Floer homology in disc bundles and symplectically twisted geodesic flows

We show that if K: P \to R is an autonomous Hamiltonian on a symplectic manifold (P,\Omega) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the Hamiltonian flow of K has contractible periodic orbits with bounded period on all sufficiently small energy levels. As a special case, if the geodesic flow on the cotangent bundle of M is twisted by a symplectic magnetic field form, then the resulting flow has contractible periodic orbits on all low energy levels. These results were proven by Ginzburg and G\"urel when \Omega|_M is spherically rational, and our proof builds on their work; the argument involves constructing and carefully analyzing at the chain level a version of filtered Floer homology in the symplectic normal disc bundle to M.


Introduction
Consider a symplectic manifold (P, Ω) containing a closed, connected symplectic submanifold M , with 2m = dim M , 2n = dim P , and r = n − m. In recent years, there has been significant interest in the following question: Question 1.1. If K : P → [0, ∞) is a proper smooth function with K −1 ({0}) = M , must it be the case that the Hamiltonian vector field of X K (which in our convention is given by ι XK Ω = dK) has periodic orbits on all regular energy levels K −1 ({ρ}), provided that ρ > 0 is sufficiently small? Already in the case that M = {pt}, V. Ginzburg and B. Gürel show in [12] that their negative resolution of the Hamiltonian Seifert conjecture ( [9], [11]) implies that the answer to Question 1.1 is negative for some smooth Hamiltonians when n ≥ 3 and for some C 2 -Hamiltonians when n ≥ 2 unless the requirements are weakened in some way.
One way of weakening the requirements is to require existence of periodic orbits only on a suitably large set of low energy levels, where "large" might be interpreted as meaning either "dense" or "full measure." A considerable amount of work was done on this version of the problem (e.g., in [14], [17], [15], [2], [12], [18]), culminating in results of L. Macarini [22] and G. Lu [20] which show that, for some ρ 0 > 0, it is the case that the level surfaces K −1 ({ρ}) contain periodic orbits for Lebesgue-almost-every regular value ρ ∈ [0, ρ 0 ]. Further, these periodic orbits are all contractible within a tubular neighborhood of M .
Another way of weakening the requirements of Question 1.1 is to retain the requirement that orbits exist on all low energy levels, but to constrain the form of the function K in some way. It is this version of the question that we shall consider in this paper, with K required to attain a Morse-Bott nondegenerate minimum along M (in other words, the Hessian of K is to restrict nondegenerately to the normal bundle of M ). An interesting special case of this (which historically served as much of the motivation for Question 1.1) is the following.
Where M is a closed Riemannian manifold, the motion of a particle of unit mass and unit charge in a magnetic field on M is modeled by setting P = T * M and letting Ω = Ω σ = ω can + τ * σ where ω can is the standard symplectic form on T * M , τ : T * M → M is the bundle projection, and σ is a closed 2-form on M which represents the magnetic field. The phase-space trajectory of the particle is then given by the Hamiltonian vector field X K of the standard kinetic energy Hamiltonian K(q, p) = 1 2 |p| 2 . Thus, M is a symplectic submanifold of (T * M, Ω σ ) (so that this fits into the framework of Question 1.1) if and only if the magnetic field 2-form σ on M is symplectic. Of course, the case σ = 0 just corresponds to the geodesic flow on M ; accordingly the Hamiltonian flow of K on (T * M, Ω σ ) is sometimes called the σ-twisted geodesic flow. The search for periodic orbits of twisted geodesic flows was initiated by V. Arnol'd (for the case M = T 2 ) in [1], and has continued in, e.g., [7], [8], [10], [19], [21], [3], [4], [6], [27], [31], [13]. In particular, it is shown by F. Schlenk in [31] that there is ρ 0 > 0 such that X K has contractible periodic orbits on Lebesgue-almost-every energy level ρ ∈ [0, ρ 0 ], provided merely that σ does not vanish identically (thus in Schlenk's result σ need not be symplectic, putting this result in a somewhat different category than those of [20], [22]).
Note that choosing a compatible almost complex structure on (P, Ω) makes T P into a complex vector bundle, which in particular has Chern classes c k (T P ) ∈ H 2k (P ; Z). Our main result is the following. Since c 1 (T (T * M )) = 0, it immediately follows that: Corollary 1.3. If σ ∈ Ω 2 (M ) is symplectic, there are ρ 0 , T 0 > 0 such that, for every ρ ∈ (0, ρ 0 ) the Hamiltonian flow of the function K(q, p) = 1 2 |p| 2 on (T * M, Ω σ ) has a periodic orbit on K −1 ({ρ}) which is contractible in T * M and has period no larger than T 0 .
The restriction to sufficiently small energy levels is necessary: as is explained for instance in [8], already in the case where M is a hyperbolic surface with area form given by σ, there is an energy level c 0 such that the magnetic flow has no contractible periodic orbits of energy larger than c 0 , and no periodic orbits at all on the energy level c 0 . Theorem 1.2 was proven in [13] in the special case that Ω| M is spherically rational (i.e., in the case that { S 2 u * Ω|u : S 2 → M } is a discrete subgroup of R); in broad outline, our approach is similar to that of [13]. An important ingredient in the proof of the corresponding theorem in [13] is a result (Propositions 3.1 and 3.2 of [13]) which, in certain situations which include the case where c 1 (T P )| νM is torsion, allows one to use grading information in Floer homology to bound the period of a periodic orbit. As such, in order to obtain existence of periodic orbits on all sufficiently low energy levels, in these situations it is enough to find periodic orbits on a dense set of sufficiently low energy levels corresponding to a fixed Floer homological grading, since the periodic orbits so obtained then have bounded period and so the Arzelà-Ascoli theorem can be used to obtain periodic orbits on all low energy levels, with bounds on their period.
The notation ∆([γ, w], K) should be understood as follows. Choose a symplectic trivialization of w * T P over D 2 , and for 0 ≤ t ≤ τ let φ t K denote the time-t flow of the Hamiltonian vector field X K . The linearizations at γ(0) of φ t K are then represented with respect to our trivialization by a path {Ψ(t)} 0≤t≤τ of symplectic matrices, and ∆([γ, w], K) is the Salamon-Zehnder index (which was originally defined in [30], where it is denoted ∆ τ (Ψ) and referred to as the "mean winding number") of this path Ψ. Various relevant properties of ∆ are reviewed in Section 2 of [13]. We should point out that since we define X K by ι XK ω = dK instead of ι XK ω = −dK as is done in [13], our Hamiltonian vector fields have periodic orbits which are related to those in [13] by time reversal, as a result of which the Salamon-Zehnder indices ∆([γ, w], K) of these orbits have opposite sign.
Except for the statement about the Salamon-Zehnder index, Theorem 1.4 is weaker than the previously-mentioned results of [20], [22]. However, as we shall shortly see, the information about the Salamon-Zehnder indices enables one to deduce Theorem 1.2 from Theorem 1.4 in the situation of Case (i) above. We include Case (ii) in Theorem 1.4 because it illustrates the broad applicability of our method and its proof requires only a brief digression (in the proof of Proposition 4.2) from the proof of Case (i). Note that the condition that M admit a Morse function with no index-one critical points obviously implies that π 1 (M ) = 0, and in fact if dim M = 4 this condition is equivalent to requiring that π 1 (M ) = 0 (see Theorem 8.1 of [23]). Whether or not this remains true when dim M = 4 is an open question as of this writing.
