x − , ˆ x + H ′ ⊆ H ( H − , H + ) suchthat (4) holds for every H ∈ H reg ( J ) ∩ V . Indeed, if this holds, then since the sets P er ( H ± ) are finiteand the moduli spaces M (ˆ x − , ˆ x + ; H , J ) are invariant under the Γ ω -action A · (ˆ x − , ˆ x + ) = ( A · ˆ x − , A · ˆ x + ),and in our setting any A with c ( A ) = 0 represents 0 ∈ Γ ω , the set V H ′ := \ ˆ x ± ∈ g P er ( H ) ± : µ (ˆ x − )= µ (ˆ x + ) V ˆ x − , ˆ x + H ′ = \ ˆ x ± ∈ g P er ( H ) ± : − ≤ µ (ˆ x − )= µ (ˆ x + ) ≤ V ˆ x − , ˆ x + H ′ is an intersection of finitely many open sets, and hence open (the bounds − ≤ µ (ˆ x − ) = µ (ˆ x + ) ≤ H regcst ( H lin ; J ) in H cst ( H lin ), taking U := ∪V H ′ , where the union is over all H ′ ∈ H regcst ( H lin ; J ), we obtain the desired open set.We therefore fix some H ′ ∈ H cst ( H lin ) and some ˆ x ± ∈ g P er ( H ± ) with µ (ˆ x − ) = µ (ˆ x + ), and supposethat there exists no such neighbourhood V ˆ x − , ˆ x + H ′ . Then it must be the case both that M (ˆ x − , ˆ x + ; H ′ , J ) = ∅ and that there is some sequence ( H ν ) ν ∈ N ⊆ H reg ( J ) tending to H ′ in the C ∞ -topology such that( H ν , J ) ∈ HJ reg and M (ˆ x − , ˆ x + ; H ν , J ) = ∅ , for each ν ∈ N . We thereby obtain a sequence of maps u ν ∈ M (ˆ x − , ˆ x + ; H ν , J ), ν ∈ N , which have uniformly bounded energy, owing to the fact that H ν → H ′ in the C ∞ -topology. Consequently, up to passing to a subsequence, u ν converges modulo bubbling inthe sense of Floer to a collection of broken cylinders v − . . . v − k − w v +1 . . . + v + k + , k ± ∈ Z ≥ , with v ± i ∈ f M (ˆ y ± i , ˆ y ± i +1 ; H ± , J ± ), and w ∈ M (ˆ y − k − +1 , ˆ y +1 ; H ′ , J ), where ˆ y − = ˆ x − and ˆ y + k + +1 = ˆ x + .Because the ( H ± , J ± ) are Floer non-degenerate, we must have for each i = 1 , . . . , k ± that either v ± i is constant, or that µ (ˆ y ± i ) < µ (ˆ y ± i +1 ), and similarly since ( H ′ , J ) ∈ HJ regcst ( H lin ; J ), either µ (ˆ y − k − +1 ) = µ (ˆ y +1 ) and ˆ y − k − +1 = ˆ y +1 or µ (ˆ y − k − +1 ) < µ (ˆ y +1 ); in either case, µ (ˆ y − k − +1 ) ≤ µ (ˆ y +1 ). Since there are noholomorphic spheres with negative Chern number in our setting, we have that k − X i =1 ( µ (ˆ y − i +1 ) − µ (ˆ y − i )) + ( µ (ˆ y − k − +1 ) − µ (ˆ y +1 )) + k + X i =1 ( µ (ˆ y + i +1 ) − µ (ˆ y + i )) ≤ , and consequently the Conley-Zehnder index of all the orbits involved must be equal. In particular wemust have that µ (ˆ y ± i ) = µ (ˆ y ± i +1 ) for each i = 1 , . . . , k ± , whence each v ± i is constant and thus ˆ y ± i = ˆ y ± i +1 for each i = 1 , . . . , k ± . It follows that w ∈ M (ˆ x − , ˆ x + ; H ′ , J ), which is a contradiction to our assumptionof the emptiness of M (ˆ x − , ˆ x + ; H ′ , J ). Definition 6.15. Let ( H ± , J ± ) be Floer non-degenerate. We denote by HJ pl ⊆ HJ reg the collectionof regular pairs ( H ′ , J ) such that H ′ ∈ H reg ( J ) ∩ U , where U is the neighbourhood given by the abovelemma. We will call any ( H , J ) ∈ HJ pl pseudolinear , and any H ′ ∈ H regcst ( H lin ; J ) such that (4)holds for all ˆ x ± ∈ g P er ( H ± ), ˆ x − = ˆ x + , with µ (ˆ x − ) = µ (ˆ x + ), will be called an essentially linearapproximation for H . If ˆ X ∈ murm ( H ), then by Lemma 5.4 we may choose a contracting Seidel morphism for ˆ X , S (( g, ˜ g )) : CF ∗ ( H, J ) → CF ∗ ( G H, g ∗ J ), such that ˜ g ( ˆ X ) is a trivial capped braid which lies in murm ( G H ) byCorollary 5.2. Consequently, modulo composing the Hamiltonian flow generated by H with a contractibleloop of Hamiltonian diffeomorphisms, there is no loss in generality in assuming that ˆ X is a trivial cappedbraid. 22 efinition 6.16. Let ˆ X ⊆ g P er ( H ) be a trivial capped braid. We will say that a Morse function f ∈ C ∞ (Σ) is ˆ X -dominating if ˆ X ( k ) ⊆ g P er ( f ) ( k ) for all k ∈ Z .For the remainder of this section, we fix the following setting: ( H, J + ) is Floer non-degenerate,ˆ X ∈ murm ( H ) is a trivial capped braid, ( f, J − ) is Floer-regular, with f a C -small ˆ X -dominating Morsefunction, and ( H , J ) ∈ HJ pl is such that ( ¯ H , ¯ J ) ∈ HJ pl ( H, J + ; f, J − ), where ¯ H ( s, t, x ) = H ( − s, t, x ),¯ J ( s, t ) = J ( − s, t ). With this setting understood, we have Proposition 6.17. The continuation map h H : CF ∗ ( f, J − ) → CF ∗ ( H, J + ) satisfies the following.1. For all ˆ x ∈ ˆ X (1) ∪ ˆ X ( − , h H (ˆ x ) = ˆ x + σ , where supp σ ⊆ P os ∗ ( ˆ X ) .2. For all ˆ p ∈ g P er ( f ) \ ˆ X with µ (ˆ p ) ∈ {± } , h H (ˆ p ) ∈ P os ∗ ( ˆ X ) .3. P os ∗ ( ˆ X ) ⊆ ker h ¯ H ,4. for all ˆ p ∈ g P er ( f ) with µ (ˆ p ) ∈ {− , , } , h H (ˆ p ) ∈ Z h ˆ x i ˆ x ∈ ˆ X ⊕ P os ∗ ( ˆ X ) .Proof. Item (1) follows immediately from Corollary 6.11, Corollary 6.12 and Lemma 4.21. Item (2)follows from Corollary 6.12 and Lemma 4.21. For items (3) and (4) we let H ′ ∈ H regcst ( H lin ; J ) be suchthat M (ˆ x − , ˆ x + ; H ′ , J ) = ∅ whenever M (ˆ x − , ˆ x + ; H , J ) = ∅ , for ˆ x − = ˆ x + , and µ (ˆ x − ) = µ (ˆ x + ).Item (3) then follows by remarking that by our choice of ( H , J ) we may choose an essentially linearapproximation ¯ H ′ ∈ H regcst ( ¯ H lin ) for ( ¯ H , ¯ J ). Therefore if ℓ (ˆ x + , ˆ x ) > x ∈ ˆ X , then the existenceof some u ∈ M (ˆ x + , ˆ x − ; ¯ H ′ , ¯ J ) would imply, by taking v ∈ M (ˆ x, ˆ x ; ¯ H ′ , ¯ J ), that ℓ (ˆ x + , ˆ x ) ≤ ℓ ∞ ( u, v ) andthis latter is either equal to 0 (if ˆ x + = ˆ x , since all capped orbits of a C -small Morse function in theMorse range are unlinked), or bounded above by a (ˆ x ), which is itself bounded above by 0 when ˆ x lies inthe Morse range. In either case, we