PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 102(116) (2017), 17–47 DOI: https://doi.org/10.2298/PIM1716017D

PIUNIKHIN–SALAMON–SCHWARZ ISOMORPHISMS AND SPECTRAL INVARIANTS

FOR CONORMAL BUNDLE

Jovana Ðuretić

Abstract. We give a construction of the Piunikhin–Salamon–Schwarz isomorphism between the Morse homology and the Floer homology generated by Hamiltonian orbits starting at the zero section and ending at the conormal bundle. We also prove that this isomorphism is natural in the sense that it commutes with the isomorphisms between the Morse homology for different choices of the Morse function and the Floer homology for different choices of the Hamiltonian. We define a product on the Floer homology and prove triangle inequality for conormal spectral invariants with respect to this product.

1. Introduction and main results

Let M be a compact smooth manifold. The cotangent bundle T^{∗}M of M carries a natural symplectic structure ω = dλ, where λ is the Liouville form. Let

ν^{∗}N = {α ∈ T_{p}^{∗}M | p ∈ N, α|_{T_{p} N} = 0} ⊂ T^{∗}M,

be a conormal bundle of a closed submanifold N ⊆ M. Let H be a time-dependent smooth compactly supported Hamiltonian on T^{∗}M such that the intersection ν^{∗}N ∩ ϕ_{H}^{1} (o_{M}) is transverse. Here, ϕ_{H}^{t} : T^{∗}M → T^{∗}M denotes the Hamiltonian flow of the Hamiltonian vector field X_{H}. Floer chain groups CF_{∗}(o_{M} , ν^{∗}N : H) are ℤ_{2}-vector spaces generated by the finite set ν^{∗}N ∩ ϕ_{H}^{1} (o_{M}) (see [26] for more details). The Floer homology HF_{∗}(o_{M} , ν^{∗}N : H) is defined as the homology group of (CF_{∗}(o_{M} , ν^{∗}N : H), ∂_{F}) where ∂_{F} is a boundary operator

∂_{F} (x) = ∑_{y∈ν^{∗} N∩ϕ_{H}^{1} (o_{M} )} n(x, y; H)y,

and n(x, y; H) is the (mod 2) number of solutions of the system

2010 Mathematics Subject Classification: Primary 53D40; Secondary 53D12, 57R58, 57R17. Key words and phrases: Conormal bundle, Floer homology, spectral invariants, homology

product.

This work is partially supported by Serbian Ministry of Education, Science and Technological Development, project #174034.

Communicated by Vladimir Dragović.

\frac{∂u}{∂s} + J( \frac{∂u}{∂t} − X_{H} (u)) = 0, u(−∞, t) = ϕ_{H}^{t} ((ϕ_{H}^{1} )^{−1} )(x),

u(s, 0) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N, u(+∞, t) = ϕ_{H}^{t} ((ϕ_{H}^{1} )^{−1} )(y), (1.1)

x, y ∈ ν^{∗}N ∩ ϕ_{H}^{1} (o_{M}).

This homology was introduced by Floer [7], developed by Oh [23] and Fukaya, Oh, Ohta and Ono in the most general case [11]. For a convenience, these groups will be denoted by HF_{∗}(H). Although it is well known that these groups do not depend on H, we will keep H in the notation, since in many practical applications it is useful to keep track on the Hamiltonian used in their definition. For two regular pairs of parameters (H^{α} , J^{α}) and (H^{β}, J^{β}) the isomorphism S^{αβ} : HF_{∗}(H^{α}) → HF_{∗}(H^{β}) between corresponding the Floer homology groups is induced by the chain homomorphism

σ^{αβ} : CF_{∗}(H^{α}) → CF_{∗}(H^{β}), σ^{αβ} (x^{α}) = ∑_{x^{β}} n(x^{α} , x^{β}; H^{αβ} )x^{β},

that counts the number n(x^{α} , x^{β}; H^{αβ} ) of solutions of the system

\frac{∂u}{∂s} + J^{αβ} ( \frac{∂u}{∂t} − X_{H^{αβ}} (u)) = 0, u(s, 0) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N,
(1.2)  u(−∞, t) = ϕ_{H^{α}}^{t} ((ϕ_{H^{α}}^{1} )^{−1} )(x^{α}),  x^{α} ∈ ν^{∗}N ∩ ϕ_{H^{α}}^{1} (o_{M}),
u(+∞, t) = ϕ_{H^{β}}^{t} ((ϕ_{H^{β}}^{1} )^{−1} )(x^{β}),  x^{β} ∈ ν^{∗}N ∩ ϕ_{H^{β}}^{1} (o_{M}).

Here H_{s}^{αβ} and J_{s}^{αβ} are s-dependent families such that for some R > 0

H_{s}^{αβ} = \{ \begin{matrix}H^{α} , & s ⩽ −R \\ H^{β}, & s ⩾ R, \end{matrix} J_{s}^{αβ} = \{ \begin{matrix}J^{α} , & s ⩽ −R \\ J^{β}, & s ⩾ R. \end{matrix}

We define the action functional 𝒜_{H} on the space of paths

Ω(o_{M} , ν^{∗}N) = {γ : [0, 1] → T^{∗}M | γ(0) ∈ o_{M} , γ(1) ∈ ν^{∗}N}

by 𝒜_{H} (γ) = − ∫ γ^{∗}λ+∫_{0}^{1} H(γ(t), t) dt. Critical points of 𝒜_{H} are Hamiltonian paths with ends on the zero section and the conormal bundle, i.e., CF_{∗}(H). Now we can define filtered Floer homology. Denote CF_{∗}^{λ}(H) = ℤ_{2}⟨x ∈ CF_{∗}(H) | 𝒜_{H} (x) < λ⟩. Since the action functional decreases along holomorphic strip (see [23] for details) the differential ∂_{F} preserves the filtration given by 𝒜_{H}. Its restriction ∂_{F}^{λ} = ∂_{F} |_{CF _{∗}^{λ} (H)} defines a boundary operator on the filtered complex CF_{∗}^{λ}(H). The filtered Floer homology is now defined as the homology of the filtered complex

HF_{∗}^{λ}(H) = H_{∗}(CF_{∗}^{λ}(H), ∂_{F}^{λ}).

Note that the filtered Floer homology depends on the Hamiltonian H.

Let us recall the definition of the Morse homology. For a Morse function f : N → ℝ the Morse chain complex, CM_{∗}(N : f), is a ℤ_{2}–vector space generated by the set of critical points of f. Morse homology groups HM_{∗}(N : f) are the homology groups of CM_{∗}(N : f) with respect to the boundary operator

∂_{M} : CM_{∗}(N : f) → CM_{∗}(N : f), ∂_{M} (p) = ∑_{q∈Crit (f)} n(p, q; f)q,

where n(p, q; f) is the number of gradient trajectories that satisfy

(1.3)  \frac{dγ}{ds} = −∇f(γ), γ(−∞) = p, γ(+∞) = q.