Incidentally, the Morse-Bott assumption on K plays a fairly modest role in the proof of Theorem 1.4; it facilitates somewhat the construction of the functions f a , f b of Section 4, but functions with the same essential properties could be constructed with some additional effort for many other classes of K. However, the Morse-Bott assumption is vital in the proof that Theorem 1.4 implies Theorem 1.2, due to its use in Proposition 3.2 of [13].
Proof of Theorem 1.2, assuming Theorem 1.4. Since c 1 (T P )| νM is torsion, c 1 (T P ) is represented in real cohomology by a 2-form which vanishes throughout νM . Let γ be any of the periodic orbits produced by Theorem 1.4, and denote its period by T γ . Then since the disc w is contained in νM , Proposition 3.2 of [13] gives that (possibly after shrinking ρ 0 , and taking into account that our differing conventions on the signs of Hamiltonian vector fields results in a sign reversal for the Salamon-Zehnder index ∆([γ, w], K)), there are γ-independent constants a, c > 0 with Thus given ρ < ρ 0 , where T 0 = 1 a (2r + c) it follows that for each ǫ ∈ (0, ρ 0 − ρ) X K has a contractible-in-νM periodic orbit of period at most T 0 in K −1 ((ρ − ǫ, ρ + ǫ)). The Arzelà-Ascoli theorem applied to the orbits so obtained from a sequence ǫ k ց 0 then shows that X K has a contractible-in-νM periodic orbit of period at most T 0 in K −1 ({ρ}). Now Theorems 1.2 and 1.4 depend only on the behavior of K in a (sufficiently small) tubular neighborhood νM of M . The Weinstein neighborhood theorem (see, e.g., Theorem 3.30 of [24]) asserts that if (P 1 , Ω 1 ) and (P 2 , Ω 2 ) both contain M as a symplectic submanifold, with Ω 1 | M = Ω 2 | M and with the symplectic normal bundles to M in P 1 and P 2 isomorphic as symplectic vector bundles, then there are tubular neighborhoods ν i ⊂ P i of M in P i (i = 1, 2) which are symplectomorphic by a symplectomorphism restricting to M as the identity. Where E → M denotes an arbitrary sympelctic vector bundle over M (which of course necessarily admits complex vector bundle structure and a compatible Hermitian metric), at the beginning of Section 2 we shall, for suitably small R > 0, equip the radius R disc bundle E(R) with a symplectic form ω which restricts to the zero section M as an arbitrary given symplectic form ω 0 . Given (P, Ω) as in the statement of Theorem 1.4, use for E the symplectic normal bundle to M in P and use for ω 0 the restriction Ω| M . A tubular neighborhood of M in P is then symplectomorphic to a tubular neighborhood of M in E(R), and the Morse-Bott condition for K is obviously preserved via this symplectomorphism, so to prove Theorem 1.4 it is enough to prove it when (P, Ω) is the symplectic disc bundle (E(R), ω) and M is the zero-section. The rest of the paper is devoted to this latter task.
The proof uses a version of (filtered) Floer homology for the disc bundle E(R). Since E(R) is not closed, there are subtleties involved in the definition this Floer homology, because of a need for compactness results. As far as we know, the only case in which such Floer homology groups have been constructed in the literature without any constraint on the base M is when E satisfies a negative curvature hypothesis (this hypothesis allows one to use a maximum principle to obtain compactness; see [26]); however such a hypothesis is not natural in our context. In Section 2, we are able to address the compactness problem because the only Floer homology groups we need are groups of the form HF [a,b] (H) where H is a Hamiltonian which behaves in a certain standard way outside a small neighborhood of the zero section M , and where the "action window" [a, b] is small. Thus all of the cylinders which one needs to use in the definition of the Floer complex have low energy; we prove a result (Theorem 2.4) showing roughly speaking that, for the Hamiltonians which we consider, a Floer connecting orbit with small energy stays entirely within a small neighborhood of the zero section. (The main subtlety here is that the Hamiltonians can behave quite wildly very close to the zero section, so we need our constants to depend only on the behavior of the Hamiltonians away from the zero section.) This enables us to define Floer groups HF [a,b] (H) in a fairly standard way. Similar constructions are carried out in [13] in the spherically rational case when the action interval [a, b] does not intersect the image of [ω] on π 2 (E(R)); there the compactness results are somewhat easier because in that context one can use certain compactly supported Hamiltonians in place of the Hamiltonians that we use.
Where H is a reparametrized version of the Hamiltonian K that we are interested in and where H ′ is a small nondegenerate perturbation of H, the strategy is then to use a commutative diagram where F 0 ≤ H ′ ≤ F 1 and F 0 and F 1 are Hamiltonians with comparatively easyto-understand Floer complexes, in order to obtain information about HF [a,b] (H ′ ) and hence about the periodic orbits of H ′ . F 0 and F 1 are perturbed versions of Hamiltonians which depend only on the distance from the zero section, and Section 3 is concerned with learning about the Floer homologies of such Hamiltonians. The crucial results in this direction are Lemma 3.8 and its Corollary 3.10, which lead to significant topological restrictions on the cylinders that are involved in the boundary operators of the [a, b]-Floer complexes of F 0 and F 1 and in the chain map relating the complexes when b − a is sufficiently small; these topological restrictions are the primary factors that enable us to deal with the case that Ω| M is not spherically rational. In spirit, what makes these restrictions possible is that the Hamiltonian vector fields of the Hamiltonians considered in Section 3 are very nearly vertical, so that the cylinders u : R × (R/Z) → E(R) that are involved in the boundary operators and chain maps have π • u nearly pseudoholomorphic (where π : E(R) → M is the disc bundle projection). Thus requiring u and hence π • u to have low energy restricts the topology of π • u, and hence also that of u since π induces an isomorphism on π 2 . In Section 4 we fairly explicitly construct the Hamiltonians denoted above by F 0 and F 1 (and in Section 4 by F ǫ b and F ǫ a ), and then we leverage the results of Section 3 to prove Theorem 1.4. Here is an algebraic summary of the argument, with the geometry underlying the algebra deferred to Sections 3 and 4. For a judiciously chosen action interval [a, b] (equal to [c(ρ), d(ρ)] in the notation of Section 4), in grading 2r one has, for i = 0, 1, CF [a,b] 2r (F i ) = Z 2 ⊕ N i , where the Z 2 is generated by a "fiberwise-capped" (in the terminology of Section 3) periodic orbit x i , while the (typically infinitely-generated) summand N i is generated by periodic orbits with a capping other than the fiberwise capping. There are chain maps Φ F1 2r (F i ) are contained in N i . We then show in the proof of Proposition 4.1 that, if our result were false, one could construct a cycle c ′ ∈ N 0 with the property that would need to be a boundary, which cannot be the case since the foregoing implies that Φ F1 F0 (x 0 − c ′ ) / ∈ N 1 . The appendix is concerned with some facts about local Hamiltonian Morse-Bott Floer homology, which are probably at least similar to results known to experts, and which are needed in the proof of Proposition 4.2.