obtain a contradiction.To prove item (4), we note first that items (1) and (2) suffice to establish (4) in the event that µ (ˆ p ) = ± 1, so we can, and do, assume that µ (ˆ p ) = 0. In this case, we let ( H ′ , J ) be an essentially linearapproximation for ( H , J ) and take ˆ x ∈ g P er ( H ) \ ˆ X with µ (ˆ x ) = 0, and suppose that M (ˆ p, ˆ x ; H , J ) = ∅ .Therefore either ˆ p = ˆ x , or M (ˆ p, ˆ x ; H ′ , J ) = ∅ . The former case is absurd, since this would imply that ˆ x is a trivially capped constant orbit, and therefore unlinked with ˆ X , which contradicts the maximality ofˆ X . Thus, M (ˆ p, ˆ x ; H ′ , J ) = ∅ , and ˆ X ∪ { ˆ x } is linked. There are two possibilities; either ˆ p ∈ ˆ X or ˆ p ˆ X .In the former case, Lemma 6.5 immediately gives that ˆ x ∈ P os ∗ ( ˆ X ). If ˆ p ˆ X , then consideration ofthe braid cobordism between the unlinked capped braid ˆ X ∪ { ˆ p } and the linked capped braid ˆ X ∪ { ˆ x } given by the taking the constant cylinders between strands of ˆ X and by u ∈ M (ˆ p, ˆ x ; H , J ) between ˆ p and ˆ x yields that the graph of u must intersect the graph of some constant cylinder at least once, andany intersections may only contribute positively to the change in linking number as all the maps solveEquation 2 for ( H ′ , J ). Consequently, ˆ x ∈ P os ∗ ( ˆ X ). Proposition 6.18. The maps π ˆ X ◦ h H and h ¯ H | CF ∗ ( ˆ X ; H,J ) are morphisms of chain complexes.Proof. As in the proof of Theorem 6.8, it suffices to prove that the maps are chain maps in the Morserange. Thus, we may assume that ˆ p ∈ CF k ( f, J − ) for k ∈ {− , , } . We note that h H (ˆ p ) ∈ Z h ˆ x i ˆ x ∈ ˆ X ⊕ P os ∗ ( ˆ X ) by Proposition 6.17 and ∂ H,J P os ∗ ( ˆ X ) ⊆ P os ∗ ( ˆ X ) ⊆ ker π ˆ X by Lemmas 6.4 and 6.7. Conse-quently, we see that( ∂ ˆ X ◦ h H )(ˆ p ) = ( ∂ ˆ X ◦ π ˆ X ◦ h H )(ˆ p ) + ( ∂ ˆ X ◦ π P os ∗ ( ˆ X ) ◦ h H )(ˆ p )= ( ∂ ˆ X ◦ π ˆ X ◦ h H )(ˆ p ) + ( π ˆ X ◦ ∂ H,J + ◦ π P os ∗ ( ˆ X ) ◦ h H )(ˆ p )= ( ∂ ˆ X ◦ π ˆ X ◦ h H )(ˆ p )Thus, since h H is a chain map with respect to the full Floer differential, we compute( π ˆ X ◦ h H )( ∂ f,J − ˆ p ) = ( π ˆ X ◦ ∂ H,J + )( h H (ˆ p )) =( ∂ ˆ X ◦ h H )(ˆ p ) = ∂ ˆ X (( π ˆ X ◦ h H )(ˆ p )) , which shows that π ˆ X ◦ h H is a chain map. Parallel reasoning shows that h ¯ H ◦ π ˆ X is a chain map, fromwhich it immediately follows that h ¯ H | CF ∗ ( ˆ X ; H,J ) is a chain map.23 orollary 6.19. H ∗ ( CF ( ˆ X ; H, J )) ≃ QH ∗ (Σ) .Proof. Once more, it suffices to show that π ˆ X ◦ h H induces an isomorphism on homology in degreeslying in the Morse range. Since, for any ˆ p ∈ CF k ( f, J − ), with k ∈ {− , , } , we have that h H (ˆ p ) ∈ Z h ˆ x i ˆ x ∈ ˆ X ⊕ P os ∗ ( ˆ X ), and P os ∗ ( ˆ X ) ⊆ ker π ˆ X ∩ ker h ¯ H , so we compute( h ¯ H ◦ h H )(ˆ p ) = h ¯ H (( π ˆ X ◦ h H )(ˆ p ) + ( π P os ∗ ( ˆ X ) ◦ h H )(ˆ p ))= ( h ¯ H ◦ π ˆ X ◦ h H )(ˆ p )= ( h ¯ H ◦ π ˆ X ◦ π ˆ X ◦ h H )(ˆ p )= h ¯ H | CF ∗ ( ˆ X ; H,J + ) ◦ ( π ˆ X ◦ h H ) . But it is a standard fact in Floer theory that h ¯ H ◦ h H induces the identity map on homology, and so itmust be that case that the composition HF k ( f ) ( π ˆ X ◦ h H ) ∗ −−−−−−−→ HF k ( ˆ X ; H ) ( h ¯ H ) ∗ −−−−→ HF k ( f )is the identity map for k ∈ {− , , } (and analogous reasoning shows that the same holds for the abovediagram with the arrows reversed), consequently, ( π ˆ X ◦ h H ) ∗ is an isomorphism on homology, and theclaim follows. Corollary 6.20. For any α ∈ QH ∗ (Σ) , there exists σ ∈ Λ ω h ˆ x i ˆ x ∈ ˆ X ⊕ P os ∗ ( ˆ X ) ⊆ CF ∗ ( H, J ) such that [ σ ] = α ∈ HF ∗ ( H ) ≃ QH ∗ (Σ) and [ π ˆ X ( σ )] = α ∈ HF ∗ ( ˆ X ; H ) ≃ QH ∗ (Σ) , where these identificationswith the quantum homology is understood to be those naturally induced by Floer continuation maps. F ˆ X Our construction of the singular foliation in Theorem A proceeds by establishing that a generic point in S × Σ lies inside the foliated sector W (ˆ x, ˆ y ) for some ˆ x ∈ ˆ X (1) , ˆ y ∈ ˆ X ( − . The remaining points lie inthe closure of these sectors, and so lie either on leaves parametrized by broken cylinders, or the graphsof orbits in X . To establish existence of the requisite leaves, we make use of the cap action of a point onthe Floer complex, first introduced in detail in [16] (see also [24]).Given a singular homology class α ∈ H k ( M ; Z ), we represent α by a smooth chain α : ∪ ∆ k → M ,and for any ( H, J ) and any t ∈ S , we may consider, for any ˆ x, ˆ y ∈ g P er ( H ), the moduli space M α ,t (ˆ x, ˆ y ; H, J ) := { u ∈ f M (ˆ x, ˆ y ; H, J ) : u (0 , t ) ∈ im α } . For t ∈ S , we will say that the smooth chain α is ( H, J ; t ) -generic when the evaluation map ev t ( u, q ) :=( u (0 , t ) , α ( q )) ∈ M × M , ( u, q ) ∈ f M (ˆ x, ˆ y ; H, J ) × ∪ ∆ k , is transversal to the diagonal whenever µ (ˆ x ) − µ (ˆ y ) ≤ (2 n − k ) + 1. Such chains form a residual set for fixed H if we permit generic perturbations of J ,and in such a case we define the cap product of α on HF ∗ ( H ) (at time t ) at the chain level by defining,for ˆ x ∈ g P er ( H ), α ∩ t ˆ x := X ˆ y ∈ g P er ( H ): µ (ˆ x ) − µ (ˆ y )=2 n − k n α ,t (ˆ x, ˆ y )ˆ y, where n α ,t (ˆ x, ˆ y ) is the mod 2 count of the number of elements in M α ,t (ˆ x, ˆ y ; H, J ). The cap actiondescends to homology, and is independent at the homology level of all choices. Moreover, for genericadapted homotopies of Floer data, the cap action commutes with continuation maps at the chain level .That is, for generic ( H , J ) we have h H ( α ∩ t ˆ x ) = α ∩ t h H (ˆ x ) , (5)24henever the Floer pairs ( H ± , J ± ) at the ends of the homotopy