Here, γ is a negative g-gradient trajectory of f and g is a Riemannian metric on N such that (f, g) is the Morse–Smale pair. In a way analogous to S^{αβ} , we can define an isomorphism T^{αβ} : HM_{∗}(f^{α}) → HM_{∗}(f^{β}) between Morse homologies of two different Morse functions f^{α} and f^{β}. For given Morse–Smale pairs (f^{α} , g^{α}) and (f^{β}, g^{β}), we choose a homotopy of the Riemannian metrics g_{s}^{αβ} such that

g_{s}^{αβ} = \{ \begin{matrix}g^{α} , & s ⩽ −R \\ g^{β}, & s ⩾ R. \end{matrix}

The isomorphism T^{αβ} is generated by the chain homomorphism

τ^{αβ} : CM_{∗}(f^{α}) → CM_{∗}(f^{β}) where τ^{αβ} (p^{α}) = ∑_{p^{β}} n(p^{α} , p^{β}; f^{αβ} )p^{β},

that counts the number n(p^{α} , p^{β}; f^{αβ} ) of solutions of the system

(1.4)  \frac{dγ}{ds} = −∇_{g_{s}^{αβ}} f^{αβ} (γ), γ(−∞) = p^{α} , γ(+∞) = p^{β},

(see [32] for details). We use a brief notation HM_{∗}(f) or HM_{∗}(N) instead of HM_{∗}(N : f). Morse homology groups HM_{∗}(f) are isomorphic to singular homology groups H_{∗}(N; ℤ_{2}) [21, 29, 32] (we will sometimes identify Morse and singular homologies).

Our first theorem gives isomorphisms between the Morse homology HM_{∗}(N : f) and the Floer homology HF_{∗}(o_{M} , ν^{∗}N : H). These isomorphisms are essentially different from ones defined in [26].

Theorem 1.1. There exist isomorphisms

Φ : HF_{k}(o_{M} , ν^{∗}N : H) → HM_{k}(N : f),
Ψ : HM_{k}(N : f) → HF_{k}(o_{M} , ν^{∗}N : H),

that are inverse to each other: Φ ∘ Ψ = 𝕀|_{HM} and Ψ ∘ Φ = 𝕀|_{HF} .

In order to obtain isomorphisms on homology level, we consider homomorphisms on chain complexes defined by counting the intersection number of the space of gradient trajectories of function f and the space of perturbed holomorphic discs with boundary on the zero section o_{M} and the conormal bundle ν^{∗}N (see Figure 1).

The main problem we need to overcome is that we have singular Lagrangian boundary conditions on holomorphic discs since an intersection o_{M} |_{N} = o_{M} ∩ ν^{∗}N is not transverse.

Motivation for this isomorphism was the paper by Piunikhin, Salamon and Schwarz [25], where they considered the Floer homology for periodic orbits, and the paper by Katić and Milinković [15], where they gave a construction of Piunikhin– Salamon–Schwarz isomorphisms in Lagrangian intersections Floer homology for a cotangent bundle. They worked with the Floer homology generated by Hamiltonian Figure 1. Intersection of gradient trajectory and perturbed holomorphic disc

orbits that start and end on zero section o_{M}. We obtain that isomorphism as special case for N = M. Albers [2] constructed a PSS-type homomorphism (which is not necessarily an isomorphism) in a more general symplectic manifold.

In [26] Poźniak constructed a different type of isomorphism between the Morse homology HM_{∗}(N : f) and the Floer homology HF_{∗}(o_{M} , ν^{∗}N : H_{f} ). Namely, he used Hamiltonian H_{f} that is an extension of a Morse function f. We do not have that kind of restriction, our Hamiltonian H does not have to be an extension of a Morse function f.

Another advantage of using our isomorphism is its naturalness. When using Poźniak’s type isomorphism, it is not obvious whether the diagram

HF_{∗}(H^{α}) −−−−→^{S^{αβ}} HF_{∗}(H^{β})
↑  ↑
HM_{∗}(f^{α}) −−−−→^{T^{αβ}} HM_{∗}(f^{β})

commutes, because different type of equations are used in definitions of S^{αβ} and T^{αβ} . If we use our, PSS–type, isomorphisms as vertical arrows, then we obtain commutativity of the diagram above.

Theorem 1.2. The diagram

HF_{k}(o_{M} , ν^{∗}N : H^{α}) −−−−→^{S^{αβ}} HF_{k}(o_{M} , ν^{∗}N : H^{β})
↑_{Ψ^{α}}  ↑_{Ψ^{β}}
HM_{k}(N : f^{α})  −−−−→^{T^{αβ}}  HM_{k}(N : f^{β}),

commutes.

Using the existence of PSS isomorphism, we can define conormal spectral invariants and prove some of their properties. Denote by ı_{∗}^{λ} : HF_{∗}^{λ}(H) → HF_{∗}(H) the homomorphism induced by the inclusion map ı^{λ} : CF_{∗}^{λ}(H) → CF_{∗}(H). For α ∈ HM_{∗}(N : f) define a conormal spectral invariant

l(α; o_{M} , ν^{∗}N : H) = inf{λ | Ψ(α) ∈ im(ı_{∗}^{λ})}.

Figure 2. Pair–of–pairs object that defines the product ⋆

Oh defined Lagrangian spectral invariants in [23] using the idea of Viterbo’s invariants for generating functions (see [34]). It turns out that those two invariants are the same (under some normalizaton conditions), see [19, 20].

Following [3], we can define a natural homology action homomorphism of HF_{∗}(o_{M} , o_{M}) on HF_{∗}(o_{M} , ν^{∗}N). Note that HF_{∗}(o_{M} , o_{M}) stands for the Floer homology for conormal bundle in a special case when M = N. This is a standard product in Lagrangian Floer homology. Moreover, we can relate it, via the PSS isomorphism, to the action on the Morse side where it becomes the action of HM_{∗}(M) on HM_{∗}(N) via the external intersection product. As a result we obtain a triangle inequality for spectral invariants.

Theorem 1.3. Let H_{1}, H_{2}, H_{3} ∈ C_{c}^{∞} ([0, 1] × T^{∗}M) be three Hamiltonians with a compact support. Then, there exists a natural homology action homomorphism

⋆ : HF_{∗}(o_{M} , o_{M} : H_{1}) ⊗ HF_{∗}(o_{M} , ν^{∗}N : H_{2}) → HF_{∗}(o_{M} , ν^{∗}N : H_{3}).

The product ⋆, via the PSS, induces the exterior intersection product on the Morse homology

· : HM_{∗}(M) ⊗ HM_{∗}(N) → HM_{∗}(N),

i.e., for α ∈ HM_{∗}(M) and β ∈ HM_{∗}(N) it holds Ψ(α · β) = Ψ(α) ⋆ Ψ(β).

Spectral invariants are subadditive with respect to the exterior intersection product, for α ∈ HM_{∗}(M) and β ∈ HM_{∗}(N) such that α · β ≠ 0 it holds

(1.5)  l(α · β; o_{M} , ν^{∗}N : H_{1}♯H_{2}) ⩽ l(α; o_{M} , o_{M} : H_{1}) + l(β; o_{M} , ν^{∗}N : H_{2}).

For the sake of completeness, we provide a construction of ⋆ in Section 5 although it is well known. This product is defined by counting a pair-of-pants with appropriate boundary conditions (see Figure 2). The exterior intersection product in Morse homology is defined by counting gradient trees of appropriate Morse functions (see Section 5 for the definition). The notion of the exterior intersection product was studied in [5], Subsection 4.3.

If we put α = [M] ([M] is the fundamental class) and H_{2} = 0 in (1.5), then we conclude that conormal spectral invariants are bounded for every nonzero singular homology class. The idea of this property came from Humilière, Leclercq and Seyfaddini’s paper [13]. Note that the concatenation H♯0 is just a reparametrization of H and it does not change Hamiltonian orbits, Floer strip or spectral invariants.

Figure 3. Holomorphic strip with a jump that defines the inclusion morphism

Corollary 1.1. For every α ∈ HM_{∗}(N) ∖ {0} it holds

l(α; o_{M} , ν^{∗}M : H) ⩽ l([M]; o_{M} , o_{M} : H).