Acknowledgements. I am grateful to B. Gürel for helpful conversations. Most of this work was carried out while I was at Princeton University.

Filtered Floer homology in symplectic disc bundles
As our input we take: • a closed connected symplectic manifold (M, ω 0 ) of (real) dimension 2m; • a Morse function h : M → R having just one local maximum (i.e., just one critical point of index 2m; Theorem 8.1 of [23] shows how to construct such an h). In the situation of case (ii) of Theorem 1.4 we will also assume that −h has no critical points of index 1, so that h has no critical points of index 2m − 1; • a Hermitian vector bundle π : E → M with (complex) rank r; and • a unitary connection A on E, which in particular provides a splitting T E = T hor E ⊕ T vt E. We assume that the connection A is trivial on some neighborhood of the (finitely many) critical points of h. Where ·, · denotes the Hermitian inner product on E, define L : E → R by L(x) = 1 4 x, x , where v vt denotes the vertical part of v ∈ T x E as given by the connection A. Routine computations then show that: (i) dθ ∈ Ω 2 (E) restricts to each C r fiber of E → M as the standard symplectic form on C r given by the imaginary part of the Hermitian metric; where F A ∈ Ω 2 (M, u(E)) denotes the curvature 2-form of the Hermitian connection A.
. Let J 0 be an ω 0 -compatible almost complex structure on M ; this determines a metric g 0 on M by g 0 (v, w) = ω 0 (v, J 0 w), in particular allowing us to measure the , ω := π * ω 0 + dθ defines a symplectic form on E(R), which tames the almost complex structureJ on E obtained by lifting J 0 to T hor E and using the complex vector bundle structure on T vt E. Choose an R such that this is the case. Below, unless otherwise noted, we always measure distances in E(R) using the metric g on E(R) given by g(v, w) = 1 2 (ω(v,J w) + ω(w,Jv)); by decreasing R if necessary, for convenience let us also assume that the g-distance between any two points in the same fiber of E(R) is equal to their distance as considered within their common fiber using the Hermitian metric on the fiber.
We wish to construct a version of filtered Floer homology for Hamiltonians of a particular kind on E(R). Let L be the space of equivalence classes [γ, w] where γ : and only if the classes [ω], c 1 (T E(R)) ∈ H 2 (E(R); Z) both evaluate trivially on the sphere obtained by gluing w ′ to w orientation-reversingly along their common boundary γ. Given an arbitrary Hamiltonian H : (R/Z) × E(R) → R, the action functional A H : L → R is then defined by The critical points of A H are precisely those [γ, w] where γ is a contractible 1periodic orbit of the (time-dependent) Hamiltonian vector field X H , where our sign convention is that dH = ω(X H , ·). For real numbers a < b, the filtered Floer complex CF ∂u ∂s + J t (u(s, t)) ∂u ∂t − X H (t, u(s, t)) = 0 which connect two generators of CF [a,b] (H) (J t is a t-parametrized family of almost complex structures, which in practice will be close to theJ of the previous paragraph). Of where B is a constant; supp(f ) ⊂ (R/Z) × E(α); δ is a small positive number, which in particular should satisfy δh • π C 2 ≤ α/4; and as before L(x) = 1 4 x, x and h : M → R is our fixed Morse function. Thus H has a fairly specific form on E(α; R) = E(R) \ E(α), but the presence of the term f means that we effectively assume nothing about the behavior of H on the interior of E(α); indeed our intention is to apply our work to Hamiltonians having arbitrarily large derivatives in certain subsets of E(α).

Lemma 2.2.
If H is as in (3) and if the C 1 path γ : [0, 1] → E(R) has image which intersects both E(α) and E(2α; R) then Proof. The hypothesis implies that we can choose t 0 , t 1 ∈ [0, 1] such that γ(t 0 ), γ(t 0 ) = α 2 , γ(t 1 ), γ(t 1 ) = 4α 2 ; for simplicity assume t 0 < t 1 . If necessary by increasing t 0 and decreasing t 1 we may assume that γ(t), So the Schwarz inequality gives Now a general ω-tame almost complex structure J on E(R) induces a metric , so that in our earlier notation |v| = |v|J . For a R/Z-parametrized path of ω-tame almost complex structures J t denote (We only work with almost complex structures such that this is finite).
There are constants C and α 0 , depending only on J t and the function δh : M → R, with the following property.
is a connected submanifold with boundary and that u : S → E(2α; 3α) satisfies the Floer equation (2) with H(t, x) = 2πL(x) + δh(π(x)). Suppose also that, for some Take α 0 small enough that for any x ∈ [0, 1] × (R/Z) × E(R/2) ⊂Ẽ the exponential map at x for the Riemannian manifold (Ẽ, gJ ) is an embedding on the ball of radius α 0 in T xẼ . That u : S → E(2α; 3α) satisfies the Floer equation (2) is equivalent to the statement that the mapũ : S →Ẽ defined byũ(z) = (z, u(z)) isJ-holomorphic; further we have (for any subsurface S ′ ⊂ S) But our hypothesis implies that there is z 0 ∈ S such that z 0 , z 0 Theorem 2.4. There are constants D and α 0 , depending only on J and the function δh : M → R, with the following property. Let α ≤ α 0 , and let H be of form (3). Suppose that u :

Proof. Let
obviously Z is an open subset of R. If s ∈ Z, then either γ(t) = u(s, t) satisfies the hypothesis of Lemma 2.1 or else γ satisfies the hypothesis of Lemma 2.2. So (since (2) shows that ∂u ∂s Jt = ∂u There remains the case that Z has measure less than Cα 2 /2 (by lowering α 0 if necessary, assume that is then a submanifold with boundary of R × (R/Z), at least one of whose connected components, say S, has the property that ζ( Thus taking D = min{8C/R 2 , 9C J t −2 /16} completes the proof. [32] cited therein shows that, if the J t are chosen in a suitably small C 2 -neighborhood of the standard almost complex structureJ, and if the function δh is chosen in a sufficiently small C 2 -neighborhood of 0, then the constants α 0 , C, and D of Lemma 2.3 and Theorem 2.4 can be taken independent of the particular J t and δh from within these neighborhoods. (Alternately, at least if Dα 4 is less than the minimal energy of a J-holomorphic sphere in E(R), this can be seen as a direct consequence of Gromov compactness). Also, since the proof of Theorem 2.4 makes use only of the behavior of u on u −1 (E(α; 3α)), it is enough to assume that the restriction of u to u −1 (E(α; 3α)) satisfies (2).

Floer homology.