are such that the relevant moduli spacesare transversal. It follows from the above that, under the identification of HF ∗ ( H ) with QH ∗ + n ( M ), thecap action on HF ∗ ( H ) is identified with the standard cap action of the homology of M on its quantumhomology.A rather important point for us will be that the cap action interacts nicely with the respect to thechain maps π ˆ X ◦ h H and h ¯ H ◦ π ˆ X introduced in the previous section. We take up again the settingand notation of Section 6.3, with the added stipulation that ( H , J ) ∈ HJ pl is generic in the sense thatrelation 5 holds. Proposition 7.1. Suppose that α represents α as above and α is both ( H, J + ; t ) -generic and ( f, J − ; t ) -generic for some t ∈ S , then ( π ˆ X ◦ h H )( α ∩ t ˆ p ) = π ˆ X ( α ∩ t ( π ˆ X ◦ h H )(ˆ p )) , ∀ ˆ p ∈ g P er ( f ) and ( h ¯ H ◦ π ˆ X )( α ∩ t π ˆ X (ˆ y )) = α ∩ t ( h ¯ H ◦ π ˆ X )(ˆ y ) , ∀ ˆ y ∈ g P er ( H ) Proof. Since f is C -small, we may reason similarly as in the proof of Theorem 6.8, and reduce to thecase where µ (ˆ p ) ∈ {− , , } . Note that Proposition 6.17 implies that we may write h H (ˆ p ) = σ + β for σ ∈ CF ∗ ( ˆ X ; H, J ) and β ∈ P os ∗ ( ˆ X ), so that π ˆ X ( α ∩ t ( π ˆ X ◦ h H )(ˆ p )) = π ˆ X ( α ∩ t σ ) . The central point is that capping with α preserves P os ∗ ( ˆ X ). Indeed, if ˆ y ∈ P os ∗ ( ˆ X ) and there exists u ∈ M α ,t (ˆ y, ˆ y ′ ; H, J ) for some ˆ y ′ ∈ g P er ( H ), then for each ˆ x ∈ ˆ X , Lemma 4.11 with v ( s, t ) = x ( t )implies that ℓ (ˆ y, ˆ x ) ≤ ℓ (ˆ y ′ , ˆ x ). Thus α ∩ t β ∈ P os ∗ ( ˆ X ) and we have( π ˆ X ◦ h H )( α ∩ t ˆ p ) = π ˆ X ( α ∩ t σ + α ∩ t β ) = π ˆ X ( α ∩ t σ ) , where we use Lemma 6.4 in the last equality. The second equality is proved similarly, needing only theadditional remark that for ˆ x ∈ ˆ X , α ∩ t ˆ x ∈ Z h ˆ x i ˆ x ∈ ˆ X ⊕ P os ∗ ( ˆ X ), which follows by the same reasoningas above, using that b (ˆ x ) ≥ µ (ˆ x ) lies in the Morse range.The following proposition is essentially tautological. Proposition 7.2. Let ( H, J ) be Floer non-degenerate, t ∈ S , and suppose that p ∈ Σ is ( H, J ; t ) -generic(for the point class in homology). Then p ∈ W (ˆ x, ˆ y ) implies that µ (ˆ x ) − µ (ˆ y ) ≥ , and ˆ y ∈ supp ( p ∩ t ˆ x ) if and only if ( t, p ) ∈ W (ˆ x, ˆ y ) . Combining the above with Corollary 4.25, allows us to conclude Corollary 7.3. Suppose that µ (ˆ x ) = 2 k + 1 for k ∈ Z and p is ( H, J ; t ) -generic, then ˆ y ∈ supp ( p ∩ t ˆ x ) if and only if there exists an open neighbourhood of ( t, p ) ∈ S × Σ which is foliated by leaves of F ˆ x, ˆ y . Recall from section 4.3 that W (ˆ x, ˆ y ) = { ( t, u ( s, t )) ∈ S × Σ : u ∈ f M (ˆ x, ˆ y ; H, J ) } , and write W ( ˆ X )for the union of all W (ˆ x, ˆ y ) where ˆ x ∈ ˆ X (1) , ˆ y ∈ ˆ X ( − . Lemma 7.4. Let ( H, J ) be Floer non-degenerate and ˆ X ∈ murm ( H ) , then W ( ˆ X ) is open and dense in S × Σ .Proof. By Lemma 5.4, we may suppose without loss of generality that ˆ X is trivial. Let ( H, J + ), ( f, J − )and ( H , J ) be as in the setting of Proposition 7.1. We fix t ∈ S arbitrarily and let p ∈ Σ be both( f, J − ; t )-generic and ( H, J ; t )-generic. We let σ ∈ CF ( f, J − ) represent the fundamental class [Σ] ∈ QH (Σ) ≃ HF ( f ), and we note that we must have [ p ∩ t σ ] = [ pt ] ∈ QH (Σ), which is in particular not0. But ( π ˆ X ◦ h H ) ∗ is an isomorphism of homology groups, and so0 = ( π ˆ X ◦ h H )( p ∩ t σ ) = π ˆ X ( p ∩ t ( π ˆ X ◦ h H )( σ )) , and this implies that for every such generic p , there must exist some ˆ x ∈ supp ( π ˆ X ◦ h H )( σ ) ⊆ ˆ X (1) andsome ˆ y ∈ ˆ X ( − such that ˆ y ∈ supp p ∩ t ˆ x , and hence every p which is both ( H, J ; t )-generic and ( f, J − ; t )-generic lies inside the open 3-dimensional connecting submanifold W (ˆ x, ˆ y ) for some such ˆ x, ˆ y ∈ ˆ X , whichproves the lemma. 25 emma 7.5. Let ( H, J ) be Floer non-degenerate, and ˆ X ⊆ g P er ( H ) any unlinked capped braid. Thenfor any ˆ γ ∈ g P er ( H ) such that M (ˆ x + , ˆ γ ; H, J ) × M (ˆ γ, ˆ x − ; H, J ) = ∅ for some ˆ x + ∈ ˆ X (1) , ˆ x − ∈ ˆ X ( − , ˆ X and ˆ γ are unlinked.Proof. Suppose that ˆ X ∪ { ˆ γ } is linked, then Lemma 6.5 implies that ˆ γ ∈ P os ∗ ( ˆ X ), and so ℓ (ˆ γ, ˆ x ) > x ∈ ˆ X , but then Proposition 4.20 implies that ℓ (ˆ x − , ˆ x ) > 0, which contradicts the assumptionthat ˆ X is unlinked.Inductively applying Lemma 7.5 yields Corollary 7.6. Suppose that ( H, J ) is Floer non-degenerate, and let ˆ X ⊆ g P er ( H ) be such that ˆ X isunlinked and ˆ X = ˆ X (1) ∪ ˆ X ( − , then ˆ X and ˆΥ are unlinked, where ˆΥ is the capped braid consisting ofall ˆ γ ∈ g P er ( H ) with µ (ˆ γ ) = 0 such that M (ˆ x + , ˆ γ ; H, J ) × M (ˆ γ, ˆ x − ; H, J ) = ∅ , ˆ x ± ∈ ˆ X . Finally, we are ready to prove the existence of the advertised foliation. Theorem 7.7 (Existence part of Theorem A) . Let ( H, J ) be a non-degenerate Floer pair, and ˆ X ∈ murm ( H ) , then the collection of submanifolds F ˆ X := ∪ [ u ] ∈M H,J ( ˆ X ) { im ˇ u } forms a Stefan-Sussmannfoliation of S × Σ .Proof. We adapt a strategy used in [11] that shows that the foliation e F ˆ X with leaves given by the graphs˜ u of all the u ∈ f M (ˆ x, ˆ y ; H, J ), ˆ x, ˆ y ∈ ˆ X , is a smooth 2-dimensional foliation if R × S × Σ, from which itfollows immediately that F ˆ X is a Steffan-Sussmann foliation. Indeed, in this event, F ˆ X integrates thedistribution D ˆ X = ˇ π ∗ e D ˆ X , where e D ˆ X is the distribution integrated by e F ˆ X , and this realizes D ˆ X in a waythat is manifestly smooth in the sense of generalized distributions (see Definition 2.2).To see that e F ˆ X is a smooth foliation, note that by Lemma 7.4, the set W ( ˆ X ) is open and densein S × Σ. This implies, by the R -invariance of solutions to Equation 1, that the set of points f W ( ˆ X )lying on the graph ˜ u of some u ∈ f M (ˆ x, ˆ y ; H, J ), ˆ x ∈ ˆ X (1) , ˆ y ∈ ˆ X ( − is open and dense in R × S × Σ.Consequently, the partition S e F ˆ x, ˆ y , where the union runs over all ˆ x ∈ ˆ X (1) , ˆ y ∈ ˆ X ( − , gives a smoothfoliation of an open, dense set of R × S × Σ. Consequently we may argue just as in [11] in the paragraphsfollowing the proof of lemma 6 . 