Floer theory of disjointly supported Hamiltonians on symplectically aspherical manifolds

We study the Floer-theoretic interaction between disjointly supported Hamiltonians by comparing Floer-theoretic invariants of these Hamiltonians with the ones of their sum. These invariants include spectral invariants, boundary depth and Abbondandolo-Haug-Schlenk's action selector. Additionally, our method shows that in certain situations the spectral invariants of a Hamiltonian supported in an open subset of a symplectic manifold are independent of the ambient manifold.

1 Introduction and results.
The paper deals with Hamiltonian diffeomorphisms of symplectic manifolds, which model the Hamiltonian dynamics on phase spaces in classical mechanics. A central tool for studying Hamiltonian diffeomorphisms is Floer theory, which is an infinite-dimensional version of Morse theory applied to the action functional on the space of contractible loops. As such, Floer theory associates a chain complex to each Hamiltonian, which is generated by the critical points of the action functional and whose differential counts certain negative gradient flow lines, called Floer trajectories.
Our main object of interest is Floer theory for Hamiltonians supported in pairwise disjoint open sets, namely F = F 1 + ... + F N where F i is supported in U i and U 1 , . . . , U N are pairwise disjoint. On the level of dynamics, the Hamiltonian diffeomorphisms ϕ i corresponding to F i do not interact. The Hamiltonian diffeomorphism corresponding to F is the composition ϕ = ϕ 1 • · · · • ϕ N , and the diffeomorphisms ϕ i commute. However, it is unclear a priori whether in Floer theory there is any communication between the disjointly supported Hamiltonians F i . The Floer-theoretic interaction between disjointly supported Hamiltonians was studied by Polterovich [14], Seyfaddini [18], Ishikawa [12] and Humilière-Le Roux-Seyfaddini [11], mostly through the relation between invariants of the sum of Hamiltonians and invariants of each one. These works suggest that such an interaction should be quite limited. The main finding of this paper is a construction, on symplectically aspherical manifolds and under some conditions on the domains U i , of what we call a "barricade" -a specific perturbation of the Hamiltonians F i near the boundaries of U i , which prevents Floer trajectories from entering or exiting these domains. The presence of barricades limits the communication between disjointly supported Hamiltonians as expected. The construction is motivated by the following simple idea in Morse theory.
Given a smooth function F on a Riemannian manifold, that is supported inside an open subset U , one can perturb it into a Morse function f that has a "bump" in a neighborhood of the boundary, as illustrated in Figure 1. The negative gradient flow-lines of f cannot cross the bump, and therefore a flow-line starting inside U , and away from the boundary, remains there. On the other hand, flow-lines that start on the bump can flow both in and out of U . Since the Morse differential counts negative gradient flow-lines, such constraints can be used to gain information about it. This idea can be adapted to Floer theory on symplectically aspherical manifolds (that is, when the symplectic form ω and the first Chern class c 1 vanish on π 2 (M )), and under certain assumptions on the domain U . The resulting construction can be used to study Floer-theoretic invariants, such as spectral invariants and the boundary depth, of Hamiltonians supported in such domains. Spectral invariants measure the minimal action required to represent a given homology class in Floer homology. These invariants play a central role in the study of symplectic topology and Hamiltonian dynamics. Using the barricades construction, we prove that the spectral invariants with respect to the fundamental and the point classes of Hamiltonians supported in certain domains, do not depend on the ambient manifold. This result is stated formally in Section 1.1.1. Another application of the barricades construction concerns spectral invariants of Hamiltonians with disjoint supports. This problem was studied in [14,18,12] and lastly in [11], where Humilière, Le Roux and Seyfaddini proved that the spectral invariant with respect to the fundamental class satisfies a "max formula", namely, the invariant of a sum of disjointly supported Hamiltonians is equal to the maximum over the invariants of the summands. This property does not hold for a general homology class. However, using barricades we show that an inequality holds in general, see Section 1.1.2. A third application of this method concerns the boundary depth, which was defined by Usher in [19] and measures the maximal action gap between a boundary term and its smallest primitive in the Floer chain complex, see Section 1.1.3. We prove a relation between the boundary depths of disjointly supported Hamiltonians and that of their sum. The last application concerns a new invariant that was constructed by Abbondandolo, Haug and Schlenk in [1]. We give a partial answer to a question posed by them, asking whether a version of Humilière, Le Roux and Seyfaddini's max formula holds for the new invariant, see Section 1.1.4.

Results.
The limitation in Floer theoretic interaction between disjointly supported Hamiltonians is reflected through Floer theoretic invariants of these Hamiltonians and their sum. In order to define these invariants, we briefly describe filtered Floer homology. For more details, we refer to Section 2 and the references therein. Throughout the paper, (M, ω) denotes a closed symplectically aspherical manifold, namely, ω| π 2 (M ) = 0 and c 1 | π 2 (M ) = 0, where c 1 is the first Chern class of M . Given a Hamiltonian F : M × S 1 → R, its symplectic gradient is the vector field given by the equation ω(X F , ·) = −dF (·). The 1-periodic orbits of the flow of X F , whose set is denoted by P(F ), correspond to critical points of the action functional associated to F and generate the Floer complex CF * (F ). The differential of this chain complex is defined by counting certain negative-gradient flow lines of the action functional and therefore decreases the value of the action. Note that the gradient of the action functional is taken with respect to a metric induced by an almost complex structure J on M . The homology of this chain complex, denoted HF * (F ), is known to be isomorphic to the singular homology of M up to a degree-shift, HF * (F ) ∼ = H * +n (M ; Z 2 ). The complex CF * (F ) is filtered by the action value, namely, for every a ∈ R, we denote by CF a * (F ) the sub-complex generated by 1-periodic orbits whose action is not greater than a. The homology of this sub-complex is denoted by HF a * (F ). In what follows we present four applications of the barricades construction, which is an adaptation to Floer theory of the idea presented in Figure 1 and is described in Section 1.2. The class of admissible domains for the barricade construction include symplectic embeddings of nice star-shaped 1 domains in R 2n into M . In order to present this class in full generality we need to recall a few standard notions. Let U ⊂ M be a domain with a smooth boundary. We say that U has a contact type boundary if there exists a vector field Y , called the Liouville vector field, that is defined on a neighborhood of ∂U , is transverse to ∂U , points outwards of U and satisfies L Y ω = ω. If the Liouville vector field Y extends to U , we say that U is a Liouville domain. Finally, a subset X ⊂ M is called incompressible if the map ι * : π 1 (X) → π 1 (M ), induced by the inclusion X → M , is injective. In particular, every simply connected subset is incompressible.  • The image under a symplectic embedding of a nice star-shaped domain in R 2n into M is a CIB domain.
• A non-contractible annulus in M = T 2 is a CIB domain. More generally, if M = T 2n = C n /Z 2n , then certain tubular neighborhoods of L = R n /Z n in M are CIB domains.
(a) Incompressible embedding (b) Not incompressible Figure 2: Two embeddings of the annulus into T 2 . The first is incompressible (as well as its boundary) and hence is a CIB domain. The second embedding is contractible in T 2 and therefore not incompressible.

Remark 1.3.
• Note that a disjoint union of CIB domains is again a CIB domain.
• Every incompressible Liouville domain is a CIB domain.
• Every CIB domain is incompressible, as the fact that ∂U is incompressible implies that U is incompressible, see Appendix A.

Locality of spectral invariants and Schwarz's capacities.
For a homology class α ∈ H * (M ; Z 2 ) and a Hamiltonian F , the spectral invariant c(F ; α) is the smallest action value a for which α appears in HF a * (F, J), namely, where ι a * : HF a * (F ) → HF * (F ) is induced by the inclusion ι a : CF a * (F ) → CF * (F ). The following result states that the spectral invariants with respect to the fundamental and the point classes, of a Hamiltonian F supported in a CIB domain, do not depend on the ambient manifold M . More formally, let U ⊂ M be a CIB domain and assume that there exists a symplectic embedding, Ψ : U → N , of U into another closed symplectically aspherical manifold (N, Ω), such that Ψ(U ) is a CIB domain in N . Denote by c M (·; ·), c N (·; ·) the spectral invariants in the manifolds M, N respectively.
where Ψ * F : N × S 1 → R is the extension by zero of F • Ψ −1 .
The assertion of Theorem 1 does not hold when M is not symplectically aspherical, or when U is not incompressible in M . This is shown in Example 4.6. Theorem 1 also holds for the spectral invariants defined in [8] on open manifolds obtained as completions of compact manifolds with contact-type boundaries, see Remark 5.1. Moreover, Theorem 1 can be extended to certain other homology classes, as stated in Claim 5.2. One corollary of Theorem 1 concerns Schwarz's relative capacities 2 .
In [17] Schwarz shows that if the spectral capacity of the support of X F is finite and ϕ 1 F = 1l, then the Hamiltonian flow of F has infinitely many geometrically distinct nonconstant periodic points corresponding to contractible solutions. In Section 4, we use Theorem 1 to show that, when A is a contractible domain with a contact-type boundary, its spectral capacity does not depend on the ambient manifold. Corollary 1.5. Let S be the set of contractible compact symplectic manifolds with contacttype boundaries that can be embedded into symplectically aspherical manifolds, e.g., nice star-shaped domains in R 2n . Then, Schwarz's spectral capacities, {c γ (·; M )}, induce a capacity, c γ , on S. In particular, c γ (A; M ) is finite for every A ⊂ M such that A ∈ S and can be symplectically embedded into (R 2n , ω): where e(A; R 2n ) is the displacement energy 3 of A in R 2n .
Here we used the fact that every bounded subset of R 2n is displaceable with finite energy. Another corollary of Theorem 1 concerns the notions of heavy and super-heavy sets, which were introduced by Entov and Polterovich in [6]: A closed subset X ⊂ M is called heavy if ζ(F ) ≥ inf k is the partial symplectic quasi-state associated to the spectral invariant c and the fundamental class. The following corollary was suggested to us by Polterovich. Corollary 1.6. Let A ⊂ M be a contractible domain with a contact-type boundary that can be symplectically embedded in (R 2n , ω 0 ). Then, M \ A is super-heavy. In particular, A does not contain a heavy set. Corollary 1.6 can be viewed as an extension of the results of [12] to a wider class of domains, when restricting to symplectically aspherical manifolds. Theorem 1 and Corollaries 1.5 and 1.6 are proved in Section 4.

Max-inequality for spectral invariants.
In [11], Humilière, Le Roux and Seyfaddini proved a max formula for the spectral invariants, with respect to the fundamental class, of Hamiltonians supported in disjoint incompressible Liouville domains in symplectically aspherical manifolds.
Theorem (Humilière-Le Roux-Seyfaddini, [11,Theorem 45]). Suppose that F 1 , . . . , F N are Hamiltonians whose supports are contained, respectively, in pairwise disjoint incompressible Liouville domains U 1 , . . . , U N . Then, The existence of barricades can be used to give an alternative proof for this theorem, as well as to prove a version of it for other homology classes. Clearly, other homology classes do not satisfy such a max formula -for example, by Poincaré duality the class of a point satisfies a min formula. However, an inequality does hold for a general homology class. Notice that, by definition, every incompressible Liouville domain is a CIB domain. Moreover, a disjoint union of CIB domains is again a CIB domain. Hence, the inequality for N Hamiltonians follows by induction. We also mention that a "min inequality" does not hold in general, namely, c(F + G; α) might be strictly smaller than min{c(F, α), c(G, α)} as shown in Example 6.4. Theorem 2 is proved in Section 6.

The boundary depth of disjointly supported Hamiltonians.
In [19], Usher defined the boundary depth of a Hamiltonian F to be the largest action gap between a boundary term in CF * (F ) and its smallest primitive, namely The following result relates the boundary depths of disjointly supported Hamiltonians to that of their sum, and is proved in Section 7.
Theorem 3. Let F, G be Hamiltonians supported in disjoint CIB domains, then Note that equality does not hold in (5) in general, as shown in Example 7.2.

Min-inequality for the AHS action selector.
In a recent paper, [1], Abbondandolo, Haug and Schlenk presented a new construction of an action selector, denoted here by c AHS , that does not rely on Floer homology. Roughly speaking, given a Hamiltonian F , the invariant c AHS (F ) is the minimal action value that "survives" under all homotopies starting at F . In Section 8, we review the definition of this selector and a few relevant properties. An open problem stated in [1,Open Problem 7.5] is whether c AHS coincides with the spectral invariant of the point class. As a starting point, Abbondandolo, Haug and Schlenk ask whether c AHS satisfies a min formula like the one proved by Humilière, Le Roux and Seyfaddini in [11] for the spectral invariant with respect to the point class 4 . Due to a result from [11], this will imply that c AHS coincides with the spectral invariant with respect to the point class in dimension 2 on autonomous Hamiltonians. In Section 8, we use barricades in order to prove an inequality for the AHS action selector.
1.2 The main tool: Barricades.
The central construction in this paper is an adaptation of the idea presented in Figure 1 to Floer theory, which is an infinite-dimensional version of Morse theory, applied to the action functional associated to a given Hamiltonian F : M × S 1 → R. As in Morse theory, the Floer differential counts certain negative-gradient flow lines of the action functional. These flow lines are called "Floer trajectories" and correspond to solutions u : R×S 1 → M of a certain partial differential equation, called "Floer equation" (FE), that converge to 1-periodic orbits of the Hamiltonian flow at the ends, lim s→±∞ u(s, t) = x ± (t) for x ± ∈ P(F ).
In this case we say that u connects x ± , see Section 2 for more details. Following the idea from Morse theory, given a Hamiltonian F supported in a subset U ⊂ M , we wish to construct a perturbation for which Floer trajectories cannot enter or exit the domain. Moreover, we extend this construction to homotopies of Hamiltonians, namely, smooth functions H : M × S 1 × R → R, for the following reason: Most of the results presented above compare Floer theoretic invariants of different Hamiltonians. Such a comparison is usually done using a morphism between the different chain complexes, that is defined by counting solutions of the Floer equation with respect to a homotopy between the two Hamiltonians. We consider only homotopies that are constant outside of a compact set, namely there exists R > 0 such that We denote by H ± := H(·, ·, ±R) the ends of the homotopy H. Note that we think of single Hamiltonians as a special case of this setting, by identifying them with constant homotopies, H(x, t, s) = F (x, t). Given an almost complex structure J on M , we consider solutions of the Floer equation (FE) with respect to the pair (H, J). The property of having a barricade is defined through constraints on these solutions. Definition 1.7. Let U and U • be open subsets of M such that U • U . We say that a pair (H, J) of a homotopy and an almost complex structure has a barricade in U around U • if the periodic orbits of H ± do not intersect the boundaries ∂U , ∂U • , and for every x ± ∈ P(H ± ) and every solution u : R × S 1 → M of the corresponding Floer equation, connecting x ± , we have: See Figure 3 for an illustration of solutions satisfying and not satisfying these constraints. When H is a constant homotopy, corresponding to a Hamiltonian F , the presence of a barricade yields a decomposition of the Floer complex, in which the differential admits a triangular block form. To describe this decomposition, let us fix some notations: For a subset X ⊂ M denote by C X (F ) ⊂ CF * (F ) the subspace generated by orbits contained in X, and by ∂| X the map obtained by counting only solutions that are contained in X. Then, for a Floer regular pair (F, J) with a barricade in U around U • , The block form (7) implies that the differential restricts to the subspace C U• (F ). We study the homology of the resulting subcomplex (C U• (F ), ∂| U• ) in Section 5.1. Given a homotopy H that is compactly supported in a CIB domain, we construct a small perturbation h of H and an almost complex structure J, such that (h, J) has a barricade.
Theorem 5. Let U be a CIB domain and let H : M × S 1 × R → R be a homotopy of Hamiltonians, supported in U ×S 1 ×R, such that ∂ s H is compactly supported. Then, there exist a C ∞ -small perturbation h of H and an almost complex structure J such that the pairs (h, J) and (h ± , J) are Floer-regular and have a barricade in U around U • . In particular, when H is independent of the R-coordinate (namely, it is a single Hamiltonian), h can be chosen to be independent of the R-coordinate as well 5 .
This result is proved in Section 3, by an explicit construction of the perturbation h and the almost complex structure J. We remark that the assumptions on (M, ω) being symplectically aspherical and U having either incompressible boundary or being an incompressible Liouville domain are crucial for this construction. See the proofs of Lemmas 3.3-3.4 for details.