With the above C 0 -estimate established, the definition of our Floer groups becomes an application of standard machinery. Let be equal to one-half of the minimal energy of a nonconstantJ-holomorphic sphere in E(R); Gromov compactness of course implies that this is a positive number and that, if J t is a t-parametrized family of almost complex structures which are sufficiently C 2close toJ then, for all t, any nonconstant J t -holomorphic sphere will have energy at least . Choose any α > 0 with the property that α < α 0 and Dα 4 < (where, as in Remark 2.5, α 0 and D are chosen to satisfy Theorem 2.4 for any J t and δh sufficiently C 2 -close toJ and 0). Let H be a Hamiltonian of the form (3) and let a < b be real numbers such that b − a < Dα 4 . Note that the form we are assuming for H implies that all 1-periodic orbits γ : R/Z → E(R) of the Hamiltonian vector field X H of H are contained within the region E(α); assume furthermore that H has the property that all of its one-periodic orbits γ are nondegenerate in the sense that, where φ H is the time-one map of X H , the linearization dφ H : T γ(0) E(R) → T γ(0) E(R) does not have 1 as an eigenvalue. Use the notation [γ, w] to denote the equivalence class of a (nondegenerate) contractible one-periodic orbit γ of X H together with a nullhomotopy w : and c 1 vanish on the sphere obtained by gluing w and w ′ orientation-reversingly along their common boundary γ. Since γ is nondegenerate, any such object [γ, w] has a well-defined Maslov index µ H ([γ, w]); we adopt the conventions of Section 2 of [29] for the definition of this index. For any integer k, define In the usual way, one then defines the matrix elements of the Floer boundary operator ∂ and otherwise equal to the number modulo two of solutions (modulo s-translation) u : R × S 1 → E(R) to (2) for a generic path J t of almost complex structures C 2 -close toJ, having the property that u(s, ·) → γ ± as s → ±∞ and [γ + , w − #u] = [γ + , w + ] where γ − #u denotes the disc obtained by gluing the cylinder u to the disc w − along their common boundary component γ − . Note that any such u has The fact that Energy(u) < precludes the bubbling off of holomorphic spheres in sequences of such u, while the fact that Energy(u) < Dα 4 implies, via Theorem 2.4, that all such u are a priori contained in the region E(3α). Gromov-Floer compactness and gluing then yield in the standard way that (∂  (3), and let J ± t be families of almost complex structures close toJ. A monotone  (3) with the function δh independent of s (in particular, the vector fields X H s all restrict in the same way to E(α; R)); since we assume that ∂H s ∂s ≥ 0. In particular, since Energy(u) ≥ 0 we have (H ± ) and so have actions differing by at most b − a < Dα 4 < , we have Energy(u) < min{ , Dα 4 }. So, just as with the definition of the Floer boundary operator, Gromov compactness together with the radius-energy estimate Theorem 2.4 establish compactness of the space of such u (since, as in the last sentence of Remark 2.5, the fact that X H s is independent of s outside E(α) means that u actually satisfies (2) on u −1 (E(α; 3α))). Provided that J ± t ∈ J reg (H ± ) and that J s,t is chosen from a certain setJ reg (H s ) having residual intersection with a small neighborhood of the constant path atJ, this allows us to define a map Φ H s ,Js,t : CF (H + ) is independent of the choices of H s and J s,t (in fact, any two choices of (H s , J s,t ) induce chain homotopic maps Φ H s ,Js,t ); and We remark at this point in the situation of greatest interest to us, the "period map" [ω] : π 2 (E(R)) → R will have dense image, as a result of which for each periodic orbit γ of X H there will be infinitely many different choices of w for which [γ, w] is a generator of CF (H + ) will typically be far from being either injective or surjective. However, in suitable situations, it will still be possible to obtain useful information about Ψ H + H − . The following simple proposition will be helpful to us in the proof of Proposition 4.1 below.
Proof. First we note that the hypothesis implies that Indeed, where φ s,t is the time-t map of the Hamiltonian flow of H s , and φ ±,t is the time-t map of the Hamiltonian flow of H ± , one has is the path of symplectic matrices obtained by setting A ± (t) equal to the linearization at γ(0) of the map φ ±,t , as measured via a symplectic trivialization of w * T M , and µ CZ is the Conley-Zehnder index (see [29], Section 2.6). But, where A s : [0, 1] → Sp(2n) is defined similarly with φ ±,t replaced by φ s,t , the paths A s give a homotopy from A − to A + ; moreover the fact that γ is nondegenerate for each H s shows that none of the matrices A s (1) has 1 as an eigenvalue. Hence the homotopy invariance of the Conley-Zehnder index ( [29], Section 2.4) shows that Given this, the fact thatγ(t) = X H s (γ(t)) for every s implies that setting u(s, t) = γ(t) gives an index-zero solution to (4), regardless of the choice of J s,t . Meanwhile, if u is any solution to (4) which is not of the form u(s, t) = γ(t), asymptotic say to [γ ′ , w ′ ] as s → ∞ and to [γ, w] as s → −∞, then we must have ∂u ∂s (s, t) = 0 for some (s, t) ∈ R × (R/Z), and hence R×(R/Z) ∂u ∂s , the proposition follows directly from the definition of the map Φ H s ,Js,t .
(No considerations of sign are needed since we are working modulo two.)

Special features of the Floer complexes of certain Hamiltonians
The proof of Theorem 1.4 requires us to understand certain properties of the Floer complexes of Hamiltonians on E(R) having a particular form.
As a first step, we prove the following elementary fact.
Then there is δ 0 > 0 having the following significance. If 0 < δ < δ 0 , and if then x is the constant loop at some zero of V .
Remark 3.2. When ρ = 0, the Yorke estimate [33] shows that δ 0 can be taken equal is then an open neighborhood of V −1 ({0}) with the property that, as long as δ 0 ≤ β 0 , if for some t 0 we have x(t 0 ) ∈ U ′ , then for all t we have x(t) ∈ U (and soẋ(t) = V (x(t)) by the hypothesis on x). Thus if x(R/Z) ∩ U ′ = ∅, then x actually satisfies the hypothesis of the proposition with ρ = 0, so that (using any δ 0 ≤ min{β 0 , 2π/C}) the proposition holds for x by the Yorke estimate mentioned in Remark 3.2.
As such, it suffices to consider those x : R/Z → M whose images are contained is a neighborhood of the origin containing a ball of radius injrad(M, g 0 ), let φ : W → M be a normal coordinate chart with φ( 0) = x(0). Let B 0 = φ(W ). Let V 0 denote the vector field on B 0 obtained by the parallel transport of V (x(0)) along geodesics in and where ζ > 0 is some number depending only on the metric g 0 (in particular, ζ can be taken independent of x(0)). So, for each t, we have (by the hypothesis on x) provided that δ is less than some constant δ 0 which depends only on ζ, γ 0 , ρ, ǫ, is thus a path contained entirely within a neighborhood W of the origin in R dim M whose velocity vector has strictly positive inner product with the nonzero constant vector fieldV 1 on W , and this precludes the possibility that x(1) = x(0), contrary to the hypothesis of the theorem. This shows that in fact if δ 0 is small enough every x satisfying the hypotheses of the theorem must pass through U ′ , and so must be a constant loop at a zero of V by our earlier remarks.