Observing perturbed holomorphic strips with a jump on the upper boundary (see Figure 3), we can define the inclusion morphism of the Floer homologies. Using the PSS isomorphism, we obtain the inclusion morphism on the Morse side and the appropriate inequality among spectral invariants.

Theorem 1.4. Let H ∈ C_{c}^{∞} ([0, 1] × T^{∗}M) be a compactly supported Hamiltonian. There exists a morphism m : HF_{∗}(o_{M} , ν^{∗}N : H) → HF_{∗}(o_{M} , o_{M} : H) in Floer homology. On Morse homology level it holds Φ ∘ m ∘ Ψ = i_{∗}, where i_{∗} is the morphism induced by the inclusion i : N ↪ M in the sense of Schwarz [32, Auxiliary Proposition 4.22]. This gives rise to the following inequality among spectral invariants

(1.6)  l(i_{∗}(α); o_{M} , o_{M} : H) ⩽ l(α; o_{M} , ν^{∗}N : H),

for every α ∈ HM_{∗}(N) ∖ {0}.

Inequality (1.6) is expected because of the next observation. If α is realized at level λ in the filtered Lagrangian Floer homology HF_{∗}^{λ}(o_{M} , ν^{∗}N), then it is also realized, via the inclusion, at the same level, in the homology HF_{∗}^{λ}(o_{M} , o_{M}).

It is obvious that the composition of morphisms ⋆ and m lead to the product on Lagrangian Floer homology. Via the PSS, we obtain the operation on Morse homology.

Corollary 1.2. Let H_{1}, H_{2}, H_{3} ∈ C_{c}^{∞} ([0, 1] × T^{∗}M) be three Hamiltonians

with compact support. Then, there exists a product

∗ : HF_{∗}(o_{M} , ν^{∗}N : H_{1}) ⊗ HF_{∗}(o_{M} , ν^{∗}N : H_{2}) → HF_{∗}(o_{M} , ν^{∗}N : H_{3}),

in homology, defined by ∗ = ⋆ ∘ (m ⊗ 𝕀). The product ∗ induces the operation on HM_{∗}(N) via the PSS isomorphism as α • β = Φ(Ψ(α) ∗ Ψ(β)), for α, β ∈ HM_{∗}(N).

As a special case, when N = M, we obtain the product defined in [24] (also discussed in [16]). We can see that ∗ counts pair-of-pants with a boundary on o_{M} ∪ ν^{∗}N and a jump from o_{M} to ν^{∗}N on a slit of pants (see Figure 4). The operation • on HM_{∗}(N) can be described as a composition of the inclusion and the exterior intersection product.

Figure 4. Pair–of–pants object that defines product on HF_{∗}(o_{M} , ν^{∗}N : H)

The triangle inequality for conormal spectral invariant, with respect to •, follows directly from Theorem 1.3 and Theorem 1.4. Our inequality is a generalization of the one made by Monzner, Vichery and Zapolsky in [22].

Corollary 1.3. Let us take two compactly supported Hamiltonians H, H′ and α, β ∈ HM_{∗}(N) such that α • β ≠ 0. Then

l(α • β; o_{M} , ν^{∗}N : H♯H′) ⩽ l(α; o_{M} , ν^{∗}N : H) + l(β; o_{M} , ν^{∗}N : H′).

This paper is organized as follows. In Section 2, we define diverse moduli spaces and prove some of their properties. In Section 3, we present the construction of PSS-type homomorphisms and we prove Theorem 1.1. Section 4 contains a proof of Theorem 1.2. In the last section, we provide constructions of morphisms ⋆ and m, and prove the mentioned inequalities among spectral invariants.

2. Holomorphic discs, gradient trajectories and moduli spaces

We start with a construction of mixed-type object space that we use for the definition of Ψ and Φ. Let p be a critical point of a Morse function f. Morse homology HM_{k}(f) is graded by Morse index k = m_{f} (p) of critical points.

To each element of CF_{∗}(H), we can assign a solution of the Hamiltonian equation

(2.1)  ẋ = X_{H} (x), x(0) ∈ o_{M} , x(1) ∈ ν^{∗}N.

For a solution x of (2.1), there exists a canonically assigned Maslov index

μ_{N} : CF_{∗}(H) → \frac{1}{2}ℤ,

see [23, 27, 28] for details. The Floer homology HF_{k}(H) is graded by k = μ_{N} (x)+ \frac{1}{2} dimN.

Let ℳ(p, f, g; x, H, J) be the space of pairs of maps

γ : (−∞, 0] → N, u : ℝ × [0, 1] → T^{∗}M,

that satisfy Figure 5. ℳ(p, f, g; x, H, J) and ℳ(x, H, J; p, f, g)

\frac{dγ}{ds} = −∇f(γ(s)), \frac{∂u}{∂s} + J( \frac{∂u}{∂t} − X_{ρ_{R}^{+} H} (u)) = 0,

E(u) = ∫∫_{ℝ×[0,1]} ∥∂_{s}u∥_{J}^{2} dt ds < +∞,

u(s, 0) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N, s ∈ ℝ,

γ(−∞) = p, u(+∞, t) = x(t), γ(0) = u(−∞),

where R is a positive fixed number and ρ_{R}^{+} : ℝ → ℝ is a smooth function such that

ρ_{R}^{+}(s) = \{ \begin{matrix}1, & s ⩾ R + 1 \\ 0, & s ⩽ R. \end{matrix}

The strip u is holomorphic for s ⩽ R and has finite energy. So, u admits a unique continuous extension u(−∞) (see [18, Section 4.5] and [31, Theorem 3.1]). The extension is a point that belongs to o_{N} = ν^{∗}N ∩ o_{M}, and we can omit the second argument of u(−∞).

Let ℳ(x, H, J; p, f, g) be the space of pairs of maps

γ : [0, +∞) → N, u : ℝ × [0, 1] → T^{∗}M,

that satisfy

\frac{dγ}{ds} = −∇f(γ(s)), \frac{∂u}{∂s} + J( \frac{∂u}{∂t} − X_{ρ_{R}^{−} H} (u)) = 0,

E(u) < +∞, u(s, 0) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N, s ∈ ℝ,

γ(+∞) = p, u(−∞, t) = x(t), γ(0) = u(+∞),

where ρ_{R}^{−} : ℝ → ℝ is a smooth function such that

ρ_{R}^{−}(s) = \{ \begin{matrix}1, & s ⩽ −R − 1 \\ 0, & s ⩾ −R. \end{matrix}

Proposition 2.1. For a generic Morse function f and a generic compactly supported Hamiltonian H, the set ℳ(p, f, g; x, H, J) is a smooth manifold of dimension m_{f} (p) − (μ_{N} (x) + \frac{1}{2} dimN), and ℳ(x, H, J; p, f, g) is a smooth manifold of dimension μ_{N} (x) + \frac{1}{2} dimN − m_{f} (p).

Proof. Let W^{u}(p, f) be the unstable manifold associated to a critical point p of a Morse function f. We know that dimW^{u}(p, f) = m_{f} (p) [21].

Let ℳ_{+} (H, J; x) be the set of solutions of

u : ℝ × [0, 1] → T^{∗}M, \frac{∂u}{∂s} + J( \frac{∂u}{∂t} − X_{ρ_{R}^{+} H} (u)) = 0, E(u) < +∞,

u(s, 0) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N, s ∈ ℝ, u(+∞, t) = x(t).