Related works.
There have been several works studying the Floer-theoretic interaction between disjointly supported Hamiltonians, mainly through the spectral invariants of these Hamiltonians and their sum. Early works in this direction, mainly by Polterovich [14], Seyfaddini [18] and Ishikawa [12], established upper bounds for the invariant of the sum of Hamiltonians, which depend om the supports. Later, Humilière, Le Roux and Seyfaddini [11] proved that in certain cases the invariant of the sum is equal to the maximum over the invariants of each individual summand. The method was also conceptually different. While previous works relied solely on the properties of spectral invariants, Humilière, Le Roux and Seyfaddini studied the Floer complex itself. We also take this approach and study the interaction between disjointly supported Hamiltonians on the level of the Floer complex, but our methods are substantially different.
In a broader sense, it is worth to mention two works which regard symplectic homology. Symplectic homology is an umbrella term for a type of homological invariants of symplectic manifolds, or of subsets of symplectic manifolds, which are constructed via a limiting process from the Floer complexes of properly chosen Hamiltonians. In this setting, questions regarding disjointly supported Hamiltonians correspond to local-to-global relations, such as a Mayer-Vietoris sequence. In [5], Cieliebak and Oancea defined symplectic homology for Liouville domains and Liouville cobordisms and proved a Mayer-Vietoris relation. Their method include ruling out the existence of certain Floer trajectories, and partially rely on a work by Abouzaid and Seidel, [2]. Versions of some of these arguments are being used in Section 3 below. Another work concerning Mayer-Vietoris property is by Varolgunes, [20], in which he defines an invariant of compact subsets of closed symplectic manifolds, which is called relative symplectic homology, and finds a condition under which the Mayer-Vietoris property holds. In particular, for a union of disjoint compact sets, the relative symplectic homology splits into a direct sum.

Structure of the paper.
In Section 2 we review the necessary preliminaries from Floer theory and contact geometry. In Section 3 we construct barricades and prove Theorem 5. We then use it to prove Theorem 1 in Section 4. In Section 5, we discuss the relation to Floer homology on certain open manifolds and two extensions of Theorem 1. Sections 6-8 are dedicated to the proofs of Theorems 2-4 respectively. Finally, on Section 9 we prove several transversality and compactness claims that are required for establishing the main results. Appendix A contains a claim about incompressibility, whose proof we include for the sake of completeness.

Acknowledgements.
We are very grateful to Lev Buhovsky and Leonid Polterovich for their guidance and insightful inputs. We thank Vincent Humilière and Sobhan Seyfaddini for useful discussions and, in particular, for proposing a question that led us to Theorem 1. We thank Alberto Abbondandolo, Carsten Haug and Felix Schlenk for sharing a preliminary version of their paper [1] with us. We also thank Felix Schlenk for suggesting to consider Floer homology on open manifolds, which led us to Section 5. We thank Mihai Damian and Jun Zhang for discussions concerning the arguments appearing in Section 9.1. Finally, we thank Henry Wilton for discussions regarding Proposition A.1 over MathOverflow.
Y.G. was partially supported by ISF Grant 1715/18 and by ERC Advanced grant 338809. S.T. was partially supported by ISF Grant 2026/17 and by the Levtzion Scholarship.

Preliminaries from Floer theory.
In this section we briefly review some preliminaries from Floer theory and contact geometry on closed symplectically aspherical manifolds (namely, when ω| π 2 (M ) = 0 and c 1 | π 2 (M ) = 0, where c 1 is the first Chern class of M ). For more details see, for example, [3,13,15]. We also fix some notations that will be used later on.

Floer homology, regularity and notations.
Let F : M × S 1 → R be a Hamiltonian on M . The corresponding action functional A F is defined on the space of contractible loops in M by The critical points of the action functional are the contractible 1-periodic orbits of the flow of X F and their set is denoted by P(F ). The Hamiltonian F : M × S 1 → R is said to be non-degenerate if the graph of the linearized flow of X F at time 1 intersects the diagonal in T M × T M transversally. In this case, the flow of X F has finitely many 1-periodic orbits. The Floer complex CF * (F ) is spanned by these critical points, over Z 2 6 . A time dependent ωcompatible 7 almost complex structure J induces a metric on the space of contractible loops, in which negative-gradient flow lines of A F are maps u : R × S 1 → M that solve the Floer equation The energy of such a solution is defined to be E(u) := R×S 1 ∂ s u 2 J ds dt, where · J is the norm induced by the the inner product associated to J, ·, · J := ω(·, J·). When the Hamiltonian F is non-degenerate, for every solution u with finite energy, there exist x ± ∈ P(F ) such that lim s→±∞ u(s, t) = x ± (t), and we say that u connects x ± . The well known energy identity for such solutions is a consequence of Stokes' theorem: For two 1-periodic orbits x ± ∈ P(F ) of F , we denote by M (F,J) (x − , x + ) the set of all solutions u : R × S 1 → M of the Floer equation (FE) that satisfy lim s→±∞ u(s, t) = x ± (t). Notice that R acts on this set by translation in the s variable. We denote by M (F,J) the set of all finite energy solutions. It is well known (e.g., [3,Theorem 6.5.6] Moreover, for non-degenerate Hamiltonians one can define an index µ : P(F ) → Z, called the Conley-Zehnder index, which assigns an integer to each orbit (see e.g. [3,Chapter 7] and the references therein). The Floer complex is graded by the index µ, namely, for k ∈ Z, CF k (F ) is the Z 2 -vector space spanned by the periodic orbits x ∈ P(F ) for which µ(x) = k.
In order to define the Floer differential for the graded complex CF * (F ), one needs an almost complex structure J, such that the pair (F, J) is Floer-regular. The definition of Floer regularity concerns the surjectivity of a certain linear operator and is given in Section 9.1. When the pair (F, J) is Floer-regular, the space of solutions, Recall that an element a ∈ CF * (F ) is a formal linear combination a = x a x · x where x ∈ P(F ) and a x ∈ Z 2 . For a Floer-regular pair (F, J), the Floer differential where # 2 is the number of elements modulo 2. The homology of the complex (CF * (F ), ∂ (F,J) ) is denoted by HF * (F, J) or HF * (F ). A fundamental result in Floer theory states that Floer homology is isomorphic to the singular homology, with a degree shift, HF * (F, J) ∼ = H * −n (M ; Z 2 ). The Floer complex admits a natural filtration by the action value. We denote by CF a * (F ) the sub-complex spanned by critical points with value not-greater than a.
Since the differential is action decreasing, it can be restricted to the sub-complex CF a * (F ). The homology of this sub-complex is denoted by HF a * (F, J). It is well known that when F is a C 2 -small Morse function, its 1-periodic orbits are its critical points, P(F ) ∼ = Crit(F ), and their actions are the values of F , A F (p) = F (p). In this case, the Floer complex with respect to a time-independent almost complex structure J, coincides with the Morse complex when the degree is shifted by n (which is half the dimension of M ), since Morse-ind(p) = µ(p) + n for every p ∈ Crit(F ) ∼ = P(F ): For a proof, see, for example, [3,Chapter 10]. We conclude this section by fixing notations that will be used later on.
x be an element of CF * (H).
• We say that x ∈ a if a x = 0.
• We denote the maximal action of an orbit from a by λ H (a) := max{A H (x) : a x = 0}.
• For a subset X ⊂ M , let C X (H) ⊂ CF * (H) be the subspace spanned by the 1periodic orbits of H that are contained in X. Moreover, let π X : CF * (H) → C X (H) be the projection onto this subspace. Note that C X (H) is not necessarily a subcomplex, and π X is not a chain map in general.

Communication between Floer complexes using homotopies.
Now let H : M × S 1 × R → R denote a homotopy of Hamiltonians, rather than a single Hamiltonian. Throughout the paper, we consider only homotopies that are constant outside of a compact set. Namely, there exists R > 0 such that ∂ s H| |s|>R = 0, and we denote by H ± (x, t) := lim s→±∞ H(x, t, s) the ends of the homotopy H. Given an almost complex structure J, we consider the Floer equation (FE) with respect to the pair (H, J): where H s (·, ·) := H(·, ·, s). We sometimes refer this equation as "the s-dependent Floer equation", to stress that it is defined with respect to a homotopy of Hamiltonians. For 1-periodic orbits x ± ∈ P(H ± ), we denote by M (H,J) (x − , x + ) the set of all solutions u : As in the case of Hamiltonians, the definition of Floer-regularity concerns the surjectivity of a certain linear operator and is given in Section 9.1. For a Floer-regular pair, (H, J), the space M (H,J) (x − , x + ) is a smooth manifold of dimension µ(x − ) − µ(x + ). In this case, one can define a degree-preserving chain map, called the continuation map, between the Floer complexes of the ends, Φ : The regularity of the pair guarantees that the map Φ is a well defined chain map that induces isomorphism on homologies, see, e.g., [3,Chapter 11].

Contact-type boundaries.
In order to construct barricades for Floer solutions around a given domain, we need the boundary to have a contact structure: Let U ⊂ M be a domain with a smooth boundary. We say that U has a contact type boundary if there exists a vector-field Y , called the Liouville vector field, that is defined on a neighborhood of ∂U , is transverse to ∂U , points outwards of U and satisfies L Y ω = ω. The differential form λ := ι Y ω is a primitive of ω, namely dλ = ω. The Reeb vector field R is then defined by the following equations: where ψ τ is the flow of Y . We stress that the differential form λ and the vector field R are defined wherever the Liouville vector field Y is defined. If the Liouville vector field Y extends to U , we say that U is a Liouville domain.
3 Barricades for solutions of the (s-dependent) Floer equation.
In what follows, H : M × S 1 × R → R denotes a homotopy of (time-dependent) Hamiltonians and J denotes a (time-dependent) almost complex structure. We assume that ∂ s H is compactly supported and denote H ± := lim s→±∞ H(·, ·, s). Note the we consider the case where H is a single Hamiltonian as a particular case, by identifying it with a constant homotopy. Fix a CIB domain U ⊂ M , and denote by Y and R the Liouville and Reeb vector fields respectively. Then, λ = ι Y ω is the contact form on the boundary ∂U . The flow ψ τ of Y is called the Liouville flow, and is defined for short times.
In order to prove Theorem 5, namely, that there exist a perturbation h of H and an almost complex structure J such that (H, J) has a barricade, we construct h and J explicitly. Let us sketch the idea of this construction before giving the details.
• To construct h, we first add to H a non-negative bump function in the radial coordinate, which is defined on a neighborhood of ∂U using the Liouville flow. Then, we take h to be a small non-degenerate perturbation of it. • The almost complex structure J is taken to be cylindrical near ∂U (see Definition 3.1 below).
We want to rule out the existence of solutions violating the constrains from Definition 1.7. Suppose there exists a solution u connecting Then, the image of u intersects ∂U • , say along a loop Γ. We first bound the action of Γ (Lemma 3.2), and then conclude a negative upper bound for the action of x + (Lemma 3.3). Since h ≈ 0 on U c • ⊃ x + , the action of x + can be taken to be arbitrarily close to zero, in contradiction.

Preliminary computations.
Some of the arguments and results in this section were carried out by Cieliebak and Oancea in [5] for the setting of completed Liouville domains, instead of closed symplectically aspherical manifolds. Specifically, some of the computations appearing in the proofs of Lemma 3.   We remark that conditions 2,3 in the above definition imply that, near ∂U , H does not depend on the R-coordinate. Suppose that (H, J) is δ-cylindrical near ∂U and let u :  Lemma 3.2. Let (H, J) be a pair that is δ-cylindrical near ∂U and let u : R × S 1 be a finite-energy solution of the s-dependent Floer equation connecting x ± ∈ P(H ± ). Suppose that u intersects ∂U transversely and let Γ := Im (u)∩∂U denote the intersection, oriented as the boundary of Im (u) ∩ U c . Then, Proof. Set Σ := u −1 (U c ) ⊂ R × S 1 and denote its boundary by γ, then u(γ) = Γ, since x ± do not intersect ∂U . The orientation on Σ is given by the positive frame (∂ s , ∂ t ). Let γ i be a connected component of γ, then Γ i := u(γ i ) is connected. Let τ ∈ [0, T i ] be a unit-speed parametrization of γ i , and notice that this induces parametrization on Γ i . Denote by ν(τ ) the outer normal to Σ at γ i (τ ), thenγ i (τ ) = jν(τ ), where j is the standard complex structure on R × S 1 (i.e., j∂ s = ∂ t ). Pushing (ν(τ ),γ i (τ )) to T M we obtain We remark that N (τ ) is not necessarily normal to ∂U (with respect to the inner product induced by J), but is always pointing inwards (or tangent to the boundary), see Figure 5. The relation between N (τ ) andΓ i (τ ) goes through the Floer equation (FE), which can be written in the following form: It follows thatΓ i (τ ) can be written as a linear combination of JN (τ ), the gradient of H 16 and the symplectic gradient of H: Using this to compute the integral of λ along Γ i , we obtain Recalling our assumptions that ∇ J H = δY on ∂U and that JY is the Reeb vector-field, we obtain Let us estimate separately each term in the sum (14), starting with the first: Since JY = R, the vector field Y is perpendicular to the hyperplane T (∂U ) at each point and is pointing outwards of U . By our construction, N (τ ) points inwards to U (as it is tangent to Im (u) and points out of Im (u) ∩ U c ) and therefore Y • Γ i , N ≤ 0 for all τ . We conclude that We turn to estimate the second summand in (14): Noticing that ν, ∂ s j = jν, j∂ s j = γ i , ∂ t j = dt(γ i ), we have LetΣ be the closure of Σ in the compactification (R ∪ {±∞}) × S 1 of the cylinder, then ∂Σ ⊂ ∂Σ∪{±∞}×S 1 . Notice that ∂Σ contains {−∞}×S 1 (resp., {+∞}×S 1 ) if and only if x − ⊂ U c (resp., x + ⊂ U c ). As {±∞}×S 1 dt = ±1 and, by Stokes' theorem, ∂Σ dt = 0, we conclude that Combining (14), (15) and (16) we obtain When the homotopy H is non-increasing in U c , Lemma 3.2 can be used to bound the action of the ends of solutions that cross the boundary of U . Lemma 3.3 below is similar to a result obtained by Cieliebak and Oancea in [5, Lemma 2.2] for the setting of completed Liouville domains, using neck-stretching. The proof of Lemma 3.3 uses a different approach and is an application of Lemma 3.2 above. Lemma 3.3. Suppose that (H, J) is δ-cylindrical near ∂U and assume in addition that ∂ s H ≤ 0 on U c . For every finite-energy solution u connecting x ± ∈ P(H ± ), where c is the value of H on ∂U .
Proof. We prove the first statement, where x − ⊂ U and x + ⊂ U c . The second statement is proved similarly. As in [5, Lemma 2.2], after replacing U by its image, ψ τ U , under the Liouville flow for small time τ , we may assume that u is transverse to ∂U 8 . Note that, since ∇ J H = δY on a neighborhood of ∂U , H is constant on ∂(ψ τ U ) = ψ τ (∂U ). Moreover, choosing the sign of τ to be opposite to the sign of δ, the value of H on ψ τ (∂U ) is smaller than c (in order to prove the second statement, choose τ to be of the same sign as δ, and then the value of H on ψ τ (∂U ) will be greater than c). Denote Σ := u −1 (U c ) ⊂ R × S 1 and let us compute an energy-identity for the restriction u| Σ : where, in the last two inequalities, we used our assumption that ∂ s H ≤ 0, and the positivity of the energy, respectively. As before, denote byΣ the closure of Σ in the com- where the last equality follows from (16) for γ = ∂Σ. Therefore, using Stokes' theorem, we obtain Letx ± be capping disks of x ± respectively, and let v ⊂Ū be a union of disks capping the connected components of Γ := u(γ), such that the contact form λ is defined on v.
The existence of such disks follows from our definition of a CIB domain: If the relevant connected component of U is an incompressible Liouville domain, then we can take a capping disk that is contained in that component. Otherwise, the boundary of the relevant connected component of U is incompressible and we can take the capping disk to lie in the boundary. Since M is symplectically aspherical and ω = dλ where λ is defined, we have Combining (18) and (19) yields where the last inequality is due to (17). Using Lemma 3.2 we conclude that The following Lemma is essentially a version of [5, Lemma 2.2] for closed symplectically aspherical manifolds instead of completed Liouville domains.
Lemma 3.4. Suppose that (H, J) is δ-cylindrical near ∂U and that ∂ s H ≤ 0 on U c . Then, for every x ± ∈ P(H ± ) that are contained in U , every solution u connecting them is contained in U .
Proof. As before, after replacing U by its image, ψ τ U , under the Liouville flow for a small time τ , we may assume that u is transverse to ∂U . Setting again Σ := u −1 (U c ) ⊂ R × S 1 and computing an energy identity, as in (17), for the restriction of u to Σ, we have where, as before,Σ is the closure of Σ in the compactification of the cylinder. This time, both ends x ± are contained in U and hence ∂Σ = γ. Since H is constant on ∂U , it follows from (16) that On the other hand, taking v ⊂Ū to be a union of disks capping the connected components of Γ = u(γ) (which is oriented as the boundary of Im (u) ∩ U c ), such that λ is defined on v, the fact that M is symplectically aspherical implies that where the last inequality follows from Lemma 3.2. Combining the above two inequalities we find Since we assumed that H ± are non-degenerate and have no 1-periodic orbits intersecting ∂U , this implies Im (u) ∩ int(U c ) = ∅ and hence Im (u) ⊂Ū . Noticing that we may argue similarly for the image ψ τ U of U under the Liouville flow for small negative time, τ < 0, we conclude that Im (u) ⊂ ψ τ U ⊂ U .