Recall that at the outset we have fixed a Morse function h : M → R, and chosen our connection A to be trivial in a neighborhood of the set of critical points of h.
where f : R → R is a smooth function (and again L(x) = 1 4 x, x ). Let us consider the Hamiltionian vector fields X H of δh-Hamiltonians H. Of course X H = X f •L + δX h•π . We have noted earlier that (under the canonical identification of E(R) with the R-disc bundle in the restriction of T vt E to the zero section of E) X 2πL (x) = −πix, so that In particular We find, for w ∈ T hor x E, and so |. This yields: Proof. First note that as long as R 2 < (2 F A ∞ ) −1 , for each w ∈ T hor E(α 0 ) we have |dθ(w,J w)| ≤ 1 2 ω 0 (π * w, J 0 π * w), and so 1 2 |π * w| 2 ≤ |w| 2 ≤ 3 2 |π * w| 2 . So (6) implies that, where y(t) = π(x(t)), Now the zeros of Y h are just the critical points of h; recall that we assumed the connection A to be trivial (and hence to have vanishing curvature) on a neighborhood of each of these points. This, together with the hypothesis on R (and the fact that x, so that the time-one map φ H = φ 1 of X H restricts to the fibers E p over critical points p of h as the fixed points of φ H are precisely the points lying on a sphere of radius ℓ in the fiber over a critical point of h, where ℓ ∈ L f . In particular (unless L f = {0}) H will be a degenerate Hamiltonian, so if we wish to take its Floer homology we will need to perturb it. We assume, as will be the case in our application, that 0 < f ′ (0) < 4π and that, for each ℓ ∈ L f \ {0} we have f ′′ (ℓ) = 0, which in particular implies that L f is a discrete set. Let , where φ t denotes the time t-flow of X H . Thus (as a standard calculation shows) H ǫ has time-1 map φ H ǫ equal to φ H • ψ f,ǫ . Of course, as ǫ → 0, ψ f,ǫ converges to the identity in any C k -norm, and so any fixed points of φ H ǫ must, for sufficiently small ǫ, be close to fixed points of φ H . Since β f vanishes near the 0-section of E(R), the fixed points of φ H ǫ near the zero section coincide with those of φ H , and thus are precisely the critical points of h : M → R. Now we consider the fixed points of φ H ǫ away from the zero section. For small ǫ, all of these fixed points of φ H ǫ must be contained in the interior of a region where χ • π = β f • L = 1. Now for any z = (u, x 1 + iy 1 , . . . , x r + iy r ) in this region, we have ψ ǫ,f (u, x 1 + iy 1 , . . . , x r + iy r ) = (u, (x 1 + ǫ) + iy 1 , . . . , x r + iy r ).
Consideration of the first coordinate shows that f ′ (L(z))/2 must not be a multiple of 2π, in view of which this forces x 2 = y 2 = . . . = x r = y r = 0, and Let ℓ ∈ L f \ {0} (so f ′ (ℓ) = 4πk for some integer k); we know a priori that any fixed point z of H ǫ must (assuming ǫ is small enough) have L(z) close to one such ℓ.
In particular, where φ t H ǫ denotes the time-t flow of H ǫ , each of the 1-periodic orbits γ j,0 (t) = φ t H ǫ (v j,0 ) or γ ± j,ℓ (t) = φ t H ǫ (v ± j,ℓ ) lies entirely in one of the fibers of E(R) → M over a critical point of h. Recall that the generators of the Floer complex of H ǫ are equivalence classes [γ j,0 , w] or [γ ± j,l , w] where w : D 2 → E(R) is a nullhomotopy (or a "capping") of the orbit γ j,0 or γ ± j,l . The fact that the orbits are all contained in single fibers allows us to define, for each orbit γ, a "fiberwise capping" w 0 : D 2 → E(R) by w(se 2πit ) = sγ(t) (in particular if γ is the constant orbit at a critical point of h on the zero section then this is just the constant map at the zero section). A general generator for the Floer complex then has the form

The actions and Maslov indices are related by
(see Section 2 of [29] for the conventions we use on the Maslov index; in particular, if γ is the constant orbit at a critical point p of a Morse function G with its trivial capping, in our convention its Maslov index µ G is equal to its Morse index as a critical point of −G). Thus to understand the actions and gradings of the generators of the Floer complex it is enough to understand the actions and gradings of the generators [γ, w 0 ] where w 0 is the fiberwise capping of γ (where γ ranges among the γ j,0 and γ ± j,ℓ ). A routine calculation using the characterization of the Maslov and Conley-Zehnder indices from Sections 2.4 and 2.6 of [29] gives the following result for the Maslov indices of the [γ, w 0 ] (we leave this calculation to the reader; see Section 5.2.5 of [12] for a sketch of a similar calculation, but note that we use different conventions both for Hamiltonian vector fields and for the normalization of the Maslov index): Proposition 3.6. In the notation of Proposition 3.5, for ℓ ∈ L f \ {0} write Also let n = m + r = 1 2 dim E(R). Then the Maslov indices for the periodic orbits of H ǫ with their fiberwise cappings w 0 are, for ǫ sufficiently small, given by (In particular, for each ℓ ∈ L f \ {0} the orbits [γ ± j,ℓ , w 0 ] have Maslov indices differing from each other by 2r − 1, as would be expected since they are the two orbits that remain from an S 2r−1 -family of periodic orbits of H after perturbing H to H ǫ ).

3.2.
Restrictions on Floer trajectories. We will be needing some information about the Floer complexes CF [a,b] (H ǫ ) of perturbations H ǫ of particular δh-Hamiltonians H = f • L + δh • π; in our application the length of the interval [a, b] will be rather small. Lemma 3.8 below will be a considerable help in this direction; that result, in turn, will depend on the following: . In view of this, we can choose a parameter e ′ 0 > 0 with the property that if u is J 0holomorphic and (−1,2)×R/Z |du| 2 ≤ e ′ 0 then u([0, 1] × R/Z) has diameter at most β/2; our parameter e 0 will be equal to the minimum of e ′ 0 and half the minimal energy of a nonconstant J 0 -holomorphic sphere.
The almost complex structuresJ n have been constructed so as to ensure that the W n areJ n -holomorphic maps. Now where i is the standard almost complex structure on (−1, 2) × (R/Z), theJ n converge in C 0 -norm to the product almost complex structure i × J 0 . Hence, by Theorem 1 of [16], after passing to a subsequence the W n converge, at least modulo bubbling, to a i × J 0 -holomorphic curve. Now the fact that the W n have form (s, t) → (s, t, w n (s, t)) implies that the limiting bubble tree has a principal component of form (s, t) → (s, t, w ∞ (s, t)) (having energy, as measured by the symplectic form ds ∧ dt + ω 0 and the almost complex structure i × J 0 on (−1, 2) × (R/Z) → (−1, 2) × (R/Z) × M , equal to Area((−1, 2) × (R/Z)) + (−1,2)×R/Z |dw ∞ | 2 ), with any bubbles given by J 0 -holomorphic spheres in fibers n ≤ Area((−1, 2) × (R/Z)) + e 0 by hypothesis, so the fact that e 0 is strictly less than the minimal energy of a nonconstant J 0 -holomorphic sphere implies that no bubbles can appear in the limit. As such, we in fact have w n → w ∞ in L 1,p loc for each p < ∞ (by Corollary 1.3 of [16]), and therefore also in C α ([0, 1] × (R/Z); M ) for each 0 < α < 1 by the Sobolev lemma. By Fatou's Lemma, we have that and therefore that w ∞ ([0, 1]×(R/Z)) has diameter at most β/2. So since w n → w ∞ in C 0 it follows that, for sufficiently large n, w n ([0, 1]×(R/Z)) has diameter at most β, in contradiction with the assumption that all w n ([0, 1] × (R/Z)) had diameter larger than β. This contradiction proves the lemma.