10 (p. 231-232); all of the remaining leaves in e F ˆ X are graphs of constantorbits or of cylinders u which connect orbits of index difference equal to 1. In either case, by standardcompactness theorems of Floer theory those graphs which form the leaves of the foliation of f W ( ˆ X )converge modulo reparametrization in the C ∞ loc -topology either to the graphs ( s, t ) ( s, t, x ( t )) of theorbits x for ˆ x ∈ ˆ X , or to graphs of cylinders connecting orbits of index difference 1, which come in pairs( u, v ) ∈ f M (ˆ x, ˆ γ ; H, J ) × f M (ˆ γ, ˆ y ; H, J ), ˆ x ∈ ˆ X (1) , ˆ γ ∈ ˆ X (0) , ˆ y ∈ ˆ X ( − . By Corollary 7.6 and Lemma 7.5,the capped braid formed by the collection of all the ˆ γ ∈ g P er ( H ) on which such pairs break are unlinkedwith ˆ X , and so lie in ˆ X by maximality. Consequently, the graphs of such broken trajectories cannotintersect, nor can they intersect any leaf of e F ˆ X in the dense set f W ( ˆ X ). Since every point in R × S × Σlies in the closure of f W ( ˆ X ), every such point much lie on the graph of an orbit in X or the graph ofsuch a broken cylinder. It follows that the union of all the leaves in e F ˆ X thus fits together into a smoothfoliation on all of R × S × Σ, and so the theorem follows. F ˆ X as negative gradient flow-lines of the restricted action functional For ˆ X ∈ murm ( H ), denote M ˆ X = M ˆ X ; H,J := { ˆ α ∈ e L (Σ) : ∃ ˆ x, ˆ y ∈ ˆ X, ∃ u ∈ f M (ˆ x, ˆ y ; H, J ) , such that ˆ u s = ˆ α, for some s ∈ R } Proposition 7.8. The map Ev : S × M ˆ X → S × Σ , given by Ev ( t, ˆ α ) = ( t, α ( t )) is a diffeomorphism.Proof. The generalized distribution D ˆ X ( t,u ( s,t )) = h ∂ s u, ∂ t ⊕ ∂ u i which is integrated by F ˆ X contains theone-dimensional distribution D M ( t,u ( s,t )) = h ∂ t ⊕ ∂ t u i , which is easily seen to be smooth near the singularfibers by employing the local model for leaves of a Stefan-Sussmann foliation of section 2. Consequently, D M is a smooth foliation which integrates precisely to the graphs of the maps α : S → Σ for ˆ α ∈ M ˆ X .This is obviously equivalent to the proposition. 26 efinition 7.9. For ˆ X ∈ murm ( H ), define the ( ˆ X -)restricted action functional A ˆ X ∈ C ∞ ( S × Σ)by A ˆ X := A H ◦ Ev − . Additionally, for each t ∈ S , we define A ˆ Xt := ι ∗ t A ˆ X , where ι t : Σ ֒ → S × Σ isthe inclusion of the fiber over t ∈ S .Note that each A ˆ Xt is automatically Morse, since the Hessian of A ˆ Xt at x ( t ) for ˆ x ∈ ˆ X obviouslyinherits the non-degeneracy of the Hessian of A H at ˆ x . In fact, our construction clearly identifies Floertrajectories connecting orbits in ˆ X with negative gradient flow lines of the A ˆ Xt , giving us Morse modelsfor the foliation F ˆ X . Proposition 7.10. If ( H, J ) is Floer non-degenerate, ˆ X ∈ murm ( H ) and ǫ > is sufficiently small, thenfor every t ∈ S , and every ˆ x, ˆ y ∈ ˆ X , there is a natural identification f M (ˆ x, ˆ y ; H, J ) ∼ = f M (ˆ x, ˆ y ; ǫA ˆ Xt , J t ) given by u ( s, t ) u ( ǫs, t ) . Corollary 7.11. Let ( H, J ) be Floer non-degenerate and ˆ X ∈ murm ( H ) . Then for every t ∈ S , andany ǫ > sufficiently small, CF ∗ ( ˆ X ; H, J ) ∼ = C Morse ∗ +1 ( A ˆ Xt , g J t ) ⊗ Λ ω ∼ = CF ∗ ( ǫA ˆ X , J t ) . F ˆ X as a positively transverse foliation Each regular leaf of the foliation F ˆ X arises naturally as the image of an embedding ˆ u : R × S ֒ → S × Σfor u , the standard orientation on the cylinder induces the orientation ∂ s u ∧ ∂ t u on each regular leaf, sowe may view F ˆ X in a natural way as an oriented singular foliation. Definition 7.12. Let F be an oriented codimension 1 Steffan-Sussmann foliation of an oriented d -dimensional manifold ( M d , o M ). We will say that a smooth path α : [0 , → M is positively transverse to F if the following dichotomy holds, either1. α is contained in a singular leaf of F , or2. for every t ∈ [0 , { ( ∂ t α ) t , v , . . . , v d − } is an oriented basis for ( T α ( t ) M, o M ), where { v , . . . , v d − } is an oriented basis for the tangent space of the regular leaf of F passing through α ( t ). Definition 7.13. Let F be an oriented codimension 1 Steffan-Sussmann foliation on an oriented d -dimensional manifold ( M d , o M ) and let X ∈ X ( M ) be a vector field generating an isotopy ( φ Xt ) t ∈ R .We say that X (or ( φ Xt ) t ∈ R ) is positively transverse to F if every integral curve of X is positivelytransverse to F . Proposition 7.14 (Positive transversality part of Theorem A) . Let ( H, J ) be Floer non-degenerate, ˆ X ∈ murm ( H ) , and ˇ X H := ∂ t ⊕ X H ∈ X ( S × Σ) , then F ˆ X is positively transverse to ˇ X H .Proof. As the singular leaves of F ˆ X are orbits of ˇ X H , it suffices to consider points ( t, p ) ∈ S × Σ lyingon regular leaves. In such a case, since u solves Equation 1, the basis formed by { ˇ X H , ∂ s ˇ u, ∂ t ˇ u } is easilyseen to be orientation-equivalent to the basis { ∂ t , ∂ s u, J t ∂ s u } , which is a positively oriented basis, as J t ∈ J (Σ , ω ) for all t ∈ S .The previous