Constructing the barricade.
As before, U denotes a CIB domain and ψ τ is the flow of the Liouville vector field Y , which is defined in a neighborhood of the boundary ∂U . Consider a pair (H, J) of a homotopy (or, in particular, a Hamiltonian) and an almost complex structure. The following definition is an adaptation of Figure 1 to Floer theory. Definition 3.5. We say that the pair (H, J) admits a cylindrical bump of width τ > 0 and slope In analogy with the discussion in Morse theory, we show that a pair with a cylindrical bump has a barricade.
Proposition 3.6. Let (H, J) be a pair with a cylindrical bump of width τ and slope δ.
Proof. The proof essentially follows from Lemmas 3.3 and 3.4, together with the fact that a pair (H, J) with a (τ, δ)-bump around ∂U is in particular cylindrical near both ∂U and ∂U • . Let u be a solution of the s-dependent Floer equation, with respect to H and J, that connects x ± ∈ P(H ± ). We need to show that u satisfies the constraints from Definition 1.7, and therefore split into two cases: Lemma 3.4 to H, J and U • , and conclude that Im (u) ⊂ U • as required. Otherwise, x + ⊂ U c • is a critical point of H + and its value lies in the interval (−δ, δ). On the other hand, applying Lemma 3.3 to H, J and U • yields that A H + (x + ) < −δ, in contradiction.
2. Suppose x + ⊂ U . As before, if x − ⊂ U then applying Lemma 3.4 to H, J and U yields u ⊂ U as required. Otherwise, x − ⊂ U c is a critical point of H − and its value lies in (−δ, δ). On the other hand, applying Lemma 3.3 to H, J and U , and noticing that In order to prove Theorem 5, it remains to guarantee the regularity assertion, for which we use the result from Section 9.3.1 below.
Proof of Theorem 5. Let H be a homotopy of Hamiltonians that is supported in U ×S 1 ×R. Then, there exists τ > 0 small enough, such that H is supported inside ψ −τ U =: U • . Fix an almost complex structure J that is cylindrical near both ∂U and ∂U • (see Item 2 of Definition 3.5 above), and let h be a C ∞ -small perturbation of H such that the pair (h, J) admits a (τ, δ)-bump around ∂U and h ± are non-degenerate. Notice that, by definition, the pairs (h ± , J) also admit a (τ, δ)-bump around ∂U . By Proposition 3.6, the pairs (h, J) and (h ± , J) have a barricade in U around U • .
The pairs (h, J), (h ± , J) constructed above are not necessarily Floer-regular. In order to achieve regularity, we perturb the homotopy h and its ends. Proposition 9.21 below states that for a homotopy h that satisfies P(h ± ) = P(h ± ) and supp(∂ s h ) ⊂ M × S 1 × I for some fixed finite interval I, if h is close enough to h, then (h , J) also has a barricade in U around U • . Therefore, it remains to describe a perturbation that satisfies the above constraints, and ensures regularity. Starting with the ends and recalling that h ± are nondegenerate, we perturb them without changing their periodic orbits to guarantee that the pairs (h ± , J) are Floer-regular (the fact that this is possible is a well known result from Floer theory, cited in Claim 9.1 below). If the homotopy h is constant, that is, corresponds to a single Hamiltonian, we are done. Otherwise, let us perturb h so that its ends will agree with the regular perturbations of h ± . Finally, we perturb the resulting homotopy on the set M × S 1 × I, for some fixed finite interval I, to make the pair (h, J) Floer-regular. This is possible due Proposition 9.2 below, which is a slight modification of standard claims from Floer theory and is proved in Section 9.1.
Remark 3.7. Proposition 3.6 suggests that, when given a homotopy (or a Hamiltonian) H that is supported in U × S 1 × R, we have some freedom in choosing the pair (h, J) from Theorem 5. Let us mention some additional properties that can be granted for the perturbation h and the almost complex structure J, and will be useful in applications.
1. The almost complex structure J can be taken to be time-independent. Moreover, if one of the ends of H, say H − , is zero, then h can be chosen such that h − is any time-independent small Morse function that has a cylindrical bump around ∂U . To see this, choose h ≈ H and J such that (h, J) has a cylindrical bump around ∂U , and J, h − are time-independent. Then, the pair (h − , J) is Floer-regular and, by perturbing h + first and then replacing the homotopy by a compactly supported perturbation, we end up with a pair (h, J) that is Floer regular, as well as its ends, and (h − , J) is time-independent.
2. When the homotopy H is constant on some domain, we can choose the perturbation h such that, on this domain, its ends, h ± , agree on their 1-periodic orbits up to second order. This follows from the use of Claim 9.1 in the proof of Theorem 5.

Given an interval
. This follows from the use of Proposition 9.2 in the proof of Theorem 5.

Locality of spectral invariants, Schwarz's capacities and super heavy sets.
In this section we use barricades to prove Theorem 1 and derive Corollaries 1.5 and 1.6. We will use the definitions and notations from Section 2, in particular Notations 2.1 and Formula (11). We will also use the following properties of spectral invariants (see [15,Proposition 12.5.3], for example): 1. (spectrality) c(F ; α) ∈ spec(F ).

(energy-capacity inequality) If the support of
In this case, the displacement energy of X is given by Let us sketch the idea of the proof of Theorem 1 before giving the details. We will prove the statement for the class of a point, and use Poincaré duality to deduce the same for the fundamental class. We start by showing that the spectral invariant, with respect to [pt], of a Hamiltonian supported in a CIB domain is non-positive (Lemma 4.1). Then, after properly choosing regular perturbations with barricades (Lemma 4.4), we consider a representative of [pt] of negative action on M . Such a representative must be a combination of orbits in U • and thus can be pushed to a cycle on N . Finally, we use continuation maps, induced by homotopies to small Morse functions, to conclude that the cycle on N represents [pt] there. As mentioned above, our first step towards proving Theorem 1 is showing that the spectral invariant with respect to [pt] of a Hamiltonian supported in a CIB domain is always non-positive. Proof. Let H be a linear homotopy 9 from H − := 0 to H + := F . By Theorem 5, there exist a small perturbation h of H and an almost complex structure J such that (h, J) and (h ± , J) are Floer-regular and have a barricade in U around U • , where U • contains the support of F . By Remark 3.7, item 1, we can choose J to be time independent and h such that h − is a time-independent small Morse function. Moreover, we may assume that h − has a minimum point p that is contained in U c . Since the Floer complex and differential of (h − , J) agree with the Morse ones, the point p represents . Indeed, otherwise, we would have a continuation solution starting at p ⊂ U c and ending at some • is a small Morse function. Its 1-periodic orbits there are critical points and their actions are the critical values. Therefore, using the stability property of spectral invariants, we conclude that c(F ; • Using Poincaré duality for spectral invariants, the above lemma implies that c(F ; [M ]) ≥ 0 for every Hamiltonian F supported in a CIB domain. This is already known for incompressible Liouville domains. Indeed, it follows easily from the max formula, proved in [11], when applied to the functions F 1 = F and F 2 = 0: • Lemma 4.1 does not hold if M is not symplectically aspherical. For example, the equator in S 2 is known to be super-heavy. Therefore, if F is a Hamiltonian on S 2 which is supported on a disk containing the equator, then is not greater than the maximal value that F attains on the equator, see [15,Chapter 6]. Therefore, one can construct a Hamiltonian supported in a disk on S 2 with a negative spectral invariant with respect to the fundamental class.
Our next step towards the proof of Theorem 1 is choosing suitable perturbations for the Hamiltonians F and Ψ * F , as well as homotopies from them to small Morse functions. Before that, we use the embedding Ψ to define a linear map between subspaces of Floer complexes of Hamiltonians on M and on N , that agree on U through Ψ.
that is a combination of orbits contained in U , we define its pushforward with respect to the embedding Ψ to be 4. The differentials and continuation maps commute with the pushforwad map Ψ * when restricted to U • : and We postpone the proof of Lemma 4.4, and prove Theorem 1 first.
Proof of Theorem 1. We will prove that c • , its 1-periodic orbits there are its critical points, and their actions are the critical values. As a consequence, a representative a ∈ CF ) is a combination of orbits that are contained in U • , namely a ∈ C U• (f M ). Therefore, the pushforward Ψ * a ∈ CF * (f N ) is defined, and by (22), Ψ * a is closed in CF * (f N ). To see that Ψ * a represents the class of a point, we will use (21). Indeed, since a represents [pt] on M , and continuation maps induce isomorphism on ho- Morse function (and J M is time-independent), its Floer complex and differential coincide with the Morse ones, is also a sum of an odd number of minima, and as such, represents the point class in CM * +n (h N + ) ∼ = CF * (h N + ). Since Ψ * a is closed, we conclude that it represents [pt] in CF * (f N ). Together with the fact that, in Ψ(U ), f M • Ψ −1 and f N agree on their 1-periodic orbits, this implies that where the equality λ f N (Ψ * a) = λ f M (a) follows from the fact that U is incompressible, see We turn to construct the pairs on N . Let h N be an extension to N of the homotopy h M • Ψ −1 , which is defined on Ψ(U ). Notice that by replacing h M with a smaller perturbation of h M if necessary, h N can be taken to be arbitrarily close to h N . This way, we can use Proposition 9.21 again to conclude that (h N , J N ) has a barricade in Ψ(U ) around Ψ(U • ). Finally, we repeat the arguments made above and perturb h N to make all of the pairs on N Floer-regular. We obtain a homotopy h N that is constant for s / ∈ [0, 1], approximates h M • Ψ −1 on Ψ(U ) and such that the pairs (h N , J N ) and (h N ± , J N ) are all Floer-regular and have barricades in Ψ(U ) around Ψ(U • ).
It remains to prove that, in U • , the pushforward map commutes with the continuation maps and the differentials for the homotopies h M , h N and their ends, respectively. We will write the proof for the continuations maps, the proof for the differentials is analogous. We first show that the continuation maps of h N and h N agree on Ψ(U • ), and then prove that the commutation relation (21) holds for h M and h N , which agree on U through Ψ. Proposition 9.31 (for the differentials, Proposition 9.25) states that the restriction of the continuation map to C Ψ(U•) does not change under small perturbations, when the pairs have a barricade and satisfy a certain regularity assumption on Ψ(U ). This assumption holds for Floer-regular pairs, as well as for pairs that coincide on U with a Floer-regular pair. Therefore, recalling that h N is a small perturbation of h N , and that the pair (h N , J N ) agrees, on Ψ(U ), through a symplectomorphism, with the Floer-regular pair (h M , J M ), we may apply Proposition 9.31 and conclude that recall the definitions of Ψ * and the continuation maps (11). We need to show that for every ). This essentially follows from the fact that both pairs (h M , J M ) and (h N , J N ) have barricades, and that respectively. The symplectic embedding Ψ induces a bijection between these two sets, and so it follows that the counts of their elements coincide.
Having established Theorem 1, we now explain how to derive Corollaries 1.5, 1.6. Let us start by recalling the definition of a symplectic capacity: Definition 4.5 (See, for example, [4,10]). Given a class S of symplectic manifolds, a symplectic capacity on S is a map c : S → [0, ∞] that satisfies the following properties: Let us use Theorem 1 to show that Schwarz's relative capacities, which are defined for subsets of a given closed symplectically aspherical manifold, induce a capacity on the class of contractible compact symplectic manifolds with contact-type boundaries that can be embedded into symplectically aspherical manifolds.
Proof of Corollary 1.5. Let A ∈ S be a contractible symplectic manifold with a contacttype boundary that can be embedded into a symplectically aspherical manifold (M, ω).
Abusing the notations, we write A ⊂ M . Recalling the definition of Schwarz's relative capacity (2), we consider a Hamiltonian F on M such that X F is supported in A × S 1 . Since A is contractible, its boundary connected and therefore F is constant on ∂A, as well as on the complement, M \ A. Denoting C := F | M \A , the difference F − C is supported in A. Moreover, it follows from the spectrality and stability of spectral invariants that ) and hence, by replacing F with F − C, we may assume that F is supported in A. Suppose that A can be embedded into another symplectically aspherical manifold (N, Ω). Since A is contractible, its boundary is simply connected, and in particular, incompressible in both M and N . Since ∂A is of contact-type, we conclude that A ⊂ M and A ⊂ N are CIB domains. By Theorem 1, the spectral invariants of Hamiltonians supported in A on M and N coincide, and therefore the relative capacities of A with respect to M and N agree, and we can define We may extend this definition to unbounded domains U ⊂ R 2n by taking the supremum over all A ∈ S that can be embedded into U . Before proving that c γ satisfies the axioms of a symplectic capacity, let us prove the second assertion of the corollary. Given A ∈ S that can be symplectically embedded into (R 2n , ω 0 ), we need to show that By the energy-capacity inequality, for every Hamiltonian F supported in the embedding of A into N , and for every homology class α one has c(F ; α) ≤ e(A; N ) = e(A; R 2n ). Using Theorem 1 we conclude that for every symplectically aspherical M and an embedding of We now briefly explain why c γ satisfies the axioms of a capacity. Nontriviality follows from the fact that Schwarz's capacities are not smaller than the Hofer-Zehnder capacity, and are not greater than twice the displacement energy, see [17]. Monotonicity follows from the definition of c γ (·; M ), together with the fact that the image of every embedding of a domain in S into a symplectically aspherical manifold is a CIB domain. To prove the conformality property, suppose (A, Ω) ∈ S is embedded into (M, ω), then (A, τ · Ω) is embedded into (M, τ · ω). In order to prove that we show that for every F such that supp(X F ) ⊂ A × S 1 and for every homology class α ∈ H * (M ), it holds that Starting from the case where τ > 0, we notice that the action functional with respect to the form τ ω and the Hamiltonian τ F is proportional to the action functional with respect to ω and F . The Floer complexes of (ω, J, F ) and (τ ω, J, τ F ) coincide, while the action filtration is rescaled by τ , and therefore (23) holds. It remains to deal with τ = −1. In this case, the Floer complexes of (ω, J, F ) and (−ω, −J, F ) are isomorphic via the map Proof of Corollary 1.6. Let A ⊂ M be a contractible domain with a contact-type boundary that can be symplectically embedded in (R 2n , ω 0 ). As in the proof of Corollary 1.5, let Q ⊂ R 2n be a large enough cube such that the image of A in R 2n is displaceable in Q. Embedding Q into a large torus, N ∼ = T 2n , we denote by Ψ : A → N the composition of the embeddings. As Ψ(A) is displaceable in N , it follows from non-nativity of c(·; [M ]) (Lemma 4.1), Theorem 1 and the energy capacity inequality that for every Hamiltonian As a consequence, the partial symplectic quasi-state, ζ, associated to c vanishes on functions supported in A. The fact that the complement of A is super-heavy follows from the following equivalent description of super-heavy sets.
[15, Definition 6.1.10]: A closed subset X ⊂ M is super-heavy if ζ(F ) = 0 for every Hamiltonian F that vanishes on X.
The fact that A cannot contain a heavy set can be seen directly from the definition. Alternatively, this fact follows from the intersection property of heavy and super-heavy sets, established by Entov and Polterovich in [6]: Every super-heavy set intersects every heavy set.
We conclude this section with two examples, showing that Theorem 1 does not hold in a more general setting. • The condition on M and N being symplectically aspherical in Theorem 1 is necessary.
A simple example is to embed the unit disk D ⊂ R 2 into a small sphere and into a large sphere. Namely, take M and N to be spheres of areas 1.5π and 2π respectively. Then, there exist Hamiltonians, supported in the embedding of D into M , with arbitrarily large spectral invariants with respect to the fundamental class. This follows from the fact that the embedding of D into M contains the equator, which is a heavy set, see [15,Chapter 6]. A Hamiltonian F that attains large values on the equator in M has a large spectral invariant.
On the other hand, the spectral invariant of any Hamiltonian that is supported in the embedding of D into N is bounded by the displacement energy of this embedded disc in N , which is equal to π.
• The condition on ∂U to be incompressible is also necessary. One can construct two different embeddings of the annulus A := int(D \ 1 2 D) into a torus of large area, such that the image under one embedding is heavy (and the boundary is incompressible), and the image under the other embedding is displaceable (and the boundary is not incompressible). As mentioned above, in the first case one can construct Hamiltonians with arbitrarily large spectral invariants (with respect to the fundamental class), and in the second case, the spectral invariant is bounded by the (finite) displacement energy. In particular, the assertion of Theorem 1 cannot hold in this case.