The dimension of ℳ_{+} (H, J; x) is dimℳ_{+} (H, J; x) = \frac{1}{2} dimN − μ_{N} (x), see [23] for details. We used the definition of Maslov index μ_{N} (x) = μ(B_{Φ} (ℝ^{m}), V^{Φ} ), where Φ : x^{∗}T (T^{∗}M) → [0, 1] × ℂ^{m} is any trivialization and

V^{Φ} = Φ(T_{x(1)} ν^{∗}N), B_{Φ} (t) = Φ ∘ T ϕ_{H}^{t} ∘ Φ^{−1} .

For a generic choice of parameters, the evaluation map

Ev : W^{u}(p, f) × ℳ_{+} (H, J; x) → N × N, Ev(γ, u) = (γ(0), u(−∞)),

is transversal to the diagonal, thus ℳ(p, f, g; x, H, J) = Ev^{−1} (△) is a smooth manifold of dimension

m_{f} (p) + \frac{1}{2} dimN − μ_{N} (x) − (2 dimN − dimN) = m_{f} (p) − \frac{1}{2} dimN − μ_{N} (x).

The proof for ℳ(x, H, J; p, f, g) is similar. □

We need some additional properties of the manifolds ℳ(p, f, g; x, H, J) and ℳ(x, H, J; p, f, g). The set of solutions of (1.1) is denoted by ℳ(x, y; H) and ℳ(p, q; f) denotes the set of solutions of (1.3) (modulo ℝ-action).

Proposition 2.2. Let f be a generic Morse function and H a generic compactly supported Hamiltonian. If m_{f} (p) = μ_{N} (x)+ \frac{1}{2} dimN, then ℳ(p, f, g; x, H, J) is a finite set. If m_{f} (p) = μ_{N} (x) + \frac{1}{2} dimN + 1, then ℳ(p, f, g; x, H, J) is onedimensional manifold with topological boundary

∂ℳ(p, f, g; x, H, J) = ⋃_{m_{f} (q)=m_{f} (p)−1} ℳ(p, q; f) × ℳ(q, f, g; x, H, J)
∪ ⋃_{μ_{N} (y)=μ_{N} (x)+1} ℳ(p, f, g; y, H, J) × ℳ(y, x; H).

Proof. Let (γ_{n}, u_{n}) be a sequence in ℳ(p, f, g; x, H, J) that has no W^{1,2}- convergent subsequence. Since N is compact, γ_{n}(t) is bounded for every t. The sequence γ_{n} is equicontinuous because

d(γ_{n}(t_{1}), γ_{n}(t_{2})) ⩽ ∫_{t_{1}}^{t_{2}} ∥γ̇(s)∥ ds

⩽ \sqrt{t_{2} − t_{1}} \sqrt{∫_{t_{1}}^{t_{2}} ∥γ̇(s)∥^{2} ds} = \sqrt{t_{2} − t_{1}} \sqrt{∫_{t_{1}}^{t_{2}} \frac{∂}{∂s} f(γ_{n}(s)) ds}

⩽ \sqrt{t_{2}−t_{1}} \sqrt{max_{x∈N}f(x)−f(γ_{n}(−∞))} = \sqrt{t_{2}− t_{1}} \sqrt{max_{x∈N}f(x)−f(p)} .

It follows from the Arzelà–Ascoli theorem that γ_{n} has a subsequence converging uniformly on compact sets. Since the sequence γ_{n} is a solution of the equation γ̇_{n} = −∇f(γ_{n}), and the function f is smooth, γ_{n} converges with all its derivatives on compact subsets of (−∞, 0].

The energy of u_{n} is uniformly bounded since

𝒜_{H} (x(t)) = 𝒜_{ρ_{R}^{+} H} (u_{n}(+∞), t) − 𝒜_{ρ_{R}^{+} H} (u_{n}(−∞), t) =
= −E(u_{n}) + ∫_{−∞}^{+∞} ∫_{0}^{1}(ρ_{R}^{+}(s))′H(u_{n}(s, t), t) dt ds.

The Hamiltonian H has a compact support, (ρ_{R}^{+}(s))′ is nonzero only on [R, R + 1], so the last integral is uniformly bounded

| ∫_{−∞}^{+∞} ∫_{0}^{1}(ρ_{R}^{+}(s))′H(u_{n}(s, t), t) dt ds| ⩽ C.

We have a sequence u_{n} whose energy is uniformly bounded. From the Gromov compactness [12], it follows that u_{n} has a subsequence that converges together with all derivatives on compact subsets of (ℝ × [0, 1]) ∖ {z_{1}, …, z_{m}}. Bubbles can occur at z_{i} if it is an interior point of ℝ × [0, 1]. It is also possible that a bubble appears at the boundary point z_{k} as holomorphic disc with the boundary conditions on zero section and conormal bundle. But in our case neither holomorphic spheres nor discs appear. If v : S^{2} → T^{∗}M is a holomorphic sphere, then

∫_{S^{2}} ∥dv∥^{2} = ∫_{S^{2}} v^{∗}ω = ∫_{∂S^{2}} v^{∗}λ = 0.

If v : ℝ × [0, 1] → T^{∗}M is a holomorphic disc, then

∫_{ℝ×[0,1]} ∥dv∥^{2} = ∫_{ℝ×[0,1]} v^{∗}ω = ∫_{∂(ℝ×[0,1])} v^{∗}λ = 0,

since λ = 0 on o_{M} and ν^{∗}N.

So, (γ_{n}, u_{n}) has a subsequence which converges with all its derivatives uniformly on compact sets. From C_{loc}^{∞}-convergence it follows W^{1,2}-convergence. Thus, (γ_{n}, u_{n}) has a subsequence that converges to some element of ℳ(p^{m}, f, g; x^{0}, H, J). Similarly as in [8, 14, 17, 30, 32], we conclude that the only loss of compactness is a “trajectory breaking" in the following way

⋃ ℳ(p, p^{1}; f) × ⋯× ℳ(p^{m−1} , p^{m}; f) × ℳ(p^{m}, f, g; x^{0}, H, J) (2.2)

× ℳ(x^{0}, x^{1}; H) × ⋯× ℳ(x^{l−1} , x; H) .

Here, p, p^{1}, …, p^{m} are critical points of f and x^{0}, …, x^{l−1} , x are Hamiltonian paths with decreasing Morse and Maslov indices such that m_{f} (p^{m}) ⩾ μ_{N} (x^{0}) + \frac{1}{2} dimN.

Therefore, we have that the boundary ∂ℳ(p, f, g; x, H, J) is a subset of union (2.2). The other inclusion follows from the standard gluing arguments.

If m_{f} (p) = μ_{N} (x) + \frac{1}{2} dimN, then ℳ(p, f, g; x, H, J) is a compact, zero-dimen-sional manifold, so ℳ(p, f, g; x, H, J) has a finite number of elements.

If m_{f} (p) = μ_{N} (x) + \frac{1}{2} dimN + 1 then the boundary of ℳ(p, f, g; x, H, J) can contain an element of a set ℳ(p, q; f)×ℳ(q, f, g; x, H, J) for some q ∈ Crit(f) such that m_{f} (q) = m_{f} (p) − 1 or an element of a set ℳ(p, f, g; y, H, J) × ℳ(y, x; H) for some Hamiltonian orbit y, such that μ_{N} (y) = μ_{N} (x) + 1. □

We have a similar proposition for ℳ(x, H, J; p, f, g).