proposition tells us that to any non-degenerate Hamiltonian H and each ˆ X ∈ murm ( H ),we may associate a foliation on S × Σ with respect to which the graph of the isotopy is well-behaved ina certain sense. However, if we’re willing to modify the isotopy by a contractible loop, then we can infact do better and obtain a positively transverse singular foliation on Σ itself.To see this, consider the distribution D M ( t,u ( s,t )) = h ∂ t ⊕ ∂ t u i introduced in the proof of Proposition 7.8.As noted therein, D M integrates to a smooth 1-dimensional foliation by the graphs of the loops t u s ( t )for ˆ u s ∈ M ˆ X . This induces a natural loop of diffeomorphisms ( ψ ˆ Xt ) t ∈ S given by sliding the fiber { } × Σalong the foliation which integrates D M . In other words, we have the isotopy ψ ˆ Xt ( p ) = u p ( s, t ), t ∈ S ,where u p ∈ f M (ˆ x, ˆ y ; H, J ), ˆ x, ˆ y ∈ ˆ X , is any Floer cylinder such that u p ( s, 0) = p . It follows fromCorollary 4.25 and the fact that if ˆ x = ˆ y then u ( s, t ) = x ( t ) that ψ ˆ X is well-defined. Let us establishsome elementary properties of this isotopy. Proposition 7.15. ψ := ( ψ ˆ Xt ) t ∈ S is a contractible Hamiltonian loop. Moreover, if ˜ ψ : e L (Σ) → e L (Σ) is the lift of ψ ∗ which sends [ x (0) , x (0)] to ˆ x for ˆ x ∈ ˆ X , then S ( ψ − , ˜ ψ − ) is a contracting Seidel morphismfor ˆ X . roof. ψ defines a loop of diffeomorphisms based at the identity by construction, and it follows easilyfrom the fact that ∂ t u verifies −∇ A ˆ X + J ( ∂ t u − X H ) = 0 that the generating Hamiltonian function is givenby H t − A ˆ Xt . To see that ψ is contractible, we may as well-suppose that Σ = S , or else contractibilityis immediate, and up to composing with a contracting Seidel morphism, we may suppose that ˆ X is atrivial capped braid, then clearly ψ fixes x = x (0) for all t ∈ S , for each ˆ x ∈ ˆ X and we may simplycompute the Maslov index of the linearization of ψ about some such x , relative the trivial capping,but this Maslov index turns out to vanish and so ψ is necessarily contractible. Indeed, suppose withoutloss of generality that µ ([ x , x ]) = 1 and let ξ , ξ ∈ x ∗ T Σ be a basis of eigenvectors of A x ,J withwinding number 0 relative the trivial capping, then ( Dψ t )( ξ i (0)) = ξ i ( t ) for t ∈ S by the asymptoticestimates of Theorem 4.7, and so ( Dψ t ) t ∈ S is homotopic to ( Id ) t ∈ S . Thus ψ − is contractible as welland S ( ψ − , ˜ ψ − ) is clearly a contracting Seidel morphism for ˆ X by Proposition 5.1.Note that F ˆ X is everywhere transverse to the fibers { t } × Σ of S × Σ, and so may be viewed as an S -family of (singular) foliations on Σ. Let us write F ˆ Xt for the foliation obtained on Σ by intersecting F ˆ X with { t } × Σ. Theorem 7.16. Let ( H, J ) be a Floer non-degenerate pair, ˆ X ∈ murm ( H ) , then the orbits of theHamiltonian isotopy ( ψ ˆ X ) − ◦ φ H are positively transverse to the foliation F ˆ X Proof. Writing ψ = ψ ˆ X , observe that the vector field ( Z t ) t ∈ [0 , which generates the isotopy ψ − ◦ φ H iseasily computed via the chain rule as ( Z t ) u ( s, = ( ψ − t ) ∗ ( X Ht − ∂ t u ) u ( s,t ) . Consequently, because ψ t isa Hamiltonian diffeomorphism for all t ∈ S , we see that ω u ( s, ( Z t , ∂ s u ) = ( ψ ∗ t ω )( Z t , ∂ s u ) = ω u ( s,t ) ( X Ht − ∂ t u, ∂ s u ) = ω u ( s,t ) ( − J t ∂ s u, ∂ s u )from which the claim follows.Theorem B is an immediate consequence of Proposition 7.10 and the preceding Theorem. For ( H, J ) non-degenerate and ˆ X ∈ murm ( H ), we define the Piexoto graph of F ˆ X to be the directed graph Γ( F ˆ X ) whose vertex set is ˆ X and such that there is a directed edge from ˆ x to ˆ y only if µ (ˆ x ) − µ (ˆ y ) = 1, and in this case there is an edge from ˆ x to ˆ y for each element in M (ˆ x, ˆ y ; H, J ). Remark. Note that since F ˆ Xt may be realized as the singular foliation obtained by the negative gradientflow of ( A ˆ X , g J t ), Γ( F ˆ X ) may be naturally identified with the Piexoto graph (see [23]) of ( A ˆ Xt , g J t ) Definition 7.18. Let ( H, J ) be non-degenerate and ˆ X ∈ murm ( H ). To any capped loop ˆ γ ∈ e L (Σ)such that ( γ, x ) is a braid for all ˆ x ∈ ˆ X , we may define the linking cochain ℓ ˆ γ (ˆ x ) := ℓ (ˆ γ, ˆ x ) ∈ Z forany ˆ x ∈ V (Γ( F ˆ X )) = ˆ X , as well as the intersection cochain I γ : E (Γ( F ˆ X )) → Z , where I γ ( u ) countsthe signed intersection number of (some transverse perturbation of) the maps ˇ u ( s, t ) = ( t, u ( s, t )) andˇ γ ( t ) = ( t, γ ( t )).The following relation between these two quantities is immediate from the definition of the homologicallinking number. Proposition 7.19. Let ( H, J ) be non-degenerate and ˆ X ∈ murm ( H ) . For any capped loop ˆ γ ∈ e L (Σ) such that ( γ, x ) is a braid for all ˆ x ∈ ˆ X , we have I γ = δℓ γ . For any Hamiltonian H , we write H ♮k := H . . . H for the k -fold concatenated Hamiltonian whichgenerates ( φ H ) k as its time-1 map. Positive transversality of ˇ X H to F ˆ X implies positive transversalityof ˇ X H ♮k for every k ∈ Z > and so we obtain Corollary 7.20. Let ( H, J ) be non-degenerate, ˆ X ∈ murm ( H ) and let ˆ γ ∈ g P er ( H ♮k ) for k ∈ Z > ,then ℓ ˆ γ is non-decreasing along edges of Γ( F ˆ X ) . Moreover, for every ˆ x, ˆ z ∈ ˆ X , ℓ (ˆ x, ˆ γ ) < ℓ (ˆ z, ˆ γ ) if andonly if ˇ γ ( t ) ∈ W (ˆ x, ˆ z ) for some t ∈ S . efinition 7.21. For a Hamiltonian H , we will say that a capped braid ˆ X ⊆ g P er ( H ) is stronglylinking if for any ˆ γ ∈ g P er ( H ), ℓ (ˆ γ, ˆ x ) = 0 for all ˆ x ∈ ˆ X implies that ˆ γ ∈ ˆ X . Denote by usl ( H ) thecollection