Relation to certain open symplectic manifolds.
In this section we discuss an extension of Theorem 1 to CIB domains in certain open symplectic manifolds. We start by briefly reviewing Floer homology on such manifolds, following [8] 13 . Let (W, ω) be a 2n-dimensional compact symplectic manifold with a contacttype boundary. Using the Liouville vector field Y , we can symplectically identify a neighborhood of the the boundary in W with ∂W × (ε, 0] endowed with the symplectic form d(e r λ), where λ = ι Y ω and r is the coordinate on the interval. The completion of (W, ω) is defined to be Let J be an ω-compatible almost complex structure on W that, on ∂W , maps Y to the Reeb vector field R and, on ∂W × [0, ∞), is time-independent and is invariant under rtranslations. A time dependent Hamiltonian F on W is called admissible if it coincides on ∂W × [0, ∞) with ρ(e r ) for a function ρ : [0, ∞) → R whose derivative on (0, ∞) is positive and smaller than the minimal period of a periodic Reeb orbit (note that in this case, F has no 1-periodic orbits in W × (0, ∞)). Remark 5.1. It was suggested to us by Schlenk that Theorem 1 holds for the spectral invariant with respect to the point class on the above open manifolds as well. Namely, given a CIB domain U in W and a symplectic embedding Ψ : (U, ω) → (W , ω ) whose image is a CIB domain in W , for every Hamiltonian F supported in U ,

The homology of the subcomplex C U• (f ).
In what follows, (M, ω) denotes a closed symplectic manifold, as always. Given a Hamiltonian F supported in U , let (f, J) be a Floer regular pair on M with a barricade in U around U • , for some U • U . The block form (7) of the differential implies that the differential restricts to C U• (f ) ⊂ CF (f ). In this section we study the homology of this subcomplex. We show that for a properly chosen such pair (f, J), the homology of (C U• (f ), ∂| U• ) coincides with the homology of U , namely, For that end, consider a perturbation f of F such that (f , J) has a (τ, δ)-bump around ∂U (in the sense of Definition 3.5). In particular, we assume that J is cylindrical. Let f to be a

Locality of spectral invariants with respect to other homology classes.
In this section we show how Floer homology on open manifolds is useful in the study of Floer complexes of Hamiltonians supported in CIB domains in closed manifolds 15 . In particular, we explain how to extend Theorem 1 to homology classes in the image of the map induced by the inclusion ι : U → M .
where c M and c U are the spectral invariants in the manifolds (M, ω) and ( U , ω) respectively, andF is the extension by zero of F | U to U .
Proof. The proof relies on the observations of Section 5.1: let f be a perturbation of F and J an almost complex structure such that (f, J) has a barricade in U around U • . Assume in addition that the perturbation is chosen to be arbitrarily close to some f , for which the pair (f , J) has a cylindrical bump around ∂U . As explained previously, the Floer We will show that formula (25) holds for f andf up to 2δ, for some δ which can be made arbitrarily small by shrinking the size of the perturbations. We start by noticing that given a class β ∈ ι −1 * (α), every representative b ∈ C U• (f ) of β, is a representative of α in CF * (f ). This immediately implies that c M (f ; α) ≤ min β∈ι −1 * (α) c U• (f ; β). To prove inequality in the other direction, let β ∈ ι −1 * (α) be a class on which the minimum in the RHS of (25) is attained, and let a ∈ CF * (f ) and b ∈ C U• (f ) be representatives of α and β of minimal action levels. We need to show that λ f (b) ≤ λ f (a) + 2δ, where λ f : CF * (f ) → R is the maximal action of an orbit, as defined in Notations 2.1. Notice that if a ∈ C U• (f ), then it represents in C U• (f ) a class in ι −1 * (a) and, by our choice of b, λ f (b) ≤ λ f (a), which concludes the proof for this case. Therefore we suppose that a contains critical points in M \ U • , which implies that λ f (a) > −δ. Assume for the sake of contradiction that λ f (b) > λ f (a) + 2δ, then λ f (b) > δ. Recalling that a and b are homologous in CF * (f ) (they both represent α), there exists c ∈ CF * (f )  (25) implies the min formula for the point class, which is equivalent, by Poincaré duality, to Theorem 45 in [11] (the max formula).

Spectral invariants of disjointly supported Hamiltonians.
In this section we use barricades to prove Theorem 2, which states that a max inequality holds for spectral invariants of Hamiltonians supported in disjoint CIB domains, with respect to a general class α ∈ H * (M ), and that equality holds when α = [M ]. Suppose F and G are two Hamiltonians supported in disjoint CIB domains. In order to prove the max inequality (4) for a homology class α ∈ H * (M ), we construct a representative of α in the Floer complex of (a perturbation of) the sum F + G, out of representatives from the Floer complexes of (perturbations of) F and G. The communication between the different Floer complexes is through continuation maps, corresponding to (perturbations of) linear homotopies. The barricades will be used to study the continuation maps, or, more accurately, their restrictions to the CIB domains. In particular, we will use the observation that having a barricade for a disjoint union implies having a barricade for each component: We start by arranging the set-up required for the proof of Theorem 2. 3. On U × S 1 (respectively, V × S 1 ) the homotopy h F (respectively, h G ) is a small perturbation of a constant homotopy, and its ends agree on their 1-periodic orbits up to second order. In particular, h F − and h F + (resp., h G− and h G+ ) have the same 1-periodic orbits in U (resp., V ).
Proof. Let H F and H G be linear homotopies from F and G, respectively, to the sum F + G. As in the proof of Lemma 4.4, we consider perturbations, h F and h G , of the linear homotopies, that, when paired with J, have a cylindrical bump around ∂U ∪ ∂V .
We demand in addition that all ends are non-degenerate, that the right ends coincide, h F + = h G+ , and that the homotopies are constant on U and V respectively, By proposition 3.6, these homotopies and their ends, when paired with J, have barricades in U ∪ V around U • ∪ V • . It remains to perturb again to ensure regularity. As in the proof of Theorem 5, we replace the ends with regular perturbations h F − , h G− and h F + = h G+ , without changing their periodic orbits (as cited in Claim 9.1, for example), then perturb the homotopies to glue to these regular perturbed Hamiltonians, and finally perturb the homotopies on the set M × S 1 × I for some fixed finite interval I, to obtain homotopies that are Floer regular when paired with J. The last step is possible due to Proposition 9.2) below. Proposition 9.21 states that barricades survive under perturbations that do not change the periodic orbits of the ends and are constant (as homotopies) outside of some fixed finite interval.
The following lemma is actually a part of the proof of Theorem 2, but, in our opinion, might be interesting on its own.
Proof. Let us show that c(F + G; α) ≤ c(F ; α). The result will follow by symmetry, since, if c(G; α) < c(F ; α), then it is in particular negative.
Let h F and J be the homotopy and almost complex structure from the Set-up Lemma, 6.2 (we will not use h G in this proof), and denote the left end of the homotopy by f := h F − . Then f approximates F and, since c(F ; α) < 0 and F | U c • = 0, we may assume that Recalling that, on U , the homotopy h F is a small perturbation of a constant homotopy, it follows from Corollary 9.34 that the restriction of the continuation map Φ (h F ,J) to orbits contained in U • is the identity map: The following example shows that a strict inequality can be attained in (26).
Example 6.4. Let (M, ω) be a genus-2 surface endowed with an area form, and let x, y : S 1 → M be two disjoint non-contractible loops representing two different homology classes α, β ∈ H 1 (M ; Z 2 ) respectively. Let F, G : M → R be two small Morse functions with disjoint supports, such that F vanishes on y and takes a negative value on x, whereas G vanishes on x and is negative on y. See Figure 8 for an illustration. After perturbing F , G and F + G into Morse functions, representatives of the sum α + β first appear for F and G on a sub-level set of values approximately zero. However, this sum of classes appears for F + G in a sub-level set with negative value. We therefore conclude that the spectral invariants of both F and G with respect to the sum α + β vanish. On the other hand, the spectral invariant of F + G is negative, and thus The following inequality is a simple application of Lemma 4.1 and Lemma 6.3, and will be used to prove that equality holds in (4) for the fundamental class. In what follows we prove that the spectral invariant of the sum F +G with respect to a homology class α is not greater than the maximum. The equality for the fundamental class will follow from Lemma 6.5. Consider the almost complex structure J, and the homotopies, h F and h G , from the Set-up Lemma, 6.2, and denote Set λ := max{c(f ; α), c(g; α)} and notice that, due to Lemma 6.3 and the continuity of spectral invariants, we may assume that λ ≥ −δ if δ > 0 is small enough. Letã ∈ CF * (f ), b ∈ CF * (g) be representatives of α of action levels λ f (ã), λ g (b) ≤ λ, then a := Φ (h F ,J)ã and b := Φ (h G ,J)b are both representatives of α in CF * (h + ). Notice that a and b might be of action-level higher than λ. We wish to construct out of a and b a representative of α of action level approximately bounded by λ. Let p be a primitive of a − b, and set d := ∂ (h + ,J) π V − π V ∂ (h + ,J) p. We claim that e := π V c a + π V b − d is a representative of α of the required action level. Indeed, Let us now bound the action level of e. First, notice that outside of U • ∪ V • , h + is a small Morse function (as it approximates a Hamiltonian that is supported in U • ∪ V • ). Therefore, its 1-periodic orbits there are its critical points and their actions are the critical values, which we may assume to be bounded by δ. It follows that the action level of the projection π U c • ∩V c • (e) is bounded by δ, and so it remains to bound the action levels of π U• e and π V• e. It follows form the fact that (h + , J) has a barricade in U around U • and in V around V • (more specifically, from (7)), that Using this observation, we bound the action levels of the projections of e: • λ h + (π U• e): Notice that π U• d = 0. Indeed, As a consequence, π U• e = π U• a = π U• Φ (h F ,J)ã . Since, on U , the homotopy h F is a perturbation of the constant homotopy, we can apply Corollary 9.34 and conclude that π U• • Φ (h F ,J) = π U• . Overall we obtain where we used the fact that in U , f = h F − and h + = h F + agree on their 1-periodic orbits, and hence the action of π U•ã with respect to h + coincides with the action with respect to f .
Here π V• d = 0 as well, but the computation is a little different: J)b and, since, on V , the homotopy h G is a perturbation of the constant homotopy, we apply Corollary 9.34 and conclude that where we used the fact that on V , g = h G− and h + = h G+ agree on their 1-periodic orbits, and hence the action of π V•ã with respect to h + coincides with the action with respect to g.
We conclude that

Boundary depth of disjointly supported Hamiltonians.
In this section, we use barricades to compare the boundary depths of disjointly supported Hamiltonians and that of their sum. As in the previous section, the communication between Floer complexes of different Hamiltonians is through continuation maps corresponding to homotopies that have barricades. Since we replace the Hamiltonians and their sum by regular perturbations, we will use the continuity property of the boundary-depth: As before, we use Notations 2.1. Let us start with a lemma that will enable us to push certain boundary terms from one Floer complex to another. Lemma 7.1. Let J be an almost complex structure and let h be a homotopy, such that the pairs (h, J) and (h ± , J) are Floer-regular and have a barricade in U around U • . Assume in addition that on U , h is a small perturbation of a constant homotopy, and that its ends, h ± , agree up to second order on their 1-periodic orbits in U . Then, every boundary term a ∈ ∂ (h + ,J) CF * (h + ) that is a combination of orbits in U • , namely a ∈ C U• (h + ), is also a boundary term in CF * (h − ).
Proof. We start with the observation that, since h − and h + are close on U and agree on their periodic orbits there, the vector spaces C U (h − ) and C U (h + ) coincide. Therefore, a boundary term a ∈ CF * (h + ) that is a combination of orbits from U • is also an element of C U• (h − ). Let us show that a is a boundary term in the Floer complex of (h − , J). As the homotopy h is close to a constant homotopy on U , we may use Corollary 9.34 and conclude that Φ (h,J) • π U• = π U• . Applying this equality to a, we obtain namely, a ∈ CF * (h + ) is the image of itself under the continuation map. As Φ (h,J) induces isomorphism on homologies, it is enough to show that a is closed in CF * (h − ), and it will then follow that it is a boundary term. To see that a is closed in CF * (h − ), notice that the presence of a barricade (in particular, (7) We are now ready to prove Theorem 3.