Proposition 2.3. Let f be a generic Morse function and H a generic compactly supported Hamiltonian. If m_{f} (p) = μ_{N} (x)+ \frac{1}{2} dimN, then ℳ(x, H, J; p, f, g) is a finite set. If m_{f} (p) = μ_{N} (x) + \frac{1}{2} dimN − 1, then ℳ(x, H, J; p, f, g) is onedimensional manifold with topological boundary

∂ℳ(x, H, J; p, f, g) = ⋃_{m_{f} (q)=m_{f} (p)+1} ℳ(x, H, J; q, f, g) × ℳ(q, p; f)

∪ ⋃_{μ_{N} (y)=μ_{N} (x)−1} ℳ(x, y; H) × ℳ(y, H, J; p, f, g).

Now, we define some auxiliary manifolds that we use to prove that the composition Φ ∘ Ψ is the identity (see Theorem 1.1). Let R > 0 be a fixed number. For p, q ∈ Crit(f) we define ℳ_{R}(p, q, f; H) as the set of maps

γ_{−} : (−∞, 0] → N, γ_{+} : [0, +∞) → N, u : ℝ × [0, 1] → T^{∗}M

such that

\frac{dγ_{±}}{ds} = −∇f(γ_{±} ), \frac{∂u}{∂s} + J( \frac{∂u}{∂t} − X_{σ_{R} H} (u)) = 0, E(u) < +∞,

γ_{−} (−∞) = p, γ_{+} (+∞) = q, u(s, 0) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N, s ∈ ℝ,

u(±∞, t) = γ_{±} (0),

where σ_{R} : ℝ → [0, 1] is a smooth function such that

σ_{R}(s) = \{ \begin{matrix}1, & |s| ⩽ R \\ 0, & |s| ⩾ R + 1. \end{matrix}

We also define its parameterized version

\overline{ℳ}(p, q, f; H) = \{(R, γ_{−} , γ_{+} , u) | (γ_{−} , γ_{+} , u) ∈ ℳ_{R}(p, q, f; H), R > R_{0}\},

(see Figure 6). From now on, whenever we define new moduli space, we omit the argument g and J, although we know that a moduli space depend on a Riemannian metric and on an almost complex structure. For a generic choice of parameters, the set \overline{ℳ}(p, q, f; H) is an one-dimensional manifold if m_{f} (p) = m_{f} (q), and a zerodimensional manifold if m_{f} (p) = m_{f} (q) − 1.

Knowing the definitions of a broken gradient trajectory and a weak convergence of gradient trajectories [32], we can define a broken holomorphic strip and a weak convergence of holomorphic strips [30].

Figure 6. ℳ_{R}(p, q, f; H)

Definition 2.1. A broken (perturbed) holomorphic strip v is a pair (v_{1}, v_{2}) of (perturbed) holomorphic strips such that v_{1}(+∞, t) = v_{2}(−∞, t). A sequence of perturbed holomorphic strips u_{n} : ℝ × [0, 1] → T^{∗}M is said to converge weakly to a broken trajectory v if there exists a sequence of translations φ_{n}^{i} : ℝ × [0, 1] → ℝ × [0, 1], i = 1, 2, such that u_{n} ∘ φ_{n}^{i} converges to v_{i} uniformly with all derivatives on a compact subset of ℝ × [0, 1]. We say that an element of mixed type (γ, u) is a broken element if γ is a broken trajectory or u is a broken holomorphic strip.

The following proposition gives us a boundary of a one-dimensional manifold \overline{ℳ}(p, q, f; H).

Proposition 2.4. Let p, q ∈CM_{k}(f). The topological boundary of \overline{ℳ}(p, q, f; H) can be identified with

∂\overline{ℳ}(p, q, f; H) = ℳ_{R_{0}} (p, q, f; H) ∪ ⋃_{m_{f} (r)=k−1} ℳ(p, r; f) × \overline{ℳ}(r, q, f; H)

∪ ⋃_{m_{f} (r)=k+1} \overline{ℳ}(p, r, f; H) × ℳ(r, q; f)

∪ ⋃_{μ_{N} (x)+dim N/2=k} ℳ(p, f, g; x, H, J) × ℳ(x, H, J; q, f, g).

Proof. Consider a sequence (R_{n}, γ_{−}^{n} , γ_{+}^{n} , u_{n}) in \overline{ℳ}(p, q, f; H). It either W^{1,2}- converges to an element of the same moduli space or one of the following four statements holds:

(1) There is a subsequence such that R_{n_{k}} → R_{0} and (γ_{−}^{n_{k}} , γ_{+}^{n_{k}} , u_{n_{k}} ) converges to (γ_{−} , γ_{+} , u) ∈ ℳ_{R_{0}} (p, q, f; H).

(2) There is a subsequence of (R_{n}, γ_{−}^{n} , γ_{+}^{n} , u_{n}) that converges to a broken trajectory in ℳ(p, r; f) × \overline{ℳ}(r, q, f; H). The subsequence (γ_{+}^{n_{k}} , u_{n_{k}} ) converges in W^{1,2} topology and γ_{−}^{n_{k}} converges weakly.

(3) There is a subsequence that converges to a broken trajectory in \overline{ℳ}(p, r, f; H)× ℳ(r, q; f), similarly to (2).

(4) There is a subsequence such that R_{n_{k}} → +∞ and (γ_{−}^{n_{k}} , γ_{+}^{n_{k}} , u_{n_{k}} ) converges weakly to a broken element of ℳ(p, f, g; x, H, J) × ℳ(x, H, J; q, f, g).

If R_{n} is bounded, then we can find a compact K such that {R_{n}} ⊂ K. The family ρ_{R} can be chosen to depend continuously on R, so all estimates in Proposition 2.2 hold uniformly on R ∈ K. In a similar way to Proposition 2.2, we conclude that (γ_{−}^{n} , γ_{+}^{n} , u_{n}) has a subsequence that converges locally uniformly. So, if (R_{n}, γ_{−}^{n} , γ_{+}^{n} , u_{n}) does not converge to an element of \overline{ℳ}(p, q, f; H), then R_{n} → R_{0} or R_{n} → R > R_{0} (R_{n} denotes the subsequence, as well). If the first case, (γ_{−}^{n} , γ_{+}^{n} , u_{n}) converges in W^{1,2} topology, and in the second one (γ_{−}^{n} , γ_{+}^{n} , u_{n}) converges to a broken trajectory. Since the dimension of \overline{ℳ}(p, q, f; H) is 1, it can break only once. The breaking can happen on trajectories γ_{−}^{n} or γ_{+}^{n} and not on the disc. The sequence u_{n} cannot converge to a broken disc because the nonholomorphic part of the domain is compact and u_{n} converges there. If it breaks on the holomorphic part, then we obtain a solution of a system

v : ℝ × [0, 1] → T^{∗}M, \frac{∂v}{∂s} + J \frac{∂v}{∂t} = 0,

v(ℝ × {0}) ⊂ o_{M} , v(ℝ × {1}) ⊂ ν^{∗}N.

We have already seen that all such solutions are constant, so u_{n} cannot break on the holomorphic part either. In this way, we covered the first three cases. The fourth case arises if the sequence R_{n} is not bounded. We can find a subsequence R_{n} → +∞. Then the discs

u_{n}^{−} (s, t) := u_{n}(s − R_{n} − R_{0} − 1, t), u_{n}^{+} (s, t) := u_{n}(s + R_{n} + R_{0} + 1, t),

converge locally uniformly with all derivatives to some u^{−} and u^{+} , respectively. These discs are solutions of the system

\frac{∂u^{±}}{∂s} + J( \frac{∂u^{±}}{∂t} − X_{ρ_{R_{0}}^{±}} (u^{±} )) = 0,

u^{±} (ℝ × {0}) ⊂ o_{M} , u^{±} (ℝ × {1}) ⊂ ν^{∗}N,

u^{±} (∓∞, t) = x(t), u^{±} (±∞, t) = γ_{±} (0).