of all ˆ X ⊆ g P er ( H ) such that ˆ X is both unlinked and strongly linking.Clearly, usl ( H ) ⊆ mu ( H ). Theorem 7.22 (Theorem C) . Let H be a non-degenerate Hamiltonian, then murm ( H ) ⊆ ∩ ∞ k =1 usl ( H ♮k ) .Proof. Let ˆ X ∈ murm ( H ), fix some J such that ( H, J ) is Floer non-degenerate, and suppose for acontradiction that ˆ γ ∈ g P er ( H ♮k ) \ ˆ X but ℓ (ˆ x, ˆ γ ) = 0 for all ˆ x ∈ ˆ X . We may in fact suppose thatˆ γ π (Σ) · ˆ X , since if ˆ γ = A · ˆ x for some A ∈ π (Σ), then for any ˆ y ∈ ˆ X , ˆ x = ˆ y , Proposition 3.19implies ℓ (ˆ γ, ˆ y ) = ℓ (ˆ x, ˆ y ) + c ( A )2 = c ( A )2 , since ˆ X is unlinked. So we may as well assume that γ = x forany ˆ x ∈ ˆ X . Since F ˆ X foliates S × Σ, it is necessary that ˇ γ (0) ∈ W (ˆ x, ˆ y ) for some ˆ x, ˆ y ∈ ˆ X and so byCorollary 7.20 implies that ℓ (ˆ x, ˆ γ ) < ℓ (ˆ y, ˆ γ ), a contradiction. We apply the theory developed in the preceding sections to prove Theorem D. Fix some Floer non-degenerate ( H, J ) throughout this section. For ˆ X ∈ murm ( H ), let us write f M ( ˆ X ) := [ ˆ x, ˆ y ∈ ˆ X f M (ˆ x, ˆ y ; H, J )and also ψ := ψ ˆ X , where the latter is the loop constructed in the paragraph preceding Proposition 7.15. Lemma 8.1. Let ˆ X ∈ murm ( H ) , ˆ γ = [ γ, w ] ∈ g P er ( H ) , and write w ˜ ψ − for the capping disk of ( ˜ ψ − )ˆ γ ,where ˜ ψ sends ˆ X to a trivial capped braid,1. if ˆ γ ∈ P os ( ˆ X ) , then R D ( w ˜ ψ − ) ∗ ω > , and2. if ˆ γ ∈ N eg ( ˆ X ) , then R D ( w ˜ ψ − ) ∗ ω < .Proof. We prove the first statement, with the proof of the second being entirely dual. Note that byCorollary 5.2, ˆ γ ∈ P os ( ˆ X ) if and only if ˜ ψ − ˆ γ ∈ P os ( ˜ ψ − ˆ X ). For convenience of notation, let us writeˆ γ ′ = [ γ ′ , w ′ ] = ˜ ψ − ˆ γ and remark that Proposition 3.23 implies Z D ( w ′ ) ∗ ω = Z Σ ℓ (ˆ γ ′ , ˆ p ) ω, where ˆ p denotes the trivially capped disk assigned to a point p ∈ Σ. We will therefore be done if wecan show that ℓ (ˆ γ ′ , ˆ p ) ≥ , ∀ p ∈ Σ. To see that this is so, consider the leaf F p of F ˆ X which passesthrough p ; it is parametrized by a map of the form s u ( s, s ∈ R , for some u ∈ f M ( ˆ X ). Withoutloss of generality, we may assume that u (0 , 0) = p , and so we may consider the 0-homotopy inducedby h ( s ) = ( u s , γ ′ ) ∈ L (Σ) , s ∈ ( −∞ , ψ − ˆ x, ˆ γ ′ ) = ([ x (0) , x (0)] , ˆ γ ′ ) and(ˆ p, ˆ γ ′ ). Since ψ − ◦ φ H is positively transverse to F ˆ X , any intersections between the graphs of the strandsof h must contribute positively, whence ℓ (ˆ p, ˆ γ ′ ) ≥ ℓ ( ˜ ψ − ˆ x, ˆ γ ′ ) = ℓ (ˆ x, ˆ γ ) ≥ Proposition 8.2. Let ( H, J ) be Floer non-degenerate and ˆ X ∈ murm ( H ) . For any ˆ γ ∈ g P er ( H ) , if ˆ γ ∈ P os ( ˆ X ) , then A H (ˆ γ ) < max ˆ x ∈ ˆ X (1) A H (ˆ x ) .Proof. Without loss of generality, we assume that ˆ X is a trivial braid and moreover, we may replace( H, J ) with (( ψ − X ) ∗ H, ( ψ − X ) ∗ J ) without affecting the action of any of the capped orbits or their grading,and hence we may suppose that the flow of H is positively transverse to F ˆ X , which is the singularfoliation produced by the negative gradient flow-lines of A ˆ X . Equivalently, for each t ∈ S , we have that29 H ( −∇ J A ˆ X ) ≤ 0, with equality only at critical points of A ˆ X . Because the critical points of A ˆ X areprecisely the points x (0) = x for ˆ x ∈ ˆ X , this implies that for all t ∈ S , we have that H t ( γ ( t )) < H t ( x max ) (6)for x max the maximum of A ˆ X , which is, by construction, the orbit such that R H t ( x max ) dt =max ˆ x ∈ ˆ X (1) A H (ˆ x ). Integrating (6) gives Z H t ( γ ( t )) dt < Z H t ( x max ) dt = max ˆ x ∈ ˆ X (1) A H (ˆ x ) , and since ˆ γ ∈ P os ( ˆ X ), the previous lemma implies that R D w ∗ ω > 0, from which the claim follows.For σ ∈ CF ∗ ( H, J ), write murm ( σ ) := { ˆ Y ≤ supp σ : ˆ Y is maximally unlinked relative supp σ } . Proposition 8.3. Let σ ∈ CF ∗ ( H, J ) be a cycle representing the fundamental class, then for every ˆ Y ∈ murm ( σ ) , and every ˆ X ∈ murm ( H ) such that ˆ Y ≤ ˆ X , we have ˆ Y = ˆ X (1) .Proof. Let σ be given and take any ˆ Y ∈ murm ( σ ), we will write murm ( H ; ˆ Y ) := { ˆ X ∈ murm ( H ) :ˆ Y ⊆ ˆ X } . Note that murm ( H ; ˆ Y ) is non-empty, because ˆ Y is unlinked and has all strands lying inthe Morse range. In view of a contradiction, we take an arbitrary ˆ X ∈ murm ( H ; ˆ Y ) and suppose thatˆ X ′ := ˆ X (1) \ ˆ Y is non-empty. Without loss of generality, up to applying a contracting Seidel morphism,we may henceforth assume that ˆ X (and so also ˆ Y and ˆ X ′ ) is a trivial capped braid. We adopt once morethe setting and notation of Section 6.3 with ( f, J − ) = ( ǫA ˆ X , J ), for some small ǫ > J + = J .That h ¯ H : CF ∗ ( H, J ) → CF ∗ ( f, J ) induces an isomorphism on homology, implies that h ¯ H ( σ ) = P ˆ x ∈ ˆ X ˆ x , since P ˆ x ∈ ˆ X ˆ x ∈ CF ( f, J ) is the unique cycle representing the fundamental class as f is smalland Morse. Corollaries 6.11 and 6.12, together with the observation that CF ∗ ( f, J ) = CF ∗ ( ˆ X ; H, J ),imply that h ¯ H (ˆ y ) = ˆ y for all ˆ y ∈ ˆ Y . Thus, if we write supp σ = ˆ Y ⊔ ˆ Z , we see that h ¯ H ( P ˆ z ∈ ˆ Z ˆ z ) = P ˆ x ′ ∈ ˆ X ′ ˆ x ′ . Since continuation maps are chain morphisms we see h ¯ H ( ∂ H,J X ˆ z ∈ ˆ Z ˆ z ) = ∂ f,J X ˆ x ′ ∈ ˆ X ′ ˆ x ′ , and this last quantity must be non-vanishing, because Z ≃ HF ( f ) = ker ∂ f,J is generated by P ˆ x ∈ ˆ X (1) ˆ x , and ˆ X ′ ( ˆ X (1) . We will derive a contradiction by showing that, in fact, h ¯ H ( ∂ H,J ˆ z ) = 0for each ˆ z ∈ ˆ Z .To see this, let ˆ z ∈ ˆ Z be arbitrary, fix some t ∈ S , and