Proof of Theorem 3.
In what follows we show that β(F + G) ≥ β(F ). Inequality (5) follows by symmetry. Let H be a linear homotopy from F + G to F . Notice that, since F and F + G agree on U , H is a constant homotopy there. By Theorem 5, there exist a perturbation h of H and an almost complex structure J, such that the pairs (h, J) and (h ± , J) are Floer-regular and have a barricade in V containing the supports of F , G, respectively. Since H is a constant homotopy on U , it follows from Remark 3.7, item 2, that h can be chosen such that, in U , h ± agree on their 1-periodic orbits up to second order. We stress that h − approximates F + G and that f := h + approximates F . Hence, fixing an arbitrarily small δ > 0, we may assume (by taking h to be close enough to • are small Morse functions with values in (−δ, δ). Due to the continuity of the boundary depth, it is enough to prove that β(f ) is approximately bounded by β(h − ).
Fix a boundary term a ∈ CF * (f ), and let us show that there exists a primitive of a whose action level is bounded by λ f (a) + β(h − ) + 4δ, for δ that was fixed above. We prove this claim in two steps: Step 1: Assume that a is a combination of orbits that are contained in U • , namely a ∈ C U• (f ). Applying Lemma 7.1 to (h, J), we find that a ∈ CF * (h − ) is also a boundary term. Therefore, there exists b ∈ CF * (h − ) such that ∂ (h − ,J) b = a and λ h − (b) ≤ λ h − (a) + β(h − ). Let us split into two cases: Since h − is a small Morse function outside of U • ∪ V • , its 1-periodic orbits there are its critical points, and their actions are the critical values, which are all contained in the interval (−δ, δ). As a consequence, b is necessarily a combination of orbits that are contained in , the presence of the barricade (in particular, (7)) guarantees that Replacing b by π U• b, we still have a primitive of a of non-greater action level, as Therefore, we may assume that b ∈ C U• (h − ), and so it is also an element of C U• (f ). Recalling that h is a perturbation of a constant homotopy on U , Corollary 9.34 states that Φ (h,J) • π U• = π U• , and hence Φ (h,J) b = b and Φ (h,J) a = a. Thus, i.e., b is a primitive of a in CF * (f ), with small enough action level: Turning to bound the action of the projection onto U , recall that h is a perturbation of a constant homotopy on U , and by Corollary 9.34, π U • Φ (h,J) = π U . Overall, Step 2: Let us prove the claim for general a. Note that if λ f (a) < −δ then a ∈ C U• (f ) and the claim follows from the previous step. Therefore, we assume that λ f (a) ≥ −δ. Let b be any primitive of a in CF * (f ), namely, ∂ (f,J) b = a, and write Moreover, the presence of the barricade implies that a ∈ C U• (f ). Therefore, we may apply the previous step to a and obtain b ∈ CF * (f ) such that The following example shows that equality does not hold in (5). Example 7.2. Let M = T 2 be the two dimensional torus equipped with an area form and take F and G be disjointly supported C 2 -small non-negative bumps, see Figure 9. Approximating F , G and F + G by small Morse functions, their Floer complexes and differentials are equal to the Morse complexes and differentials. Hence, the Floer differentials of both F and G vanish and in particular β(F ) = 0 = β(G). On the other hand, β(F + G) = min{max F, max G}.

Min inequality for the AHS action selector.
In this section, we use barricades to prove a "min inequality" for the action selector defined by Abbondandolo, Haug and Schlenk, in [1], on symplectically aspherical manifolds. We start by reviewing the construction of this action selector, which we denote by c AHS , and state a few of its properties.
Let H : M × S 1 × R → R be a homotopy of Hamiltonians and let J : S 1 × R → J ω be a homotopy of time-dependent almost complex structures (that are compatible with ω). Assume that ∂ s H and ∂ s J have compact support and denote by H ± , J ± the ends of the homotopies. As before, we denote by M ( In [3], Abbondandolo, Haug and Schlenk proved that the functional c AHS is continuous and monotone, and that it takes values in the action spectrum, namely c AHS (F ) ∈ spec(F ). Let us state the result establishing the continuity of c AHS : In addition, they proved that the action selector takes non-positive values on Hamiltonians supported in incompressible Liouville domains. Using these claims, the barricades construction and ideas from the proof of Proposition 3.3 from [1], one can prove that a min inequality holds for c AHS .
Proof of Theorem 4. Let F and G be Hamiltonians supported in disjoint incompressible Liouville domains, which we denote by U and V respectively. Fixing an arbitrarily small δ > 0, we will prove that c AHS (F + G) ≤ c AHS (F ) + 3δ. The claim for G will follow by symmetry. We remark that by Claim 8.3, c AHS (F + G) ≤ 0, and hence the result is immediate if c AHS (F ) ≥ −3δ. Therefore, we assume that c AHS (F ) < −3δ. We break the proof into several steps.
Step 1: Our first step is to perturb F and F + G (as well as a homotopy between them) to create barricades. Let H be a linear homotopy from F to F + G that is constant outside of [0, 1], that is, ∂ s H| s / ∈[0,1] = 0. Then, H is supported in the domain U ∪ V , which, as an incompressible Liouville domain, is also a CIB domain. Applying Theorem 5 to the homotopy H and the domain U ∪ V , we conclude that there exists a perturbation h of H, an almost complex structure J and subsets U • U , V • V , containing the supports of F , G respectively, such that the pairs (h, J ) and (h ± , J ) are Floer-regular and have a barricade in U ∪V around U • ∪V • . In particular, the ends of h are non-degenerate, f := h − approximates F and h + approximates F +G. By taking h to be close enough to H, we can assume that, outside of U • , f is a small Morse function with values in (−δ, δ). Moreover, by Remark 3.7, item 3, we can choose the perturbation h such that the homotopy h is constant outside of [0, 1], namely, ∂ s h| s / ∈[0,1] = 0. Finally, taking these perturbations to be small enough, it follows from Claim 8.2 that c AHS (f ) < −2δ, and it is sufficient to prove that c AHS (h + ) ≤ c AHS (f ) + δ.
Step 2: Recalling the definition of the action selector c AHS , we need to show that for every (K, J) ∈ D(h + ), it holds that A(K, J) ≤ c AHS (f ) + δ. Therefore, our second step is to construct pairs in D(f ) out of a given pair in D(h + ). Fix (K, J) ∈ D(h + ) and assume, without loss of generality, that K and J stabilize for s ≤ 0, namely, K(x, t, s) = h + (x, t) and J(s) = J − for s ≤ 0. We construct a sequence of pairs in D(f ) by concatenating the homotopies (K, J) with shifts the homotopy h and a homotopyJ = {J s } s∈R of almost complex structures from J to J − , that is constant outside of [0, 1], namely, ∂ sJ | s / ∈[0,1] = 0. More precisely, for s ∈ R denote by τ s the shift by s, namely, τ s h(·, ·, ·) = h(·, ·, · + s) and τ sJ (·, ·) =J(·, · + s), and consider the sequences and J n := See Figure 10 for an illustration. Noticing that (K n , J n ) ∈ D(f ) for all n, we wish to show that there exists n ∈ N for which A(K, J) ≤ A(K n , J n ) + δ.
Step 3: In this step we choose, for each n, a solution minimizing a f and extract a subsequence that partially converges to a broken trajectory. Namely, there exists a broken trajectoryv = (v 1 , . . . , v N ), whose pieces v i are solutions of (FE) with respect to the homotopies concatenated in (K n , J n ), and are obtained as limits of non-positive shifts of elements from {u n }. In particular, for each i < N , the solution v i converges to periodic orbits at the ends, that match the limits of the adjacent pieces, i.e., lim s→+∞ v i (s, t) = lim s→−∞ v i+1 (s, t). Moreover, the left end of the first piece, lim s→−∞ v 1 (s, t), coincides with the left end of each element from the subsequence. We stress that unlike the standard convergence to a broken trajectory, in our case, the right end of the last piece inv (as well as the right ends of the solutions u n ) does not necessarily converge. The notion of partial convergence to a broken trajectory is defined formally in Proposition 9.17 below. Let u n ∈ M (Kn,Jn) be a minimizer of the functional a f , namely Since the supports of the homotopies (H n , J n ) are not uniformly bounded and the ends are not all non-degenerate, the sequence of solutions {u n } n does not necessarily converge to a broken trajectory. However, noticing that for s ≤ 0, (H n , J n ) are concatenations of homotopies with non-degenerate ends, one can prove a (weaker) convergence statement, as we do in Section 9.2.2. In this case, Proposition 9.17 guarantees that there exists a subsequence of {u n }, which we still denote by {u n }, partially converging to a broken where v (·,·), ∈ M (·,·) denote solutions of s-independent Floer equations, and w (·,·) ∈ M (·,·) denote solutions of s-dependent Floer equations. Moreover, the subsequence {u n } is chosen such that for each n, lim s→−∞ u n (s, ·) = x 1,0 (·), where x 1,0 := lim s→−∞ v (f,J ),1 (s, ·) ∈ P(f ).
Step 4: We now use the barricades in order to show that the first few pieces of the broken trajectory v are contained in U • . It follows from the arguments made above that which implies, by our assumptions on f , that x 1,0 ⊂ U • . We claim that, since (f, J ) and (h, 2 , we can repeat this argument and conclude that v (f,J ),2 is contained in U • . Continuing by induction, we find that {v (f,J ), } are all contained in U • and, in particular, Now, since (h, J ) has a barricade in U around U • , we conclude that w (h,J ) ⊂ U • as well.
Step 5: Let us now show that a h + (w (K,J) ) ≤ A f (x 1,0 ) + δ = a f (u n ) + δ. For that end, we bound the action growth along the broken trajectory v: 1. Along v (·,·), : these are solutions of the s-independent Floer equations and, by the energy identity (8), the action is clearly non-increasing.
2. Along w (h,J ) : this trajectory is contained in U • , where h approximates a constant homotopy, as F | U = F +G| U . Taking h to be close enough to H, we may assume that the derivative ∂ s h| U• is bounded by δ. Denoting by x 1,L 1 ∈ P(f ) and x 2,0 ∈ P(h + ) the orbits to which w (h,J ) converges at the ends, it follows from the energy-identity (10) that Overall, we conclude that Since u n were chosen to be minimizers, a f (u n ) = A(K n , J n ) ≤ c AHS (f ). On the other hand, the fact that w (K,J) ∈ M (K,J) implies that a h + (w (K,J) ) ≥ min M(K,J) (a h + ) = A(K, J). We therefore have proved that for any (K, J) ∈ D(h + ), A(K, J) ≤ c AHS (f ) + δ, which yields that c AHS (h + ) ≤ c AHS (f ) + δ, as required. 9 The required transversality and compactness results. 9.1 Perturbing homotopies and Hamiltonians to achieve regularity.
Let (M, ω) be a closed symplectically aspherical manifold. Given a non-degenerate Hamiltonian H and an almost complex structure J, we say that a pair (H, J) is Floerregular if for every pair of 1-periodic orbits x ± of H ± and for every u ∈ M (H,J) (x − , x + ), the differential (dF) u : W 1,p (u * T M ) → L p (u * T M ) of the Floer map (see Notations 9.9 below), is surjective. In this case, the space of solutions M (H,J) (x − , x + ) is a smooth manifold of dimension µ(x − ) − µ(x + ). It is well known that for any non-degenerate Hamiltonian H and an almost complex structure J, one can perturb H, without changing its periodic orbits, in order to make the pair (H, J) Floer-regular. Let us cite a formal statement of this fact. When H is a homotopy whose ends, H ± , are Floer-regular with respect to J, one can perturb H on a compact set to guarantee that the pair (H, J) is Floer-regular. For the purposes of this paper, we need to control the size of the support of the perturbation. In this section we prove that one can take the support of the perturbation to be any closed interval with non-empty interior. Before making a formal claim, let us fix some notations. Throughout this section, we consider homotopies of Hamiltonians, H : M × S 1 × R → R, that are constant with respect to the R-coordinate, s, outside of a compact set, namely supp(∂ s H) ⊂ M × S 1 × [−R, R] for some R > 0. We assume that the ends H ± (·, ·) := lim s→±∞ H(·, ·, s) are Floer-regular with respect to a fixed almost complex structure J. For a closed finite interval I ⊂ R with non-empty interior, we consider the space C ∞ ε (I) of perturbations with support in M × S 1 × I, whose definition is given in Section 9.1.1 below. Our main goal for this section is to prove the following proposition.
Proposition 9.2. Let H be a homotopy such that (H ± , J) are Floer-regular, where J is an almost complex structure on M , and let I ⊂ R be a closed, finite interval with a non-empty interior. Then, there exists a residual subset H reg ⊂ C ∞ ε (I), such that for every h ∈ H reg , the pair (H + h, J) is Floer-regular.
The proof of this proposition is postponed to Section 9.1.2. We start by describing the space of perturbations and its relevant properties.
In this section we define the perturbations space C ∞ ε (I) and prove useful properties. Definition 9.3.
• Let ε = {ε n } be a sequence of positive numbers. For h ∈ C ∞ (M × S 1 × R), Floer's ε-norm is defined to be see [3, p.230] for details. For a proof that it is a norm, see [22,Theorem B.2] • For a closed and finite interval I ⊂ R with a non-empty interior, let C ∞ ε (I) be the In what follows we identify between the tangent space T h C ∞ ε (I) at a point h, and the space C ∞ ε (I) itself. The following claims guarantee that the properties that are required of a space of perturbations hold for C ∞ ε (I). Claim 9.4. There exists a sequence ε for which C ∞ ε (I) is dense in C ∞ (I). Claim 9.5. The Banach space C ∞ ε (I) is separable. In order to prove these claims we first state and prove two lemmas. We use notations and ideas from [3,Section 8.3] and [22,Appendix B]. Lemma 9.6. Let E be a finite dimensional vector bundle over M ×S 1 ×R, then, the space C 0 I (E) of continuous sections of E that are supported in M × S 1 × I is second countable with respect to the uniform norm. 17 . By Weierstrass approximation theorem, the latter space is separable, and hence (being a normed space) is also second countable. We conclude that the same holds for the closed subspace C 0 I (E). Following [22,Appendix B], set E (0) := E and E (k+1) := Hom T (M × S 1 × R); E (k) , then, fixing connections and bundle metrics on both T (M × S 1 × R) and E, any section η ∈ Γ(E (k) ) has a covariant derivative ∇η ∈ Γ(E (k+1) ). Set F (k) := E (0) ⊕ · · · ⊕ E (k) and consider the countable product k∈N C 0 I (F (k) ), endowed with the product topology. By Lemma 9.6, each factor is second countable and therefore so is the product.
Lemma 9.7. The space C ∞ (I) of smooth functions M × S 1 × R → R supported on M × S 1 × I, is separable with respect to the C ∞ -topology.
As explained above, the product k∈N C 0 I (F (k) ) is second countable and hence so is any closed subspace of it. In particular, C ∞ (I) is separable.
We are now ready to prove Claim 9.4. The proof is exactly that of [3, Proposition 8.3.1].
Proof of Claim 9.4. Let f n ∈ C ∞ (I) be a dense sequence, whose existence is guaranteed by Lemma 9.7. Let For this choice of a sequence ε, it holds that f n ε < ∞ for all n, namely, f n ∈ C ∞ ε (I).
The proof of Claim 9.5 is essentially that of Lemma B.4 and Theorem B.5 from [22], we include it for the convenience of the reader.
Proof of Claim 9.5. Consider again the product k∈N C 0 I (F (k) ) and let X ε be the space of sequences ξ := (ξ 0 , ξ 1 , ξ 2 , . . . ) ∈ k∈N C 0 I (F (k) ) such that We will first show that X ε is separable and then embed C ∞ ε (I) into X ε in order to prove the claim. Indeed, since C 0 I (F (k) ) is separable for each k (by Lemma 9.6), we can fix a dense countable subset P k ⊂ C 0 I (F (k) ). The set P := (ξ 0 , . . . , ξ N , 0, 0 . . . ) ∈ X ε | N ≥ 0 and ∀ 0 ≤ k ≤ N, ξ k ∈ P k is countable and dense in X ε . Now consider the injective linear map It is an isometric embedding, and hence we may view C ∞ ε (I) as a closed subspace of the separable space X ε . The latter is also second countable (being a normed space) and hence so is C ∞ ε (I).
Remark 9.8. The proof of Claim 9.5 shows that spaces of perturbations with compact support are separable in general. This observation will be used in Section 9.3.2.