The sequences γ_{±}^{n} cannot break because of dimensional reason, so they converge to some trajectories γ_{±} .

Conversely, for each broken trajectory of some of the types (γ, γ_{−} , γ_{+} , u) ∈ ℳ(p, r; f) × \overline{ℳ}(r, q, f; H),

(γ_{−} , γ_{+} , u, γ) ∈ \overline{ℳ}(p, r, f; H) × ℳ(r, q; f),
(γ_{1}, u_{1}, γ_{2}, u_{2}) ∈ ℳ(p, f, g; x, H, J) × ℳ(x, H, J; q, f, g),

there is a sequence in \overline{ℳ}(p, q, f; H) that converges weakly to a corresponding broken trajectory. The proof is based on the implicit-function theorem and pregluing and gluing techniques. □

We continue with the construction of the auxiliary manifold, again with the variable domain, that now connects the Hamiltonian orbits. Fix an ε > 0. Consider

Figure 7. ℳ_{ε}(x, y, H; f)

the moduli space ℳ_{ε}(x, y, H; f) defined as the set of maps

u_{±} : ℝ × [0, 1] → T^{∗}M, γ : [−ε, ε] → N

that satisfy

\frac{∂u_{±}}{∂s} + J( \frac{∂u_{±}}{∂t} − X_{ρ_{R}^{±} H} (u_{±} )) = 0, \frac{dγ}{ds} = −∇f(γ),

E(u_{±} ) < +∞, u_{±} (s, 0) ∈ o_{M} , u_{±} (s, 1) ∈ ν^{∗}N, s ∈ ℝ,

u_{−} (−∞, t) = x(t), u_{+} (+∞, t) = y(t), u_{∓} (±∞) = γ(∓ε),

(see Figure 7) and consider the moduli space

\underline{ℳ}(x, y, H; f) = \{(ε, u_{−} , u_{+} , γ) | (u_{−} , u_{+} , γ) ∈ ℳ_{ε}(x, y, H; f), ε ∈ [ε_{0}, ε_{1}]\},

where ε_{0} and ε_{1} are fixed positive numbers.

For μ_{N} (y) = μ_{N} (x) + 1, \underline{ℳ}(x, y, H; f) is a zero-dimensional manifold. If μ_{N} (y) = μ_{N} (x), then \underline{ℳ}(x, y, H; f) is a one-dimensional manifold and we can describe its boundary.

Proposition 2.5. Let x, y ∈ CF_{k}(H). Then the topological boundary of \underline{ℳ}(x, y, H; f) can be identified with

∂\underline{ℳ}(x, y, H; f) = ℳ_{ε_{1}} (x, y, H; f) ∪ ℳ_{ε_{0}} (x, y, H; f)

∪ ⋃_{μ_{N} (z)=μ_{N} (x)−1} ℳ(x, z; H) × \underline{ℳ}(z, y, H; f)

∪ ⋃_{μ_{N} (z)=μ_{N} (x)+1} \underline{ℳ}(x, z, H; f) × ℳ(z, y; H).

Proof. Let us take a sequence (ε_{n}, u_{−}^{n} , u_{+}^{n} , γ_{n}) ∈ \underline{ℳ}(x, y, H; f) that has no convergent subsequence in W^{1,2}-topology. Since a sequence ε_{n} is bounded, all uniform estimates for u_{±}^{n} , γ_{n} hold uniformly on ε (see Proposition 2.2). Hence, the sequences u_{−}^{n} , u_{+}^{n} and γ_{n} converge locally uniformly and (u_{−}^{n} , u_{+}^{n} , γ_{n}) can break only once (for dimensional reason). The domain of γ_{n} is bounded, so the trajectory γ_{n} cannot break. The only remaining possibilities are:

(1) There is a subsequence which converges to an element of ℳ_{ε_{1}} (x, y, H; f) or ℳ_{ε_{0}} (x, y, H; f).

(2) There is a subsequence which converges weakly to an element of ℳ(x, z; H)× \underline{ℳ}(z, y, H; f).

(3) There is a subsequence which converges weakly to an element of \underline{ℳ}(x, z, H; f) × ℳ(z, y; H). □

Now, we define moduli space similar to \overline{ℳ}(p, q, f; H), except that we are not using a fixed Hamiltonian H, but a homotopy of Hamiltonians H_{δ}, 0 ⩽ δ ⩽ 1, that connects the given Hamiltonians H_{0} and H_{1},

\overline{ℳ}(p, q, f; H_{δ}) = \{(δ, γ_{−} , γ_{+} , u) | (γ_{−} , γ_{+} , u) ∈ ℳ_{R_{0}} (p, q, f; H_{δ})), 0 ⩽ δ ⩽ 1\}.

The dimension of this manifold is m_{f} (p)−m_{f} (q)+1, and its boundary is described in the following proposition.

Proposition 2.6. Let p, q ∈ CM_{k}(f). Then the topological boundary of the one-dimensional manifold \overline{ℳ}(p, q, f; H_{δ}) can be identified with

∂\overline{ℳ}(p, q, f; H_{δ}) = ℳ_{R_{0}} (p, q, f; H_{0}) ∪ ℳ_{R_{0}} (p, q, f; H_{1})

∪ ⋃_{m_{f} (r)=k−1} ℳ(p, r; f) × \overline{ℳ}(r, q, f; H_{δ})

∪ ⋃_{m_{f} (r)=k+1} \overline{ℳ}(p, r, f; H_{δ}) × ℳ(r, q; f).

Proof. The proof is essentially the same as for Proposition 2.4. □

So far, we have discussed moduli spaces defined by a family of Hamiltonians with a fixed Morse function f. It will be useful to consider moduli spaces similar to ℳ(p, f, g; x, H, J), that depend on a family of Morse functions and a family of Hamiltonians. Let (f_{s,δ}^{αβ} , H_{s,δ}^{αβ} ), 0 ⩽ δ ⩽ 1, be a homotopy connecting (f^{α} , H_{s}^{αβ} ) for δ = 0 and (f_{s}^{αβ} , H^{β}) for δ = 1. Here

f_{s}^{αβ} = \{ \begin{matrix}f^{α} , & s ⩽ −T − 1 \\ f^{β}, & s ⩾ −T \end{matrix} and H_{s}^{αβ} = \{ \begin{matrix}H^{α} , & s ⩽ T \\ H^{β}, & s ⩾ T + 1 \end{matrix}

are homotopies connecting the Morse functions f^{α}, f^{β}, and the Hamiltonians H^{α}, H^{β}, respectively

We choose a homotopy (f_{s,δ}^{αβ} , H_{s,δ}^{αβ} ) such that for any δ and s negative (positive) enough, f_{s,δ}^{αβ} is equal to f^{α} (H_{s,δ}^{αβ} is equal to H^{β}). In the same way we choose a homotopy (g_{s,δ}^{αβ} , J_{s,δ}^{αβ} ). Let ℳ̂(p^{α} , f_{s,δ}^{αβ} ; x^{β}, H_{s,δ}^{αβ} ) be the set of the triples (δ, γ, u) such that

γ : (−∞, 0] → N, u : ℝ × [0, 1] → T^{∗}M, \frac{dγ}{ds} = −∇_{g_{s,δ}^{αβ}} f_{s,δ}^{αβ} (γ(s)),

\frac{∂u}{∂s} + J_{s,δ}^{αβ} ( \frac{∂u}{∂t} − X_{ρ_{R}^{+} H_{s,δ}^{αβ}} (u)) = 0, E(u) < +∞, γ(−∞) = p^{α} , (2.3)

u(s, 0) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N, s ∈ ℝ, u(+∞, t) = x^{β}(t), γ(0) = u(−∞).