let β : D → Σ be an embedding of a closeddisk into Σ such that α := ∂β : S → Σ is ( H, J ; t )-regular (and so a fortiori ( f, J )-regular), and suchthat β ( D ) is an isolating neighbourhood for z ( t ) in the sense that for any γ ∈ P er ( H ), γ ( t ) ∈ im β implies γ = z . Because β is isolating for z ( t ), we have immediately that ∂ H,J ˆ z = α ∪ t ˆ z , and so, up toperturbing ( ¯ H , ¯ J ) slightly to achieve regularity of the evaluation maps relative to α , we compute h ¯ H ( ∂ H,J ˆ z ) = h ¯ H ( α ∩ t ˆ z ) = α ∩ t h ¯ H (ˆ z ) = ∂ f,J ( β ∩ t h ¯ H (ˆ z )) + β ∩ t ∂ f,J h ¯ H (ˆ z ) , where in the last equality, we have used that if α = ∂β (as smooth cycles), then α ∩ σ = ∂ ( β ∩ σ ) + β ∩ ∂σ ,since the cap product is independent of the representing smooth cycle at the level of homology. However,since β ( D ) is isolating for z ( t ), in particular Crit ( A ˆ X ) ∩ im β = ∅ , and so both β ∩ t h ¯ H (ˆ z ) = 0,and β ∩ t ∂ f,J h ¯ H (ˆ z ) = 0, from which we readily conclude that h ¯ H ( ∂ H,J ˆ z ) = 0, which gives the desiredcontradiction. Proof of Theorem D. By the behaviour of the filtered Floer complexes under Poincar´e duality (see [4],Lemma 2.2), we have that for surfaces, c ( H ; [ pt ]) = − c ( ¯ H ; [Σ]), where ¯ H t ( x ) := − H t ( φ Ht ( x )), and itis easy to see that murm ( H ) = murm ( ¯ H ). It therefore suffices to prove that for any non-degenerateHamiltonian H , c ( H ; [Σ]) = min ˆ X ∈ murm ( H ) max ˆ x ∈ ˆ X A H (ˆ x ). To bound the spectral invariant from above,we may apply Corollary 6.20, so as to see that for any ˆ X ∈ murm ( H ), we may represent the fundamentalclass of CF ∗ ( H, J ) by a cycle of the form P ˆ x ∈ ˆ X (1) ˆ x + β , where β ∈ P os ∗ ( ˆ X ), consequently Proposition30.2 implies that λ H ( β ) < max ˆ x ∈ ˆ X (1) A H (ˆ x ) = max ˆ x ∈ ˆ X A H (ˆ x ). Since ˆ X ∈ murm ( H ) was arbitrary, wededuce that c ( H ; [Σ]) ≤ min ˆ X ∈ murm ( H ) max ˆ x ∈ ˆ X A H . To obtain the opposite inequality, we let σ ∈ CF ∗ ( H, J ) be a tight cycle for [Σ]. Applying Proposition8.3 to σ , we may take any ˆ Y ∈ murm ( σ ) and extend it to some ˆ X ∈ murm ( H ) with ˆ Y ⊆ ˆ X andˆ X (1) = ˆ Y , and thereby immediately obtainmin ˆ X ∈ murm ( H ) max ˆ x ∈ ˆ X A H ≤ c ( H ; [Σ]) . It is clear that the methods used in this paper have no chance of establishing the existence of singularfoliations on S × M of the type described here when ( M, ω ) is a symplectic manifold of dimensiongreater than 2; linking becomes a vacuous notion and one loses the positivity of intersections which isessential for controlling the behaviour of Floer trajectories. However, it is at the same time evident thatat least some pairs ( H, J ) and some collections of their orbits do admit such foliations: namely, if H is any Morse function (even if not C -small) and J is also autonomous, then restricting our attentionto Floer cylinders which are actually negative gradient trajectories of H with respect to g J between thecritical points of H provides such a ( S -invariant) foliation (in this case, A ˆ Xt simply recovers H for every t ∈ S ).Let us define , therefore, for a Hamiltonian H on an arbitrary symplectic manifold M n the set murm ( H ) := (cid:26) ˆ X ⊆ g P er ( H ) (cid:12)(cid:12)(cid:12)(cid:12) µ (ˆ x ) ∈ {− n, . . . , n } , ∀ ˆ x ∈ ˆ X,Ev : S × M ˆ X → S × M is a diffeomorphism for some J (cid:27) , where M ˆ X is defined as in Section 7.1, with the possible caveat that one may need to take only certainconnected components of the moduli spaces f M (ˆ x, ˆ y ; H, J ), ˆ x, ˆ y ∈ ˆ X . The non-emptiness of murm ( H )may then be viewed as containing interesting information about the degree to which H behaves ‘roughlylike an autonomous Morse Hamiltonian’ with the non-triviality of the associated Hamiltonian loop ψ ˆ X measuring an associated obstruction to something like H actually being an autonomous Morse functionwith critical points given by ˆ X (more specifically, the non-triviality of ψ ˆ X is equivalent to the failureof the foliating Floer cylinders to be S -invariant). In any case, according to this definition, on anysymplectic manifold the set of Hamiltonians with non-empty murm ( H ) is itself non-empty, and so itmay be an interesting question to investigate under what conditions one can guarantee that murm ( H ) isnon-empty for a given H , and what the structure of murm ( H ) is for interesting classes of Hamiltonians.One of the main obstructions to attacking this problem lies in finding good conditions on orbitsˆ x, ˆ y ∈ P er ( H ) and selection principles for connected components C ⊂ f M (ˆ x, ˆ y ; H, J ), such that, for anyFloer cylinder u ∈ C , one can guarantee that every element in ker( d F ) u is everywhere non-vanishing. It’sclearly necessary that any pair of orbits ˆ x, ˆ y ∈ ˆ X and certain components of their Floer moduli space musthave such a property if ˆ X ∈ murm ( H ). Let us provisionally call this the linearized non-vanishingproperty . One would need moreover a condition on collections of capped orbits ˆ Y ⊆ g P er ( H ) whichguarantees that the graphs ˜ u : R × S → R × S × M of Floer cylinders running between orbits of ˆ Y do not intersect. Let us unimaginatively call such a property the non-intersecting Floer cylindersproperty . We may then hope to equivalently characterize murm ( H ) as the set of collections of orbitsˆ X with each ˆ x ∈ ˆ X having Conley-Zehnder index in the Morse range, such that each pair of orbits hasthe linearized non-vanishing property, while the entire collection has the non-intersecting Floer cylindersproperty, where