Proof of Proposition 9.2.
We follow the proofs from Chapters 8 and 11 of [3] and make the necessary changes. Let us start by recalling the relevant notations. Notation 9.9. Let H be a homotopy, let J be an almost complex structure, and let x ± be 1-periodic orbits of H ± respectively.
• We denote by M (H,J) (x − , x + ) the set of solutions of the (s-dependent) Floer equation with respect to H, J that converge to x ± at the ends. We denote by M (H,J) the set of all finite energy solutions.
for Y ∈ W 1,p (w * T M ) and w ∈ C ∞ (x − , x + ). The latter is the space of smooth maps R × S 1 → M converging to x ± at the ends with exponentially decaying derivatives. We denote by L p (x − , x + ) the fiber bundle over P(x − , x + ) whose fiber at u is L p (u * T M ).
• The Floer map with respect to H is where (grad u H)(s, t) is the gradient of H(·, t, s) with respect to J, restricted to u. In unitary (i.e., symplectic, orthonormal) coordinates, the differential of the • Set The main ingredients in the proof of Proposition 9.2 are the following two lemmas.
Lemma 9.10. The set Z(x − , x + ) is a Banach manifold.
Lemma 9.11. The projection π : Z(x − , x + ) → C ∞ ε (I), (u, h) → h, is a Fredholm map. The outline of the proof is as follows: We first prove that the set Z(x − , x + ) is a Banach manifold (Lemma 9.10), and then we show that the projection π : Z(x − , x + ) → C ∞ ε (I) is a Fredholm map (Lemma 9.11). Taking H reg to be the set of regular values of π, the Sard-Smale theorem guarantees that it is a residual set. We will use the following claim from [3]. In order to prove Lemma 9.10, we present Z(x − , x + ) as an intersection of a certain section with the zero section in a certain vector bundle. The following lemma will be used to guarantee that this intersection is transversal. Its proof, which is a combination of the proofs of [3, Propositions 8.1.4, 11.1.8], contains the main difference between the proof of Proposition 9.2 and that of [3, Theorem 11.1.6].
Lemma 9.13. For (u, h) ∈ Z(x − , x + ), the linear operator is surjective and has a continuous right inverse.
Proof. Assume for the sake of contradiction that Γ is not surjective. By [3, Lemma 8.5.1] 18 , there exists a non-zero vector field Z ∈ L q (R × S 1 ; R 2n ) (here 1 p + 1 q = 1), of class C ∞ , such that for every Y ∈ W 1,p (R × S 1 ; R 2n ) and η ∈ C ∞ ε (I), where ·, · denotes the pairing of L q and L p . As mentioned above, the differential of the Floer map can be written in unitary coordinates as ∂ + S(s, t). Since Z is of class C ∞ , it follows from (34) that Z is a zero of the dual operator of (dF) u , which is of a "perturbed Cauchy-Riemann"-type. The continuation principle ([3, Proposition 8.6.6]) now implies that if Z has an infinite-order zero, then it is identically zero, Z ≡ 0. Therefore, let us show that (35) guarantees that Z vanishes on I × S 1 , and conclude that it vanishes identically, since we assumed that the interior of I is not empty. The proof is roughly the same as that of [3, Lemma 11.1.9], but we include it for the sake of completeness. An equivalent reformulation of (35) is: Consider the mapũ : , t), s, t). It is easy to see thatũ is an embedding. Viewing Z as a vector field alongũ on M × R × S 1 that does not have components in the directions ∂/∂t ∈ T S 1 and ∂/∂s ∈ T R, we see that it is not tangent toũ at the points where it is not zero. Assume for the sake of contradiction that there exists a point (s 0 , t 0 ) ∈ I × S 1 at which Z does not vanish. Since Z is continuous, there exists a small neighborhood C δ of (s 0 , t 0 ), in which Z(s, t) does not vanish and therefore is transversal toũ for all (s, t) ∈ C δ . Notice that if (s 0 , t 0 ) is not in the interior of I × S 1 , we may replace it with a point in C δ ∩ (int(I) × S 1 ), and then replace C δ by a smaller neighborhood that is contained in int(I) × S 1 . Therefore we assume, without loss of generality, that C δ ⊂ int(I) × S 1 . Let β : R × S 1 → R be a smooth function supported in C δ , whose integral is not zero, R×S 1 β(s, t) ds dt = 0. Define η : M × S 1 × R → R with support in a tubular neighborhood B ofũ(C δ ) in such a way that if γ (s,t) (σ) is a parametrized integral curve of Z passing throughũ(s, t) at σ = 0, then η(γ (s,t) (σ), t, s) := β(s, t) · σ, for |σ| ≤ .
The fact that Z is transversal toũ(C δ ) guarantees that η is well defined. We also assume that B ∩ Im (ũ) =ũ(C δ ), which means that supp(η) ∩ Im (ũ) ⊂ũ(C δ ). Let us compute the integral of dη(Z): As we chose β to be a function with a non-vanishing integral, we find that (35) does not hold for the function η constructed above. Note that η is a smooth function, supported in M × S 1 × I, but its ε-norm is not necessarily finite. Therefore, to arrive at a contradiction, it remains to approximate η by η ∈ C ∞ ε (I). This is possible due to Claim 9.4. When η is close to η, the integral of dη (Z) will be close to that of dη(Z) (since their supports are contained in the compact set M × S 1 × I), and hence equality (35) will not hold for η ∈ C ∞ ε (I), in contradiction. This shows that Γ is surjective. The fact that it has a continuous right inverse follows from [3, Lemma 8.5.6] and Claim 9.12.
Having Lemma 9.13, the proof of Lemma 9.10, which asserts that Z(x − , x + ) is a Banach manifold, is precisely that of [3, Proposition 8.1.3]: Proof of Lemma 9.10.
and consider the section induced by F H+h : Notice that the space Z(x − , x + ) is the intersection of σ with the zero section in E. Therefore, in order to prove that Z(x − , x + ) is a Banach manifold, it is sufficient to show that σ intersects the zero section transversally, or, equivalently, that dσ composed with the projection onto the fiber is surjective and has a right inverse, at all points for which σ(u, h) = 0. But, this composition is precisely the operator Γ whose surjectivity and right-invertability are guaranteed by Lemma 9.13.
Our next goal is to show that π is a Fredholm map, that is, to prove Lemma 9.11.
Proof of Lemma 9.11. The projection π : Z(x − , x + ) → C ∞ ε (I), π(u, h) = h, is clearly smooth. Let us show that its differential, dπ, has a finite dimensional kernel and a closed image of finite co-dimension.
and therefore, the kernel of (dπ) (u,h) agrees with the kernel of (dF H+h ) u , which, is finite dimensional by Claim 9.12.
By Claim 9.12, the image of (dF H+h ) u is closed and of finite co-dimension. Let us show that the same hold for the image of (dπ) (u,h) . Consider the map induced by G on the quotients, which is well defined due to (36). It is easy to see that G is injective and, together with the fact that B is finite dimensional, this yields that codim(Im (dπ) (u,h) ) = dim(A) is finite. This now implies that the image of (dπ) (u,h) is also closed and hence (dπ) (u,h) is a Fredholm operator.
Having proved Lemmas 9.10 and 9.11, we are ready to prove the main proposition.
Proof of Proposition 9.2. By Lemma 9.11, the projection π : Z(x − , x + ) → C ∞ ε (I) is a (smooth) Fredholm map. By Claim 9.5, the space C ∞ ε (I) is separable. To see that Z(x − , x + ) is a separable Banach manifold, recall that it is modeled over a subspace of the Banach space W 1,p (R × S 1 ; R 2n ) × C ∞ ε (I). The latter is a separable metric space, and therefore second-countable. As any subspace of a second-countable space is also secondcountable, and, in particular, separable, we conclude that Z(x − , x + ) is separable. It follows that we may apply Sard-Smale's theorem to π and conclude that the set of regular values of π is a countable intersection of open dense sets in C ∞ ε (I). The set H reg ⊂ C ∞ ε (I) is defined to be the intersection of the regular values of the projections for all choices of 1-periodic orbits, x ± .
Let us show that for each h ∈ H reg , the pair (H + h, J) is Floer-regular. Fix 1-periodic orbits x ± , then h is a regular value of the projection π : Z(x − , x + ) → C ∞ ε (I). Let us show that for every u ∈ M (H+h,J) (x − , x + ), the differential of the Floer map, (dF H+h ) u , is surjective. Indeed, otherwise, arguing as in the proof of Lemma 9.13, there exists Z ∈ L q (R×S 1 ; R 2n ), where 1 p + 1 q = 1, such that Z, (dF H+h ) u (Y ) = 0 for all Y . Since (dπ) (u,h) is surjective, for every η ∈ C ∞ ε (I), there exists Y such that grad u η = −(dF H+h ) u (Y ), and hence Z, grad u η = 0 as well. We conclude that Z satisfies both equations (34) and (35), and, proceeding as in the proof of Lemma 9.13, we find Z = 0. Thus (dF H+h ) u is indeed surjective.
It remains to show that M (H+h,J) (x − , x + ) is a smooth manifold of the correct dimension. The inverse image π −1 (h) is the space of maps u ∈ P(x − , x + ), of class W 1,p , that are solutions of the Floer equation, F H+h (u) = 0. By elliptic regularity, these solutions are all smooth, and hence π −1 (h) = M (H+h,J) (x − , x + ). Since h is a regular value of π, we therefore conclude that M (H+h,J) (x − , x + ) is indeed a smooth manifold. Its dimension is where the last equality follows from Claim 9.12 above.

Convergence to broken trajectories.
A well known phenomenon in Floer theory on symplectically aspherical manifolds is the convergence of sequences of solutions to a broken trajectory. In this section we formulate and prove results of this sort for the settings that are considered throughout the paper.