The dimension of this manifold is m_{f^{α}} (p^{α}) − (μ_{N} (x^{β}) + \frac{1}{2} dimN) + 1. We also define the moduli space ℳ(p^{α} , f_{s}^{αβ} ; x^{β}, H^{β}) as the set of pairs (γ, u) that satisfy

γ : (−∞, 0] → N, u : ℝ × [0, 1] → T^{∗}M, \frac{dγ}{ds} = −∇_{g_{s}^{αβ}} f_{s}^{αβ} (γ(s)),

\frac{∂u}{∂s} + J^{β}( \frac{∂u}{∂t} − X_{ρ_{R}^{+} H^{β}} (u)) = 0, E(u) < +∞, γ(−∞) = p^{α} ,

u(s, 0) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N, s ∈ ℝ, u(+∞, t) = x^{β}(t), γ(0) = u(−∞).

Let ℳ(p^{α} , f^{α}; x^{β}, H_{s}^{αβ} ) be the set of maps γ : (−∞, 0] → N, u : ℝ × [0, 1] → T^{∗}M such that

\frac{dγ}{ds} = −∇_{g^{α}} f^{α} (γ), \frac{∂u}{∂s} + J_{s}^{αβ} ( \frac{∂u}{∂t} − X_{ρ_{R}^{+} H_{s}^{αβ}} (u)) = 0,

E(u) < +∞, γ(−∞) = p^{α} , γ(0) = u(−∞) (2.4)

u(s, 0) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N, s ∈ ℝ, u(+∞, t) = x^{β}(t).

The manifolds ℳ(p^{α} , f_{s}^{αβ} ; x^{β}, H^{β}) and ℳ(p^{α} , f^{α}; x^{β}, H_{s}^{αβ} ) are the two components of a boundary ∂ℳ̂(p^{α} , f_{s,δ}^{αβ} ; x^{β}, H_{s,δ}^{αβ} ) which we completely describe in the next proposition.

Proposition 2.7. Let m_{f^{α}} (p^{α}) = μ_{N} (x^{β}) + \frac{1}{2} dimN. Then the topological boundary of one-dimensional manifold ℳ̂(p^{α} , f_{s,δ}^{αβ} ; x^{β}, H_{s,δ}^{αβ} ) can be identified with

∂ℳ̂(p^{α} , f_{s,δ}^{αβ} ; x^{β}, H_{s,δ}^{αβ} ) = ℳ(p^{α} , f_{s}^{αβ} ; x^{β}, H^{β}) ∪ ℳ(p^{α} , f^{α}; x^{β}, H_{s}^{αβ} )

∪ ⋃_{m_{f^{α}} (q^{α} )=m_{f^{α}} (p^{α} )−1} ℳ(p^{α} , q^{α}; f^{α}) × ℳ̂(q^{α} , f_{s,δ}^{αβ} ; x^{β}, H_{s,δ}^{αβ} )

∪ ⋃_{μ_{N} (y^{β} )=μ_{N} (x^{β} )+1} ℳ̂(p^{α} , f_{s,δ}^{αβ} ; y^{β}, H_{s,δ}^{αβ} ) × ℳ(y^{β}, x^{β}; H^{β}).

Proof. The proof is essentially the same as for Proposition 2.4. □

3. Isomorphism

We saw in Propositions 2.2 and 2.3 that ℳ(p, f, g; x, H, J) and ℳ(x, H, J; p, f, g) are finite sets if m_{f} (p) = μ_{N} (x) + \frac{1}{2} dimN. Denote their cardinal numbers (modulo 2) by n(p, f, g; x, H, J) and n(x, H, J; p, f, g) and define homomorphisms on generators:

ϕ : CF_{k}(H) → CM_{k}(f), ϕ(x) = ∑_{m_{f} (p)=k} n(x, H, J; p, f, g) p,
ψ : CM_{k}(f) → CF_{k}(H), ψ(p) = ∑_{μ_{N} (x)=k− \frac{1}{2} dim N} n(p, f, g; x, H, J) x.

Proposition 3.1. The homomorphisms ϕ and ψ are well defined chain maps.

Proof. It follows from Propositions 2.2, 2.3 and from the way in which the chain complexes CM_{∗}(f) and CF_{∗}(H) are graded that these homomorphisms are well defined. We prove that (ϕ ∘ ∂_{F} − ∂_{M} ∘ ϕ)(x) = 0 for all x ∈ CF_{k}(H). We have

(ϕ ∘ ∂_{F} − ∂_{M} ∘ ϕ)(x)

= ∑_{m_{f} (q)=k−1} ( ∑_{μ_{N} (y)+dim N/2=k−1} n(x, y; H) n(y, H, J; q, f, g)) q

− ∑_{m_{f} (q)=k−1} ( ∑_{m_{f} (p)=k} n(x, H, J; p, f, g) n(p, q; f)) q.

Let p ∈ CM_{k}(f), q ∈ CM_{k−1} (f) and y ∈ CF_{k−1} (H). From Proposition 2.3 it follows

∑_{μ_{N} (y)+dim N/2=k−1} n(x, y; H) n(y, H, J; q, f, g)

− ∑_{m_{f} (p)=k} n(x, H, J; p, f, g) n(p, q; f) = 0,

since it is the number (modulo 2) of ends of the one-dimensional manifold

ℳ(x, H, J; q, f, g).

So, (ϕ ∘ ∂_{F} − ∂_{M} ∘ ϕ)(x) = 0.

The proof of the identity ψ ∘ ∂_{M} = ∂_{F} ∘ ψ is analogous. □

From the previous proposition, it follows that ϕ and ψ induce homomorphisms

Φ : HF_{k}(H) → HM_{k}(f), Ψ : HM_{k}(f) → HF_{k}(H).

in homology. They are PSS-type isomorphisms. Now, we can prove Theorem 1.1. From the fact that these homomorphisms are inverse to each other, it immediately follows that they are isomorphisms. In order to show this, we prove that ϕ ∘ ψ and ψ ∘ ϕ are maps, chain homotopic to the identity.

Proof of Theorem 1.1. If we look at a composition of homomorphisms ϕ and ψ,

ϕ ∘ ψ(p) = ∑_{m_{f} (q)=k} ( ∑_{μ_{N} (x)+dim N/2=k} n(p, f, g; x, H, J) n(x, H, J; q, f, g)) q,

we can see that ∑ _{x} n(p, f, g; x, H, J)n(x, H, J; q, f, g) is the number of points of the set ⋃ _{x} ℳ(p, f, g; x, H, J) × ℳ(x, H, J; q, f, g), which is a component of the boundary ∂\overline{ℳ}(p, q, f; H).

Similarly to [15] we define homomorphisms l and j,

l : CM_{k}(f) → CM_{k}(f), l(p) = ∑_{m_{f} (q)=k} n(p, q, f; H) q,

j : CM_{k}(f) → CM_{k+1} (f), j(p) = ∑_{m_{f} (r)=k+1} \overline{n}(p, r, f; H) r.