the gap between this characterization and the above definition of murm ( H ) is preciselythe existence of Floer cylinders in the moduli spaces connecting orbits of ˆ X which pass through eachpoint of M . Presumably, this gap could be filled by the same type of argument as employed in Section7, which guarantees the existence of Floer cylinders passing through generic points of M via the capping31ction with the point — assuming that such collections could be shown to be part of the support of aFloer cycle which represents the fundamental class of M .We hasten to point out that this sketch is purely speculative, as it’s unclear how one might findinteresting and practical conditions which guarantee the linearized non-vanishing condition or the non-intersecting Floer cylinders property, but, in some sense, any approach which seeks to build these sortsof foliations in higher dimensions will have to address these issues in one form or another. References [1] M. Audin and M. Damian. Morse theory and Floer homology . Springer, 2014.[2] B. Bramham. Periodic approximations of irrational pseudo-rotations using pseudoholomorphiccurves. Annals of Mathematics , pages 1033–1086, 2015.[3] C. Debord. Holonomy groupoids of singular foliations. Journal of Differential Geometry , 58(3):467–500, 2001.[4] M. Entov and L. Polterovich. Calabi quasimorphism and quantum homology. International Math-ematics Research Notices , 2003(30):1635–1676, 2003.[5] J.-M. Gambaudo and ´E. Ghys. Commutators and diffeomorphisms of surfaces. Ergodic Theory andDynamical Systems , 24(5):1591–1617, 2004.[6] M. Gromov. Pseudo holomorphic curves in symplectic manifolds. Inventiones mathematicae ,82(2):307–347, 1985.[7] H. Hofer and D. A. Salamon. Floer homology and novikov rings. In The Floer memorial volume ,pages 483–524. Springer, 1995.[8] H. Hofer, K. Wysocki, and E. Zehnder. Properties of pseudo-holomorphic curves in symplectisationsII: Embedding controls and algebraic invariants. In Geometries in Interaction , pages 270–328.Springer, 1995.[9] H. Hofer, K. Wysocki, and E. Zehnder. Properties of pseudoholomorphic curves in symplectisationsI: Asymptotics. Annales de l’Institut Henri Poincare (C) Non Linear Analysis , 13(3):337 – 379,1996.[10] H. Hofer, K. Wysocki, and E. Zehnder. Properties of pseudoholomorphic curves in symplectizationsIII: Fredholm theory. In Topics in nonlinear analysis , pages 381–475. Springer, 1999.[11] H. Hofer, K. Wysocki, and E. Zehnder. Finite energy foliations of tight three-spheres and Hamilto-nian dynamics. Annals of Mathematics , pages 125–255, 2003.[12] V. Humili`ere, F. Le Roux, and S. Seyfaddini. Towards a dynamical interpretation of Hamiltonianspectral invariants on surfaces. Geometry & Topology , 20(4):2253–2334, 2016.[13] E. Kerman and F. Lalonde. Length minimizing Hamiltonian paths for symplectically asphericalmanifolds. Annales de l’institut Fourier , 53(5):1503–1526, 2003.[14] M. Khanevsky. Quasimorphisms on surfaces and continuity in the Hofer norm. arXiv:1906.08429,2019.[15] F. Lalonde. A field theory for symplectic fibrations over surfaces. Geometry & Topology , 8(3):1189–1226, 2004.[16] H. V. Lˆe and K. Ono. Cup-length estimate for symplectic fixed points. Contact and symplecticgeometry (Cambridge, 1994) , 8:268–295, 1996.[17] P. Le Calvez. Une version feuillet´ee ´equivariante du th´eoreme de translation de Brouwer. PublicationsMath´ematiques de l’IH ´ES , 102:1–98, 2005. 3218] P. Le Calvez and F. A. Tal. Forcing theory for transverse trajectories of surface homeomorphisms. Inventiones mathematicae , 212(2):619–729, 2018.[19] D. McDuff and D. Salamon. J-holomorphic curves and symplectic topology , volume 52. AmericanMathematical Soc., 2012.[20] E. Mora. Pseudoholomorphic Cylinders in Symplectisations . PhD thesis, New York University, 2003.[21] Y.-G. Oh. Spectral invariants and the length minimizing property of Hamiltonian paths. Asian J.Math. , 9(1):001–018, 03 2005.[22] Y.-G. Oh. Floer mini-max theory, the Cerf diagram, and the spectral invariants. J. Korean Math.Soc. , 46(2):363–447, 2009.[23] M. Peixoto. On the classification of flows on 2-manifolds. In Dynamical systems , pages 389–419.Elsevier, 1973.[24] S. Piunikhin, D. Salamon, and M. Schwarz. Symplectic Floer-Donaldson theory and quantumcohomology. Contact and symplectic geometry (Cambridge, 1994) , 8:171–200, 1996.[25] D. Salamon. Lectures on Floer homology. In L. T. Yakov Eliashberg, editor, Symplectic Geometryand Topology , pages 144–227. AMS/IAS, 1997.[26] D. Salamon and E. Zehnder. Morse theory for periodic solutions of Hamiltonian systems and theMaslov index. Communications on pure and applied mathematics , 45(10):1303–1360, 1992.[27] M. Schwarz. Cohomology Operations from S -cobordisms in Floer homology . PhD thesis, ETHZurich, 1995.[28] P. Seidel. π of symplectic automorphism groups and invertibles in quantum homology rings. Geo-metric & Functional Analysis GAFA , 7(6):1046–1096, 1997.[29] R. Siefring. Relative asymptotic behavior of pseudoholomorphic half-cylinders. Communications onPure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences ,61(12):1631–1684, 2008.[30] R. Siefring. Intersection theory of punctured pseudoholomorphic curves. Geom. Topol. , 15(4):2351–2457, 2011.[31] P. Stefan. Accessible sets, orbits, and foliations with singularities. Proceedings of the LondonMathematical Society , 3(4):699–713, 1974.[32] M. Usher. Spectral numbers in Floer theories. Compositio Mathematica , 144(6):1581–1592, 2008.[33] J. van den Berg, R. Ghrist, R. Vandervorst, and W. W´ojcik. Braid Floer homology.