Convergence for homotopies with non-degenerate ends.
In what follows we consider homotopies with non-degenerate ends. We remark that the same arguments apply for non-degenerate Hamiltonians, when one considers them as constant homotopies. Let H be a homotopy that is constant outside of M × S 1 × [−R, R] for some fixed R > 0, namely, ∂ s H| |s|>R = 0. Let H n be a sequence of homotopies converging H, such that for each n, Proposition 9.14. Let H be a homotopy with non-degenerate ends, and let H n be a sequence converging to H in C ∞ (M × S 1 × R) that satisfies (37) for each n. Given a sequence u n ∈ M (Hn,J) (x − , x + ) of solutions and a sequence of real numbers {σ n }, there exist: • Subsequences of {u n } and {σ n }, which we still denote by {u n } and {σ n }, • periodic orbits x − = x 0 , x 1 , . . . , x k ∈ P(H − ) and y 0 , y 1 , . . . , y = x + ∈ P(H + ), • sequences of real numbers {s i n } n for 1 ≤ i ≤ k and {s j n } n for 1 ≤ j ≤ , and the sequence u n (· + σ n , ·) converges to one of v i , w, v j , perhaps up to a shift in the s-coordinate.
The finite sequence (v 1 , . . . , v k , w, v 1 , . . . , v ) is called a broken trajectory of (H, J). Before proving the above proposition, we state and prove two lemmas. The first is an analogous statement to [3,Theorem 11.2.7], and gives a uniform bound for the J-gradient of a solution u of the Floer equation with respect to (H, J) or (H n , J). Proof. For convenience we set H 0 := H. Let x ± ∈ P(H ± ) be periodic orbits such that u ∈ M(x − , x + ), then, by the energy identity (10), where and R > 0 is the constant from (37). The fact that C is finite follows from the uniform convergence (with derivatives) of H n to H 0 = H. Setting we obtain a uniform bound for the energy, E(u) ≤ C, for all u ∈ M. As in [3, Propositions 6.6.2, 11.1.5], we conclude that there exists A > 0 such that ∂u ∂s 2 J + ∂u ∂t 2 J ≤ A. The next lemma uses Arzelá-Ascoli theorem and elliptic regularity to show that every sequence of shifted solutions has a converging subsequence. It is an adjustment of Theorem 11.3.7 and Lemma 11.3.9 from [3] to our setting.
Lemma 9.16. Let u n ∈ M (Hn,J) (x − , x + ) be a sequence of solutions and let s n ∈ R be a sequence of numbers. Then, the sequence of shifted solutions τ sn u n (·, ·) = u n (· + s n , ·) has a subsequence that converges in the C ∞ loc topology to a limit v. Moreover: 1. If s n → σ ∈ R, then v ∈ M (τσH,J) , where τ σ H(x, t, s) := H(x, t, s + σ).
Proof. Lemma 9.15 implies that the sequence v n := τ sn u n is equicontinuous. By Arzelá-Ascoli theorem and elliptic regularity (see [3,Lemma 12.1.1]), there exists a subsequence, which we still denote by {v n }, that converges to a limit v in the C ∞ loc topology. The fact that the energy of v is finite follows from the uniform bound (38) on the energies of u n . It remains to show that the limit v is a solution of the corresponding equation, for the above choices of shifts s n . For each n, v n is a solution of the equation 0 = ∂v n ∂s + J ∂v n ∂t + grad vn (τ sn H n ) = ∂v n ∂s + J ∂v n ∂t + grad vn (τ sn H) + grad vn (τ sn (H n − H)).
Since the sequence H n converges to H uniformly with the derivatives, for every > 0 there exists N such that for n ≥ N , Let us split into cases: when n is large enough. It follows that the limit v of the sequence v n is a solution of the s-dependent Floer equation with respect to τ σ H and J. 3. When s n → ∞, the proof is as in the previous case.
Having Lemma 9.16, the proof of Proposition 9.14 (namely, the convergence to a broken trajectory) is similar to the that of [3,Theorem 11.1.10]. We follow it and make the necessary adjustments.
Proof of Proposition 9.14. Let us prove the claim for the case where σ n → −∞, the other cases are analogous. We start by fixing > 0 small enough, such that the open balls Here LM is the space of contractible loops in M , endowed with the uniform metric d ∞ . By shrinking if necessary, we assume that the balls {B(y, )} y∈P(H + ) are also disjoint. Lemma 9.16 guarantees that after passing to a subsequence, the sequence τ σn u n converges in C ∞ loc to a finite energy solution v ∈ M (H − ,J) . Since H − is non-degenerate, there exist periodic orbits x 0 , x 1 ∈ P(H − ) such that v ∈ M (H − ,J) (x 0 , x 1 ). Moreover, applying Lemma 9.16 to the sequence u n with zero shifts, we conclude that after extracting a subsequence, it converges to a finite energy solution w ∈ M (H,J) (x k , y 0 ), for some x k ∈ P(H − ) and y 0 ∈ P(H + ). Let us find the solutions preceding to v, connecting v to w and following w in the broken trajectory: • Solutions preceding v: There exists s ≤ 0 such that for any s ≤ s , v(s, ·) ∈ B(x 0 , ).
Since v = lim τ σn u n , when n is large enough, u n (s + σ n , ·) ∈ B(x 0 , ) as well. If Let us now show that s n → −∞. Indeed, if {s n } were bounded, it would have had a subsequence converging to some s • ∈ R. Since τ σn u n converges to v in C ∞ loc and since s • ≤ s , we would get lim n→∞ u n (s n + σ n , ·) = v(s • , ·) ∈ B(x 0 , ), in contradiction to our choice of s n , namely, that u n (σ n + s n , ·) ∈ ∂B(x 0 , ). Therefore, we conclude that s n → −∞ and, in particular, s n + σ n → −∞ as well. Using Lemma 9.16 for τ sn+σn u n , we conclude that, after passing to a subsequence, this shifted sequence converges to some v −1 ∈ M (H − ,J) . We need to prove that v −1 converges to x 0 when s → ∞. Fix s > 0, then for n sufficiently large, s n < s + s n < s and τ sn+σn u n (s, ·) ∈ B(x 0 , ).
Continuing in this way we find v −2 , v −3 and so on, until x −k = x − . This process is finite, since there are finitely many orbits in P(H − ) and the action is strictly decreasing in each step, namely, • Solutions connecting v to w: Recall that τ σn u n converges to v ∈ M (H − ,J) (x 0 , x 1 ) and that u n converges to w ∈ M (H,J) (x k , y 0 ). Let us find the solutions that connect v to w (or prove that x 1 = x k ). In analogy with the previous case, pick s ≥ 0 such that v(s, ·) ∈ B(x 1 , ) for all s ≥ s . Then, for n large enough, u n (s + σ n , ·) ∈ B(x 1 , ) as well. Arguing similarly for w ∈ M (H,J) (x k , y 0 ), there exists s † ≤ 0 such that w(s, ·) ∈ B(x k , ) for all s ≤ s † and, since u n converge to w, for n large enough, u n (s † , ·) ∈ B(x k , ) as well. As σ n → −∞, we have s + σ n < s † for large n.
Consider the first exit of u n from B(x 1 , ), then, repeating the arguments from the previous step, one sees that s n → ∞. Moreover, it follows from the definitions of s n and s † that s n + σ n < s † . Therefore, the sequence {σ n + s n } is either bounded or tends to −∞. In the first case, it converges, after passing to a subsequence, to some number s • ∈ R. Moreover, since u n converges to w on compacts, we conclude that τ σn+sn u n converges to τ s• w.
In particular, this implies that x 1 = x k . Indeed, for every s < s • and n sufficiently large, s ∈ [σ n + s , σ n + s n ], and thus u n (s, ·) ∈ B(x 1 , ). As a consequence, w(s, ·) ∈ B(x 1 , ) for all s < s • , which means that x k = x 1 and we are done. Let us now deal with the case where s n + σ n → −∞. By Lemma 9.16, there exists a subsequence of τ sn+σn u n that converges to a finite energy solution v 1 ∈ M (H − ,J) . We need to show that the left end of v 1 converges to x 1 , namely, that v 1 ∈ M (H − ,J) (x 1 , x 2 ) for some x 2 ∈ P(H − ). Fix s < 0 and let us show that v 1 (s, ·) ∈ B(x 1 , ). Since s n → ∞, when n is large enough, s + s n ∈ [s , s n ]. As we saw above, this implies that τ σn+sn u n (s, ·) = u n (s + s n + σ n , ·) ∈ B(x 1 , ), and thus v 1 (s, ·) ∈ B(x 1 , ) as required. Repeating this process, we find solutions v 2 , · · · v k−1 such that v i ∈ M (H − ,J) (x i , x i+1 ), and therefore these connect v to w. As in the previous case, this process is finite since every solution v i is action decreasing and H − has finitely many 1-periodic orbits.
• Solutions following w: The right end of w converges to y 0 ∈ P(H + ), and hence there exists s ≥ 0 such that for every s ≥ s , w(s, ·) ∈ B(y 0 , ). As u n converge to w in C ∞ loc , for n large enough, u n (s , ·) ∈ B(y 0 , ) as well. Assume that y 0 = x + , otherwise there is nothing to prove. Then, since u n converge to x + for each n, is must leave the ball B(y 0 , ) at some point. Consider the first exit, s n := sup{s ≥ s | u n (s , ·) ∈ B(y 0 , ) for s ∈ [s , s]}, then, arguing as above, s n → ∞. Applying Lemma 9.16 to the sequence u n shifted by s n , it converges (up to a subsequence) to a finite energy solution v 1 ∈ M (H + ,J) . We need to show that the left end of v 1 converges to y 0 , namely, that v 1 ∈ M (H + ,J) (y 0 , y 1 ) for some y 1 ∈ P(H + ). As before, fix any s < 0, then when n is large enough, s + s n ∈ [s , s n ] and therefore, τ sn u n (s, ·) = u n (s + s n , ·) ∈ B(y 0 , ). Again, we conclude that v 1 (s, ·) ∈ B(y 0 , ), which guarantees that v 1 converges to y 0 . Continuing by induction and using the fact that each v j reduces the action concludes the proof.
x k−1 Figure 11: An illustration of the broken trajectory as constructed in the Proof of Proposition 9.14.

Concatenation of homotopies with possibly degenerate ends.
In what follows, we study the breaking mechanism for solutions of (FE) with respect to homotopies of Hamiltonians, that are obtained as concatenations of finitely many given homotopies, with possibly degenerate ends. In addition, we consider homotopies of almost complex structures, as opposed to the constant structures considered previously. When the ends of the first few concatenated homotopies are non-degenerate, we prove what we call a partial convergence to a broken trajectory.
Let n } n , . . . , {σ K n } n be monotone sequences of real numbers, such that for each n, σ 1 n < · · · < σ K n and for each k = j, the sequence of differences {σ k n − σ j n } n is unbounded. For the rest of this section, we consider the sequences {H n } and {J n } of homotopies of Hamiltonians and almost complex structures obtained by concatenating the shifts of {H k } and {J k } by the sequences {−σ k n }. More formally, H n and J n are the sequences satisfying for each k = 1, . . . , K, and are locally constant elsewhere, see Figure 12. Since the ends of the homotopies H k might be degenerate, a sequence of solutions u n ∈ M (Hn,Jn) does not necessarily admit a subsequence converging to a broken trajectory. However, when some of the homotopies have non-degenerate ends, a slightly weaker statement holds: Proposition 9.17. Assume that there exists 1 < K ≤ K, such that for every k < K , the ends of the homotopy H k are non-degenerate. Then, for every sequence u n ∈ M (Hn,Jn) , there exist: • A subsequence of {u n }, which we still denote by {u n }, • periodic orbits x k, ∈ P(H k+1 − ) for = 0, . . . , L k and k = 1, . . . , K , where x 1,0 = lim s→−∞ u n (s, ·) for all n, • real numbers s k, n ∈ R for = 1, . . . , L k and k = 1, . . . , K , such that s k, n < σ k+1 n < s k+1, n for all = 1, . . . , L k and = 1, . . . , L k+1 , lim n→∞ u n (· + σ k n , ·) = w k and lim n→∞ u n (· + s k, n , ·) = v k, .
In this case, we say that {u n } partially converges to the broken trajectory In order to prove Proposition 9.17, we need statements analogous to Lemma 9.15 and Lemma 9.16 adapted for the current setting. Notice that due to our assumption, that H k has non-degenerate ends for 1 ≤ k < K , the left end of the homotopies H n , which is equal to H 1 − , is non-degenerate. On the other hand, the right end, H n+ = H K + , might be degenerate. A solution u of the Floer equation with respect to a homotopy with degenerate ends does not necessarily converge to periodic orbits at the ends. However, the following lemma asserts that the action of u(s, ·) converges as s → ±∞, to a limit that belongs to the action spectrum of the corresponding Hamiltonian. The following statement is proved in the proof of Proposition 2.1, (ii) from [1] for the left end of u, namely, lim s→−∞ A H − (u(s, ·)) ∈ spec(H − ). The proof for the right end is completely analogous and we therefore omit it. Denoting by M := ∪ n M (Hn,Jn) the set of finite energy solutions, the next lemma provides a uniform bound for the energy of u ∈ M and is an adjustment of Lemma 9.15 to the current setting.
Lemma 9.19. There exists a constant A > 0 such that for every u ∈ M and (s, t) ∈ R × S 1 , one has grad (s,t) u ≤ A.
Proof. For a finite energy solution u of a homotopy with possibly degenerate ends, the limits lim s→±∞ A H ± (u(s, ·)) exist and u satisfies the following energy-identity: , t), t, s) ds dt, see, for example, [1, p.8]. When u ∈ M (Hn,Jn) , it follows from Lemma 9.18, together with the fact that the action spectrum is a compact subset of R, that where, by our construction, K bounds the area of the support of max x∈M ∂ s H n (x, t, s) in S 1 × R, and C is defined by We therefore have obtained a uniform bound on the energies of solutions in M. Arguing as in [3, Propositions 6.6.2, 11.1.5], we conclude that there exists A > 0 such that grad (s,t) u ≤ A.
The last lemma for this section is analogous to Lemma 9.16. It can be viewed as a special case of Proposition 2.1 from [1], but we include the proof for the sake of completeness.
Lemma 9.20. Let u n ∈ M (Hn,Jn) be a sequence of solutions and let s n ∈ R be a sequence of numbers such that, for some 0 ≤ k ≤ K and for every n, σ k n ≤ s n ≤ σ k+1 n , where we set σ 0 n = −∞ and σ K+1 n = +∞ to simplify the notations. Then, the sequence of shifted solutions τ sn u n (·, ·) = u n (· + s n , ·) has a subsequence that converges in the C ∞ loc topology to a limit v. Moreover: 1. If s n − σ k n → σ ∈ R, then v ∈ M (τσH k ,τσJ k ) .
3. If s n − σ k+1 n → −∞ and s n − σ k n → ∞, then v ∈ M (H k The proof is very similar to that of Lemma 9.16 and therefore we only sketch the changes. As before, Lemma 9.19 implies that the sequence v n := τ sn u n is equicontinuous, and by the Arzelà-Ascoli theorem and elliptic regularity there exists a subsequence converging to v. The maps v n solve the Floer equation with respect to the translated pair (τ sn H n , τ sn J n ): In each case, in order to prove that v is a solution of the corresponding equation, one shows that the translated homotopies (τ sn H n , τ sn J n ) converge uniformly on compacts to the required pair. For example, in the first case, where s n −σ k n → σ ∈ R, it follows from the definition of (H n , J n ) that, given r > 0, the sequence (τ σ k n H n , τ σ k n J n ) eventually stabilizes to (H k , J k ) on {|s| ≤ r}. As a consequence, (τ sn H n , τ sn J n ) We are now ready to sketch the proof of Proposition 9.17. Note that we will skip some of the details appearing in the proof of Proposition 9.14.
Proof of Proposition 9.17. As mentioned above, for each n, the left end of H n is a nondegenerate Hamiltonian. As a consequence, the left end of u n converges to a periodic orbit, namely, there exist x n− ∈ P(H n− ), such that lim s→−∞ u n (s, ·) = x n− (·) (see, for example, the proof of Theorem 6.5.6 from [3]). Since P(H n− ) = P(H 1 − ) is a finite set, we may assume, by passing to a subsequence, that x 1,0 := x n− is independent of n.
Next, let us apply Lemma 9.20 to the sequence {u n } with the shifts σ k n . Then, after passing to a subsequence, for each k = 1, . . . , K we obtain w k ∈ M (H k ,J k ) such that τ σ k n u n C ∞ loc − − → w k . Fixing 1 ≤ k ≤ K , we need to find solutions {v k, } L k =1 connecting w k−1 to w k (and x 1,0 to w 1 ). It follows from the non-degeneracy of H k − that P(H k − ) = P(H k−1 + ) is a finite set (notice that the left end of H K is non-degenerate, as it coincides with the right end of H K −1 ). Therefore, we can repeat the arguments from the proof of Proposition 9.14. For > 0 small enough, the balls {B(x − , )} x − ∈P(H k − ) are disjoint, and denoting y k := lim s→−∞ w k (s, ·) and x k,0 = lim s→∞ w k−1 (s, ·), there exists s ∈ R such that w k−1 (s, ·) ∈ B(x k,0 , ) for s ≥ s . It follows from the convergence of τ σ k−1 n u n to w k−1 that u n (s + σ k−1 n , ·) ∈ B(x k,0 , ) when n is large. Denoting by s k,1 n := sup{s ≥ s + σ k−1 n | u n (s , ·) ∈ B(y k , ) for s ∈ [s + σ k−1 n , s]} the first exit, one can argue as in the proof of Proposition 9.14 to show that s k,1 n − σ k−1 n −−−→ n→∞ ∞. Applying Lemma 9.20 to {u n } shifted by s k,1 n , we conclude that τ s k,1 n u n either converges to τ σ w k , for some σ ∈ R, or to v k, ) . In the first case, the right end of w k−1 and the left end of w k coincide, namely x k,0 = y k , and we are done. Otherwise, we continue by induction and find v k, ∈ M (H k − ,J k − ) connecting w k−1 to w k . As argued previously, this process is finite since each v k, decreases the action, and spec(H k − ) is a finite set.