Assume furthermore that H + and H − are both nondegenerate Hamiltonians, with the property that each one-periodic orbit γ ± of X H ± lies in just one fiber (depending on γ ± ) of the projection π : E(R) → M , so that in particular each γ ± has a fiberwise capping (w 0 ) γ± : D 2 → E(R). Suppose that u : R × (R/Z) → E(R) is a solution to the Floer equation such that u(s, ·) → γ ± as s → ±∞. Suppose that ker c1(T E(R)),· ∩ker [ω],· and the notation signifies equivalence inL). Then Remark 3.9. We emphasize that the constants η 1 , e 1 are independent of the functions f s .
Proof. Applying the linearization π * of the bundle map π : E(R) → M to (9) and using that π * •J = J 0 • π * and that each X fs•L is a vertical vector field, we obtain that ∂ ∂s u(s, t)).

Then
• For generic families of almost complex structures J t sufficiently close toJ, the boundary operator ∂ H,Jt : CF for some numbers a γγ ′ . • If (H s,ǫ , J s,t ) is a regular monotone homotopy with J s,t sufficiently C 2 -close to the constant path atJ, then the chain map Φ H s,ǫ ,Js,t : CF Proof. A priori, the maps ∂ H,Jt and Φ H s,ǫ ,Js,t have the form where a γγ ′ ,B counts cylinders u satisfying the appropriate equation with [γ ′ , (w 0 ) γ #u] = [γ ′ , (w 0 ) γ ′ #B]; the content of the corollary is that the only nonzero a γγ ′ ,B are those with B = 0. Since our restriction to the action window [a, b] forces the u being considered to have energy less than e 1 , this is essentially an immediate consequence of Lemma 3.8 (with the H s of Lemma 3.8 set equal to H ǫ independently of s for the first part and to H s,ǫ for the second part), except that Lemma 3.8 concerned solutions to analogues of the Floer equations used to define ∂ H,Jt and Φ H s,ǫ ,Js,t with J t and J s,t replaced by the nongeneric almost complex structureJ. However, if our corollary were false then applying Gromov compactness to solutions of the relevant Floer equation using almost complex structures equal to J n t or J n s,t where J n t , J n s,t →J would yield a solution u (possibly just one piece of a broken trajectory) to (9), whose energy and topology would contradict Lemma 3.8.

Detecting periodic orbits
We now turn to the proof of Theorem 1.4. Throughout this section, we assume that either c 1 (T E(R)) = 0 mod torsion or else that h has no critical points of index 2m − 1. Our strategy is to some extent modeled on that used in Section 6 of [13] to prove the result in the spherically rational case, though of course the possible irrationality of the symplectic form will introduce additional subtleties. Let K : E(R) → R be an autonomous Hamiltonian on E(R) which attains a Morse-Bott nondegenerate minimum (say equal to 0) along the zero section. After possibly shrinking R, this implies that there are constants C 1 , C 2 > 0 such that in particular, the only critical points of K in E(R) are the points on the zero section.
Our intention is to produce a periodic orbit for X K on an energy level in an arbitrary open interval (3ρ − β, 3ρ + β) where ρ > 0 is sufficiently small and β ≪ ρ. Put Throughout the following, ρ will be taken small enough that 30ρ C1 < α 0 and C(ρ) < min{e 1 , Dα 4 0 }, where D and α 0 are as in Theorem 2.4 and e 1 is as in Corollary 3.10.
defines a g ′ ρ,β (γ(0))-periodic orbit of X K . One property that the still-to-be-specified parameter η will have is that X K has no nonconstant periodic orbits of period at most 2η (such an η, depending only on the C 2 -norm of K, exists by the Yorke estimate [33]), so this means that any nonconstant 1periodic orbit of X H0 will correspond to a periodic orbit of X K at some energy level in (3ρ − β, 3ρ + β).
Introduce two smooth strictly increasing functions f a , f b : [0, ∞) → [0, ∞), depending on ρ but not on β, having the following properties. (see Figure 1; 'b' stands for "below," and 'a' for "above"): • For s ∈ [0, ρ/C 2 ], f a (s) = C 2 ηs; • For 2ρ/C 2 < s < 20ρ/C 1 we have 0 < f ′ a (s) ≤ C 2 η, and for 4ρ . The defining properties of the constants C 1 and C 2 , together with the above properties of f a , f b , and g ρ,ǫ , imply in particular that Choose a smooth, monotone increasing function ζ : R → [0, 1] such that ζ(s) = 0 for s ≤ 5ρ/C 1 and ζ(s) = 1 for s ≥ 10ρ/C 1 and definẽ Thus f b • L ≤H 0 ≤ f a • L, with equality in the region where L ≥ 20ρ/C 1 . Now outside the region where 3ρ − β ≤ K(x) ≤ 3ρ + β, the Hamiltonian vector field XH 0 coincides with a vector field on E(R) whose C 1 -norm is bounded by an ρ-dependent constant times η. We now specify η by requiring that this bound on the C 1 -norm be less than one-half times the minimal C 1 -norm of a vector field on E(R) having a nonconstant periodic orbit of period at most 1, as given by the Yorke estimate. Thus any periodic orbit of XH 0 with period at most 2 intersects the region {3ρ − β < K < 3ρ + β}; in fact, since XH 0 is tangent to each of the level surfaces K −1 ({y}) with y < 5ρ, any such orbit must be entirely contained in {3ρ − β < K < 3ρ + β}. We also require that η be small enough that f a and f b as constructed above have the property that all nonconstant periodic orbits of f a • L (resp. f b • L) with period less than 2 are contained in the region where For sufficiently small δ > 0, then, the Hamiltonians for ǫ sufficiently small. Also (for sufficiently small δ), all 1-periodic orbits of H are nondegenerate except for possibly those contained in the region where 3ρ − β ≤ K ≤ 3ρ + β (which in turn is contained in the region where 5ρ 2C2 ≤ L ≤ 7ρ 2C1 ), so there are arbitrarily small nondegenerate perturbations H ′ of H still having the property that F ǫ b ≤ H ′ ≤ F ǫ a . (We'll always assume that H ′ − H is supported in {3ρ − β ≤ K ≤ 3ρ + β}.) So, for d − c sufficiently small, we have a commutative diagram (10) HF .
The construction of f a and f b shows that there are unique real numbers ℓ a , ℓ b with the property that Also, since f a ( 2ρ it follows that Order the critical points p i of the Morse function h so that p 1 is the unique local maximum of h. Adapting the notation of the previous section, F ǫ a and F ǫ b then have periodic orbits γ − 1,ℓa , γ − 1,ℓ b lying in the fiber over p 1 . Proposition 3.6 then gives that, where as usual w 0 denotes the fiberwise capping, We also see that C1 we see that In particular, d(ρ) − c(ρ) = 26πρ C1 is less than the parameter e 1 of Corollary 3.10; below, we always assume that the perturbing parameters δ and ǫ are sufficiently small. The main result of this section is: There is a 1-periodic orbit γ of X H which is contained in the region {3ρ − β ≤ K ≤ 3ρ + β} and a map w : D 2 → E(R) with w| ∂D 2 = γ such that the Salamon-Zehnder index ∆([γ, w], H) satisfies −2r ≤ ∆([γ, w], H) ≤ 2m + 1. Theorem 1.4 follows quickly from Proposition 4.1: recall that H was taken to have the form H =H 0 + δh • π where δ > 0 was arbitrarily small, so applying the Arzelà-Ascoli theorem to a sequence of orbits γ δ k (with capping discs w δ k ) given by Proposition 4.1 where δ k ց 0 gives a 1-periodic orbit γ for XH 0 in the region {3ρ − β ≤ K ≤ 3ρ + β}. But in this region we haveH 0 = g ρ,β • K, so evidently γ is a 1-periodic orbit for X g ρ,β •K , which in turn implies that X K has a periodic orbit (of some, possibly large, period, but with the same Salamon-Zehnder index by Lemma 2.6 of [13]) contained in the region {3ρ − β ≤ K ≤ 3ρ + β} (for every sufficiently small parameter β). By the continuity of the Salamon-Zehnder invariant the periodic orbit γ of X K that we have obtained has a capping disc w (obtained by gluing a thin cylinder from γ δ k to γ to the end of w δ k where γ δ k is close to γ) with ∆([γ, w], K) ∈ [−2r, 2m + 1]. This proves Theorem 1.4 (interchanging our 3ρ and β with ρ and ǫ/2 in the notation of Theorem 1.2) for the case that (P, Ω) = (E(R), ω) and hence, using the Weinstein neighborhood theorem as in the introduction, for arbitrary (P, Ω).
It remains to prove Proposition 4.1. The proof depends on the following propositions (recall that p 1 is the unique local maximum of h): . So in fact generators of types (ii) and (iv) above are the only ones that actually belong to CF (F ǫ a ). Assume first that we are in the case where c 1 (T E(R)) is a torsion class. In this case, we can use the results discussed in the appendix at the end of this paper to deduce that no generators of type (ii) or (iv) can have boundary containing [γ − 1,ℓa , w 0 ] with nonzero coefficient. Indeed, in the notation of the appendix, let N denote the sphere bundle in E(R) over M of radius ℓ a , and let the (R/Z)-action be given by t · x = e −2πit x. F ǫ a is then a C 2 -small perturbation of f a • L (which plays the role of the Hamiltonian H 0 in the appendix), so for sufficiently small δ and ǫ the Hamiltonian F ǫ a can be used to compute the local Floer homology HF loc * (f a • L, U) (U is, as in the appendix, a suitable small neighborhood in the contractible loopspace of the set of 1-periodic orbits which foliate N ). Write ∂ loc for the differential on the local Floer complex CF loc * (F ǫ a , f a • L, U) (as in the appendix, this complex is generated by periodic orbits of X F ǫ a which are contained in U, with no "capping data"). Suppose that γ is an orbit of type (ii) or (iv) above. We then see that any of the [γ, w 0 ] has action differing from that of [γ − 1,ℓa , w 0 ] by an amount tending to zero with δ and ǫ, so that any cylinder u : R × (R/Z) → E(R) contributing to the matrix element 1,ℓa , w 0 ] has its energy tending to 0 as ǫ, δ → 0. So a Gromov compactness argument shows that, for ǫ, δ sufficiently small, each u(s, ·) belongs to U, and so u contributes to ∂ loc γ, γ − 1,ℓa . Conversely, if u contributes to ∂ loc γ, γ − 1,ℓa , then Lemma 3.8 implies Since the counting prescriptions for these u are identical whether they are considered as contributing to ∂ or to ∂ loc , this shows that for every orbit γ of type (ii) or (iv). But by Theorem 5.3, HF loc * (f a • L, U) is equal as a relatively Z-graded group to the singular homology of the sphere bundle N . (The relative grading is by Z because of our assumption on c 1 .) Now the (strictly) largest relative grading difference gr(γ, γ ′ ) between any two generators of CF loc * (F ǫ a , f a • L, U) is attained where γ = γ + j,ℓa and γ ′ = γ − 1,ℓa where j is chosen so that p j is a local minimum of h, and this relative grading difference is dim N = 2m + 2r − 1. So in order for HF loc * (f a • L, U) to have Z 2 -summands in gradings separated by dim N , it must be that some element of CF loc * (F ǫ a , f a • L, U) having the same relative grading as γ − 1,ℓa represents a nontrivial class in HF loc * (f a • L, U). But since γ − 1,ℓa is the only generator for CF loc * (F ǫ a , f a • L, U) in its relative grading, this is possible only if, for all γ, ∂ loc γ, γ − 1,ℓa = 0. So if γ is an orbit of type (ii) or (iv) we have ∂[γ, w 0 ], [γ − 1,ℓa , w 0 ] = 0, which completes the proof that [γ − 1,ℓa , w 0 ] has the stated properties. We now turn to the alternate case, where we make no assumption on c 1 but we do assume that h has no critical points of index 2m − 1. This assumption obviously eliminates orbits of type (ii), so the only orbit to worry about is γ = γ + 1,ℓa (in the case r = 1; if r = 1 the proof is now complete). The corresponding matrix element for ∂ can then likewise be shown to vanish by a local Floer homology argument: here we view F ǫ a as a C 2 -small perturbation of H 0 = f a •L+δh•π and use for N the circle of radius l a in the fiber over the maximum p 1 of h. CF loc * (F ǫ a , f a •L+δh•π, U) then has γ + 1,ℓa and γ − 1,ℓa as its only two generators and has homology equal (at least as a relatively (Z/2)-graded group) to the singular homology of the circle, in view of which γ − 1,ℓa represents a nontrivial class in local Floer homology and an argument identical to that used in the c 1 = 0 case then proves the result.   (3), let µ > 0, and suppose that, for all t ∈ R/Z and all x ∈ E(R), we have be a regular monotone homotopy from (G, J t ) to (G ′ , J t ), and let (H s 2 , J ′ s,t ) be a regular monotone homotopy from (G ′ , J t ) to (G + µ, J t ). Then, with respect to the identification (11), the map is chain homotopic to the homomorphism for any s this is a strictly increasing function from [0, ∞) onto itself, so let f s be the inverse of φ s . We then have f a ≤ f s ≤ f b , f s = f a for s ≥ 1, f s = f b for s ≤ 1, and ∂fs ∂s ≥ 0. For F ǫ s we take the s-parametrized family (f s • L + δh • π) ǫ (where ǫ is appropriately small and the notation means that we perturb f s • L + δh • π using the parameter ǫ as in the previous section). Now since in the only regions where its derivative ever approaches an integer multiple of 4π the function f s has a graph which is just a translated-to-the-rightversion of the graph of f a in the corresponding region, the 1-periodic orbits of any given X F ǫ s correspond precisely to the one-periodic orbits of X F ǫ a (but with generally larger values of the parameter ℓ). In particular, each F ǫ s is nondegenerate a nondegenerate Hamiltonian and has each 1-periodic orbit contained in a fiber over a critical point of h, and its only nonconstant 1-periodic orbit γ such that µ F ǫ s ([γ, w 0 ]) = 2r is the orbit γ = γ − 1,ℓ(s) , where ℓ(s) decreases from ℓ b when s ≤ −1 to ℓ a when s ≥ 1.