Barricades and perturbations.
Throughout this section, we fix an almost complex structure J on M , a CIB domain U , and U • U . We will consider non-degenerate Hamiltonians, or homotopies with non-degenerate ends, that have a barricade in U around U • , when paired with J. In this section we show that barricades survive under perturbations of H. Here H denotes a homotopy with non-degenerate ends and we consider Hamiltonians as a special case, by identifying them with constant homotopies.
Proposition 9.21. Let H be a homotopy with non-degenerate ends, such that ∂ s H| |s|>R = 0 for some R > 0 (in particular, H can be a non-degenerate Hamiltonian), and such that the pairs (H, J), (H ± , J) have a barricade in U around U • . Then, for every C ∞ -small enough perturbation H of H that satisfies P(H ± ) = P(H ± ) and ∂ s H | |s|>R = 0, the pair (H , J) has a barricade in U around U • .
In order to prove this proposition, we will use the convergence to broken trajectories, which was established in Section 9.2.1. Therefore, we start by showing that barricades also restrict broken trajectories.
Lemma 9.22. Let H be a homotopy with non-degenerate ends (or, in particular, a nondegenerate Hamiltonian) such that the pairs (H, J) and (H ± , J) have a barricade in U around U • . Then, for a broken trajectory v = (v 1 , . . . , v k , w, v 1 , . . . , v ) connecting x ± ∈ P(H ± ), we have: Proof. We prove the first statement, the second statement is completely analogous. Let v := (v 1 , . . . , v k , w, v 1 , . . . , v ) be a broken trajectory of (H, J) such that the periodic orbit x 0 := lim s→−∞ v 1 (s, ·) is contained in U • . Then, since (H − , J) has a barricade in U around U • , v 1 ⊂ U • and in particular, the periodic orbit x 1 := lim s→+∞ v 1 (s, ·) is contained in U • . By definition (see Proposition 9.14), x 1 is the negative end of v 2 , namely x 1 = lim s→−∞ v 2 (s, ·). Applying the same argument again and again, we conclude that v 2 , . . . , v k ⊂ U • . Now, x k := lim s→+∞ v k (s, ·) = lim s→−∞ w(s, ·) is also contained in U • and since (H, J) has a barricade in U around U • , this means that w ⊂ U • . Arguing the same way and using the fact that (H + , J) has a barricade in U around U • we conclude that v j ⊂ U • for all 1 ≤ j ≤ , and so the broken trajectory is completely contained in U • .
Given the above lemma, the proof of Proposition 9.21 is a simple application of Proposition 9.14.
Proof of Proposition 9.21. Let {H n } be a sequence of regular homotopies converging to H, such that for each n, P(H n± ) = P(H ± ) and ∂ s H n | |s|>R = 0. Assume for the sake of contradiction that, for each n, there exists a solution u n ∈ M (Hn,J) such that x n − := lim s→−∞ u n (s, ·) is contained in U • but u n is not. For each n, let σ n ∈ R be such that u n (σ n , ·) is not contained in U • . Since x n ± ∈ P(H n± ) = P(H ± ) are elements of a finite sets, by passing to a subsequence, we may assume that x n ± = x ± are independent of n, which means that u n ∈ M(x − , x + ) for all n. Applying Proposition 9.14 to the sequence of solutions {u n } and the sequence of shifts {σ n }, after passing again to a subsequence, {u n } converges to a broken trajectory v of (H, J), and the sequence u n (·+σ n , ·) converges to one of the solutions in v (perhaps up to a shift). Lemma 9.22, together with our assumption that x − = x 0 ⊂ U • , guarantee that the entire broken trajectory v is contained in U • , and in particular lim n→∞ u n (· + σ n , ·) ⊂ U • . Since the latter limit is uniform on compacts, it follows that lim n→∞ u n (σ n , ·) = lim n→∞ u n (0 + σ n , ·) is also contained in U • . Recalling that we chose σ n such that, for each n, the loop u n (σ n , ·) is not contained in the open set U • , we arrive at a contradiction. Similarly, one can prove that when n is large enough, every solution u n of the Floer equation with respect to (H n , J) ending in U is contained in U .

Perturbing Hamiltonians that are regular on a subset.
In this section, we define the notion of regularity on a subset, U ⊂ M , for a pair (H, J) of a Hamiltonian and an almost complex structure that has a barricade in U around some U • U . We prove that for such a pair, the restriction of the Floer differential to the set is well defined, and is stable under (regular) perturbations. Since Floer-regularity concerns the differential of the Floer map we start with a reminder. Given a Hamiltonian H and an almost complex structure J, the Floer map associated to the pair (H, J) is where grad u H := ∇ J H • u is the gradient of H with respect to J, composed on u.
2. We say that the pair (H, J) is semi-regular on U if for every x ± ∈ P(H), with µ(x − ) ≤ µ(x + ) and such that x + ⊂ U , we have: Remark 9.24.
• If (H, J) is regular on U , then it is also semi-regular on U .
• If (H, J) has a barricade in U around U • and agrees, on U , with a Floer-regular pair, then it is regular on U .
• For a pair (H, J) that is regular on U , the differential of the Floer complex might not be defined everywhere. However, using Proposition 9.14 (see also the proof of Lemma 9.26 below), one can show that when µ(x − ) − µ(x + ) = 1 and x + ⊂ U , the quotient manifold M (H,J) (x − , x + )/R is compact and of dimension 0, and hence finite. Therefore, the composition π U •∂ (H,J) can be defined by counting the elements of the latter quotients. This is a slight abuse of notations, as the map ∂ (H,J) is not defined on its own. Similarly, one can define the composition ∂ (H,J) • π U• using the fact that x − ⊂ U • implies that x + ⊂ U • ⊂ U , due to the barricade.
Our main goal for this section is to prove the following statement.
We remark that the second equation in (41) follows immediately from the first. Indeed, due to Proposition 9.21, (H , J) also has a barricade, and ∂ • π U• = π U • ∂ • π U• for both (H, J) and (H , J). In order to prove Proposition 9.25, we connect H and H by a path of Hamiltonians {H λ } λ∈[0,1] , such that, for each λ ∈ [0, 1], H λ agrees with H on the 1periodic orbits up to second order, and the pair (H λ , J) is semi-regular on U . Note that the first condition implies that, for each λ, P(H λ ) = P(H). Given x ± ∈ P(H), such that x + ⊂ U , the space is invariant under the R action u(·, ·) → u(σ +·, ·). We show that when µ(x − )−µ(x + ) = 1, the quotient M Λ (x − , x + ) = M Λ (x − , x + )/R is a smooth, compact 1-dimensional manifold with boundary, that realizes a cobordism between M (H,J) (x − , x + )/R and M (H ,J) (x − , x + )/R. We will then conclude that the number of elements in the quotients M (H,J) (x − , x + )/R and M (H ,J) (x − , x + )/R coincides modulo 2.
The existence of a semi-regular path between H and H follows from the fact that semi-regularity is an open condition.
Lemma 9.26. Suppose that (H, J) is semi-regular on U , then, for every Hamiltonian H that is close enough to H and agrees with H on P(H) up to second order, the pair (H , J) is also semi-regular on U .
Proof. Consider a sequence H n , converging to H, such that, for each n, H n agrees with H on P(H). Then, in particular, P(H n ) = P(H). Suppose that for each n, there exist a solution u n ∈ M (Hn,J) (x n − , x n + ), for some x n ± ∈ P(H n ), such that µ(x n − ) ≤ µ(x n + ) and x n + ⊂ U . Moreover, we assume that if x n − = x n + , then u n is non-constant. Since x n ± ∈ P(H) are elements of a finite set, we may assume, by passing to a subsequence, that x n ± = x ± are independent of n. By Proposition 9.14, there exists a subsequence of the solutions u n that converges to a broken trajectory v of (H, J). Moreover, the ends of the broken trajectory are x ± . Since x + is contained in U and (H, J) has a barricade in U around U • , it follows from Lemma 9.22 that the broken trajectory v is contained in U . As the pair (H, J) is semi-regular on U , for every non-constant solution in the broken trajectory, the index difference between the left end and the right end is positive. Therefore, in the notations of Proposition 9.14, we have µ(x − ) = µ(x 0 ) > µ(x 1 ) > · · · > µ(x + ).
Together with our assumption that µ(x − ) = µ(x n − ) ≤ µ(x n + ) = µ(x + ), this implies that the broken trajectory v contains only one solution: v 1 (s, t) = x − (t) = x + (t). In particular, we conclude that u n ∈ M (Hn,J) (x − , x + ) are Floer-solutions with equal ends. By the energy identity (8), the energy of u n vanishes, which guarantees that u n is a constant solution, u n (s, t) = x − (t) for all n, in contradiction.
Our next aim is to show that for a suitable choice of a path of Hamiltonians {H λ }, the set (42) is a smooth manifold. Let us start with preliminary definitions. Let {H λ } λ∈[0,1] be a path of Hamiltonians that is stationary for λ / ∈ [δ, 1 − δ] for some fixed δ > 0, and such that H λ agrees with H 0 on P(H 0 ) up to second order, for all λ ∈ [0, 1]. We will consider the space C ∞ ε ({H λ } λ ) (of perturbations) consisting of maps: with compact support in M ×S 1 ×[δ, 1−δ], that vanish up to second order on P(H 0 )×[0, 1], and such that h ε < ∞. Here · ε is Floer's ε-norm, see Definition 9.3 and [3, p.230]. We identify the map h with the path of time-dependent Hamiltonians {h λ (·, ·) := h(·, ·, λ)} λ . The next claim is an adjustment of [3, Theorem 11.3.2] to our setting and is proved similarly. For the sake of completeness we include the proof, but we postpone it until the end of this section. Claim 9.27. Let {H λ } λ∈[0,1] be a path of Hamiltonians as above, and assume that (H 0 , J) and (H 1 , J) are regular on U . Then, there exist a neighborhood of 0 in C ∞ ε ({H λ } λ ) and a residual set H reg in this neighborhood, such that if h ∈ H reg , then for Λ = ({H λ + h λ } λ , J) and every x ± ∈ P(H 0 ) with x + ⊂ U , the space M Λ (x − , x + ) is a manifold with boundary, of dimension µ(x − ) − µ(x + ) + 1, and its boundary is Proof of Proposition 9.25. Recall that H is a non-degenerate Hamiltonian such that (H, J) is regular on U . Let H be a small perturbation of H that agrees with H on P(H) up to second order, and such that the pair (H , J) is Floer-regular. We wish to show that the compositions of the differentials with respect to (H, J) and (H , J) with the projections onto C U and C U• agree. Let H λ be a linear path (or, a linear homotopy) between H and H that is stationary near λ = 0, 1, and such that for each λ, H λ agrees with H on P(H) up to second order (in particular, P(H λ ) = P(H)). Taking H to be close enough to H, and using Lemma 9.26, one can guarantee that all of the pairs (H λ , J) are semi-regular on U . By Claim 9.27, there exists a small perturbation of the path {H λ }, such that for Λ = ({H λ + h λ } λ , J) and for every x ± ∈ P(H 0 ) with x + ⊂ U , the space M Λ (x − , x + ) is a manifold with boundary, of dimension µ(x − ) − µ(x + ) + 1. Let us show that when µ(x − ) − µ(x + ) = 1, the quotient of this manifold by the R action, M Λ (x − , x + ) = M Λ (x − , x + )/R, is compact. Let (λ n , u n ) ∈ M Λ (x − , x + ) be any sequence. Since λ n ∈ [0, 1], we may assume, by passing to a subsequence, that the sequence λ n converges to a number λ ∈ [0, 1]. By the definition of the space M Λ (x − , x + ), u n ∈ M (H λn ,J) (x − , x + ) are solutions to the Floer equation with respect to Hamiltonians converging to H λ . By Proposition 9.14, there exists a subsequence of u n converging to a broken Floer trajectory v = {v 1 , · · · , v k } of (H λ , J). Since the pair (H λ , J) is semi-regular on U and x + ⊂ U , every solution in v that is non-constant (in the s-coordinate) decreases the index: µ(x − ) = µ(x 0 ) > µ(x 1 ) > · · · > µ(x k ) = µ(x + ).
Recalling that µ(x − ) − µ(x + ) = 1, we conclude that v contains exactly one non constant solution, v = v 1 ∈ M (H λ ,J) (x − , x + ). In other words, given the sequence (λ n , u n ) ∈ M Λ (x − , x + ), there exists a sequence of shifts s n ∈ R such that, after passing to a subsequence, (λ n , τ sn u n ) In particular, after dividing by the (free, proper and smooth) R-action, the subsequence (λ n , [u n ]) ∈ M Λ (x − , x + ) converges to an element of the same space, The equalities (41) now follow immediately form the definition of the differential map.
Let us sketch the proof of Claim 9.27, which follows the arguments made in [3, Chapter 11.3.b].
Remark 9.30. For a pair (H, J) that is regular on U , the continuation map might not be defined everywhere. However, using Proposition 9.14, one can see that when µ(x − ) = µ(x + ) and x + ⊂ U , the zero dimensional manifold M (H,J) (x − , x + ) is compact and hence finite. The composition π U • Φ (H,J) can be defined by counting the elements of such manifolds. We remark that this is a slight abuse of notations, as the continuation map Φ (H,J) is not necessarily defined on its own. Due to the barricade, if x − ⊂ U • then x + ⊂ U • ⊂ U . It follows that the composition Φ (H,J) • π U• is well defined as well.
Our main goal for this section is to prove the following statement. Then, the compositions of the continuation maps and projections agree: As before, the second equation in (46) follows from the first, since both (H, J) and (H , J) have a barricade in U around U • and thus Φ • π U• = π U • Φ • π U• . In analogy with the previous section, in order to prove Proposition 9.31, we connect H and H by a linear path (or, linear homotopy) of homotopies {H λ } λ∈[0, 1] , such that the pairs (H λ , J) are all semi-regular on U . Then, given x ± ∈ P(H ± ), with µ(x − ) = µ(x + ) and x + ⊂ U , we show that the space M Λ (x − , x + ) := (λ, u) : u ∈ M (H λ ,J) (x − , x + ) is a smooth, compact, 1-dimensional manifold with boundary, that realizes a cobordism between M (H,J) (x − , x + ) and M (H ,J) (x − , x + ). We will then conclude that the number of elements in M (H,J) (x − , x + ) and M (H ,J) (x − , x + ) coincides modulo 2.
As for the case of Hamiltonians, semi-regularity of homotopies is also an open condition, as the following lemma guarantees. Lemma 9.32. Suppose that (H, J) is semi-regular on U , and fix R > 0. Then, for every homotopy H that is close enough to H, such that ∂ s H | |s|>R = 0 and H ± agree with H ± on their 1-periodic orbits up to second order, the pair (H , J) is also semi-regular on U .
Proof. First, notice that by Proposition 9.21, for every homotopy H that satisfies the conditions of the lemma, the pair (H , J) has a barricade in U around U • . Assume for the sake of contradiction that there exist a sequence of homotopies H n , converging to H, such that for each n, H n satisfies the conditions of the lemma, and (H n , J) is not semi-regular on U . Then, for each n, there exist x n ± satisfying µ(x n − ) < µ(x n + ) and x n + ⊂ U , and a solution u n ∈ M (Hn,J) (x n − , x n + ). Since x n ± ∈ P(H n± ) = P(H ± ) are elements of finite sets, we may assume, by passing to a subsequence, that x n ± = x ± are independent of n. By Then, A Incompressibility of domains with incompressible boundaries.
Let M n be a smooth n-dimensional orientable manifold, and let N n be a smooth n-dimensional orientable manifold with boundary such that there exists an embedding ι : N → M . Denote by U := Im (ι (N \ ∂N )), and note that ∂U = Im (ι (∂N )).
Proof. In order to show that ι * : π 1 (U ) → π 1 (M ) is injective it is sufficient to prove that if a loop γ in U is contractible in M then it is contractible in U . Let γ : S 1 → U be a loop that is contractible in M . Then, there exists a map u : D → M such that u| ∂D ≡ γ, where D ⊂ R 2 denotes the unit disk.
Without loss of generality we may assume that γ and u are smooth, and that u ∂U . Indeed, by Whitney's smooth approximation theorem, u is homotopic to a smooth map, u. Since Im γ is compact and U is open, we can choose the smooth approximation such thatγ :=ũ| ∂D is homotopic to γ in U . Applying Thom's transversality theorem, we may assume thatũ ∂U . We replace the maps γ and u byγ andũ, in order to keep the notations.
Under the assumptions above, the preimage C = u −1 (∂U ) is a compact one dimensional submanifold of D, hence a disjoint union of embedded closed curves, C = j C j . Some of the curves C j may encompass others. We call a curve C j a maximal curve if it is not encompassed by any other component of C. More formally, for each component C j , denote by D j ⊂ D the embedded topological disk such that ∂D j = C j . The curve C j is maximal if C j D k for all k = j. We denote the set of maximal curves by C max := {C j 1 , . . . , C j }, and by D max := {D j 1 , . . . , D j } the set of the corresponding topological disks.
For every 1 ≤ i ≤ , the restriction u| C j i is a loop in ∂U which is contractible in M by u| D j i . By the incompressibility of ∂U , the loop u| C j i is contractible in ∂U , namely there exists a map v i : D j i → ∂U such that v i | C j i ≡ u| C j i . Using the maps u and v i we can define a map that contracts γ insideŪ :