Let µ > 0 have the property that c(ρ) (F ǫ s ), for all j, t, x, and let J t ∈ ∩ N +1 j=1 J reg (F ǫ sj ). Let (G s j,1 , J j,s,t ) be a regular monotone homotopy from (F ǫ sj , J t ) to (F ǫ sj+1 , J t ) and let (G s j,2 , J ′ j,s,t ) be a regular monotone homotopy from (F ǫ sj+1 , J t ) to (F ǫ sj + µ, J t ), with G s j,1 and G s j,2 chosen to be of the same form as the H s,ǫ in Corollary 3.10. We then obtain chain maps But as noted earlier, 1,ℓ(s) , w 0 ] = 0. Thus (12) shows that But since for i = j, j + 1, γ − 1,ℓ(si) is the only 1-periodic orbit γ of X F ǫ s i having both µ . Since we are working over Z 2 we must have c j = c ′ j = 1. From this it follows that Suppose, to get a contradiction, that X H had no 1-periodic orbits γ contained in {3ρ−β ≤ K ≤ 3ρ+β} having capping discs w with −2r ≤ ∆([γ, w], H) ≤ 2m+1. As noted earlier, assuming that δ > 0 is small enough, the construction of H guarantees that any 1-periodic orbit of X H which intersects {3ρ − β ≤ K ≤ 3ρ + β} is in fact contained in {3ρ − β < K < 3ρ + β}. The continuity of the Salamon-Zehnder index (and the Arzelà-Ascoli theorem) hence implies that if H ′ − H C 2 is sufficiently small then X H ′ also has no 1-periodic orbits γ intersecting {3ρ − β ≤ K ≤ 3ρ + β} and having capping discs w with −2r ≤ ∆([γ, w], H) ≤ 2m + 1. Take for H ′ a sufficiently small nondegenerate perturbation of H, with H ′ − H supported in {3ρ − β ≤ K ≤ 3ρ + β} (since the construction of H shows that its constant 1periodic orbits are all nondegenerate, while all of its nonconstant 1-periodic orbits are (for small δ) contained in the interior of {3ρ − β ≤ K ≤ 3ρ + β}, a standard argument shows that nondegenerate Hamiltonians will form a residual subset of the space of time dependent H ′ coinciding with H outside {3ρ − β ≤ K ≤ 3ρ + β}). If [γ ′ , w ′ ] is any generator of CF (where the maximum of the empty set is defined to be −∞). Note that if (G s ,J s,t ) is a regular monotone homotopy from (G − , J − t ) to (G + , J + t ), we have But then since [γ − 1,ℓ b , w 0 ] is the only generator of CF [c(ρ),d(ρ)] 2r (F ǫ b ) with the fiberwise capping (all other fiberwise-capped generators have Maslov indices unequal to 2r or actions outside [c(ρ), d(ρ)], as we have seen earlier), we have (13) [

Appendix: Background on local Floer homology for clean intersections
Let (P, Ω) be an arbitrary symplectic manifold, and suppose that H 0 : P → R is an autonomous Hamiltonian on P inducing a flow φ t H0 which has the property that fixed point set of φ 1 H0 has a connected component N ⊂ F ix(φ H0 ) such that • N is a compact submanifold of P , • there is a R/Z-action on a neighborhood of N , which preserves N , has orbits which are contractible in P , and has the property that for each x ∈ N we have t · x = φ t H0 (x); and • for each x ∈ N , ker(Id − (dφ H0 ) x ) = T x N .
Denote by L 0 P the space of contractible loops in P , N ⊂ L 0 P the subset consisting of (R/Z)-orbits through points of N , and U ⊃ N the closure of a neighborhood of N , which should be taken to be small in a sense to be specified presently. Let U ⊂ P be a tubular neighborhood of N , identified with the disc normal bundle to N with projection τ : U → N , and taken small enough that the only fixed points of φ 1 H0 in U are the points of N . U is then chosen small enough that every γ ∈ U has the properties that: (i) γ(t) ∈ U for all t, and (ii) with respect to a (R/Z)invariant Riemannian metric on N , the diameter of the loop t → (φ t H0 ) −1 (τ (γ(t))) is to be less than the injectivity radius of N . (Thus, every γ ∈ U has τ • γ close to a (R/Z)-orbit).
We've just shown that the first term on the right hand side above is zero, while the second term is zero since γ ± are loops in N and H 0 is constant on N .
Using the familiar formulas 2 Jt dsdt, this shows that our arbitrary u ∈ M J,H (U) (which is known a priori to be C 1 by standard arguments) must have ∂u ∂t = X H0 (u(s, t)) everywhere, in view of which u(0, ·) is a 1-periodic orbit of φ t H0 . Since the only such orbits belonging to U are the elements of N , this proves the proposition. Of course, Gromov compactness implies that {J t } > 0.
Proposition 5.2. (Cf. [5,Theorem 3]) Let 0 < δ < {J t }/2. Then there is η > 0 with the following properties. If H − H 0 C 2 < η, and if J ′ t is a (R/Z)-parametrized family of almost complex structures with J ′ t − J t C 2 < η for each t, then whenever ǫ < {J t } S ǫ J ′ t ,H (U) is contained in the interior of U. Further, for H − H 0 C 2 < η and J ′ t − J t C 2 < η, M ǫ J ′ t ,H (U) is independent of ǫ for all choices of ǫ ∈ [δ, {J t }). Proof. Suppose that H n → H 0 and J t,n → J t in C 2 -norm, and that u n ∈ M ǫ Jt,n,Hn (U) where ǫ < {J t }. Suppose that there were s n ∈ R having the property that u n (s n , ·) / ∈ U • . Gromov compactness applied to the maps (s, t) → u n (s − s n , t) then implies that the u n (s n , ·) converge to some element of S Jt,H0 (U) (bubbling is precluded because the u n have energy less than the minimal energy of any J tholomorphic sphere in U ). But we have shown that S Jt,H0 (U) = N , so since U • is a neighborhood of N this is a contradiction, which proves the first statement of the proposition.
For H − H 0 C 2 small, any one-periodic orbit γ of X H in U is close to a 1periodic orbit of X H0 , i.e., to a loop that belongs to N . As such, we can choose η small enough that (in addition to the first statement of the proposition holding), for H − H 0 C 2 < η, whenever γ ∈ U is a 1-periodic orbit of X H there exists a map Remark 5.4. Since, for the generators of CF loc * (H, H 0 , U), we take Hamiltonian periodic orbits without any "capping data," the grading of the local Floer chain complex should be understood as a relative grading by the cyclic group Z/Γ where Γ is twice the minimal Chern number of P . In particular, Theorem 5.3 is an isomorphism of relatively Z-graded groups when c 1 (T P ) vanishes.