Here n(p, q, f; H) is the number of intersections of the space of perturbed holomorphic discs with the unstable manifold W^{u}(p, f) and the stable manifold W^{s}(q, f). We consider discs with one half of the boundary on the zero-section o_{M}, and the other half on the conormal bundle ν^{∗}N. In other words, n(p, q, f; H) is the number of elements of ℳ_{R_{0}} (p, q, f; H). By \overline{n}(p, r, f; H) we denote the number of elements of a zero-dimensional manifold \overline{ℳ}(p, r, f; H). The sum

∑_{m_{f} (r)=k−1} n(p, r; f) \overline{n}(r, q, f; H)

corresponds to the sum that occurs in j ∘ ∂_{M}, and

∑_{m_{f} (r)=k+1} \overline{n}(p, r, f; H) n(r, q; f)

corresponds to the sum in ∂_{M} ∘ j. From Proposition 2.4, it follows

ϕ ∘ ψ − l = ∂_{M} ∘ j + j ∘ ∂_{M} .

Now, we prove that homomorphism L : HM_{k}(f) → HM_{k}(f), in homology, induced by the chain homomorphism l, does not depend on the Hamiltonian H. Let H_{0} and H_{1} be Hamiltonians and H_{δ}, 0 ⩽ δ ⩽ 1, a homotopy between them; l_{0} and l_{1} are chain homomorphisms corresponding to H_{0} and H_{1}. From Proposition 2.6 we get the relation l_{1} − l_{0} = ∂_{M} ∘ j_{δ} + j_{δ} ∘ ∂_{M} , where

j_{δ} : CM_{k}(f) → CM_{k+1} (f), j_{δ}(p) = ∑_{m_{f} (r)=k+1} \overline{n}(p, r, f; H_{δ}) r.

Here, \overline{n}(p, r, f; H_{δ}) is the number of elements of \overline{ℳ}(p, r, f; H_{δ}). If we choose a homotopy between our Hamiltonian H and 0, then we conclude that the map l is chain homotopic to the map i : CM_{k}(f) → CM_{k}(f), i(p) = ∑ _{m_{f} (q)=k} n(p, q, f; 0) q. Thus, L and the map I, induced by i, are the same maps in homology. We explained above that unperturbed holomorphic disc, with one half of the boundary on the zero-section and the other half on the conormal bundle, is constant. It follows that n(p, q, f; 0) is the number of points in W^{u}(p, f) ∩ W^{s}(q, f). Considering Morse indices of p and q, we get I = 𝕀.

We use the same idea to prove Ψ ∘ Φ = 𝕀. The composition ψ ∘ ϕ is chain homotopic to a chain homomorphism r : CF_{k}(H) → CF_{k}(H) which induces the identity in homology. If we denote by n_{ε}(x, y, H; f) the number of elements of the zero-dimensional manifold ℳ_{ε}(x, y, H; f), then the map analogous to l is

r(x) = ∑_{μ_{N} (y)=μ_{N} (x)} n_{ε}(x, y, H; f) y.

Similarly to the first part of the proof, a homomorphism in homology induced by r is independent of the choice of ε. Let r_{0} and r_{1} be homomorphisms corresponding

Figure 8. ℳ̃(x, y; H)

to the values ε_{0} and ε_{1}. We define a chain homomorphism

s : CF_{k}(H) → CF_{k+1} (H), s(x) = ∑_{μ_{N} (y)+dim N/2=k+1}  \underline{n}(x, y, H; f) y,

where \underline{n}(x, y, H; f) denotes the number of elements of \underline{ℳ}(x, y, H; f). From Proposition 2.5 we conclude that r_{0} − r_{1} = s ∘ ∂_{F} + ∂_{F} ∘ s. If we pass to the limit as ε → 0 we get that ψ ∘ ϕ if chain homotopic to the homomorphism

ĩ : CF_{k}(H) → CF_{k}(H), ĩ(x) = ∑_{μ_{N} (y)+dim N/2=k} ñ(x, y; H) y.

Here ñ(x, y; H) is the number of elements of the zero-dimensional manifold ℳ̃(x, y;H) defined as the set of pairs (u_{−} , u_{+} ) such that (see Figure 8)

u_{±} : ℝ × [0, 1] → T^{∗}M, E(u_{±} ) < ∞,

\frac{∂u_{±}}{∂s} + J( \frac{∂u_{±}}{∂t} − X_{ρ_{R}^{±} H} (u_{±} )) = 0,

u_{±} (s, 0) ∈ o_{M} , u_{±} (s, 1) ∈ ν^{∗}N, s ∈ ℝ,

u_{−} (−∞, t) = x(t), u_{+} (+∞, t) = y(t),

u_{−} (+∞) = u_{+} (−∞).

In the rest of the proof, we show that counting the number of elements of ℳ̃(x, y; H) is the same as counting the pseudo holomorphic strips between x and y (at the homology level). The main idea is to show that ℳ̃ is cobordant to the manifold that consists of appropriate pseudo holomorphic strips.

We define auxiliary manifold ℳ_{R}(x, y; H) as the set of all maps u : ℝ×[0, 1] → T^{∗}M such that

\frac{∂u}{∂s} + J( \frac{∂u}{∂t} − X_{ρ_{R} H} (u)) = 0,

u(s, 0) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N, s ∈ ℝ,

u(−∞, t) = x(t), u(+∞, t) = y(t).

Its parameterised version is

ℳ̌(x, y; H) = {(R, u) | R ⩾ R_{0}, u ∈ ℳ_{R}(x, y; H)} .

Here, ρ_{R} : ℝ → ℝ is a smooth function such that

ρ_{R}(s) = \{ \begin{matrix}1, & |s| ⩾ R + 1 \\ 0, & |s| ⩽ R. \end{matrix}

The boundary of the manifold ℳ̌(x, y; H) can be identified with

(3.1)  ∂ℳ̌ = ℳ_{R_{0}} (x, y; H) ∪ ℳ̃(x, y; H)

∪ ⋃_{z} ℳ(x, z; H) × ℳ̌(z, y; H)

∪ ⋃_{z} ℳ̌(x, z; H) × ℳ(z, y; H).

Now we explain equality (3.1). It is clear how the last two terms on the righthand side appear at the boundary. The elements from ℳ_{R_{0}} (x, y; H) appear at the boundary when R_{n} → R_{0}. The most complicated part is to prove that ℳ̃ is a part of the boundary. We show that in two steps. In Step A we explain why it holds ∂ℳ̌ ⊂ ℳ̃. And in Step B we show the opposite inclusion, ℳ̃ ⊂ ∂ℳ̌.

Step A. When R_{n} → +∞, we can identify the limit of u_{n} ∈ ℳ_{R_{n}} (x, y; H) with the element from ℳ̃(x, y; H) using the reparameterisation

u_{n}^{−} (s, t) = u_{n}(s − R_{n} + R_{0}, t), u_{n}^{+} (s, t) = u_{n}(s + R_{n} − R_{0}, t).

The strip u_{n}^{−} satisfies the equation

\frac{∂u_{n}^{−}}{∂s} + J ( \frac{∂u_{n}^{−}}{∂t} − X_{ρ_{Rn −} H} (u_{n}^{−} )) = 0,

and the boundary conditions

u_{n}^{−} (s, 0) = u_{n}(s − R_{n} + R_{0}, 0) ∈ o_{M} ,
u_{n}^{−} (s, 1) = u_{n}(s − R_{n} + R_{0}, 1) ∈ ν^{∗}N,

for s ∈ ℝ. The function ρ_{R_{n} −} is

ρ_{R_{n} −} (s) = \{ \begin{matrix}0, & −R_{0} ⩽ s ⩽ 2R_{n} − R_{0} \\ 1, & s ∈ (−∞, −R_{0} − 1] ∪ [2R_{n} − R_{0} + 1, +∞). \end{matrix}

The positive strip u_{n}^{+} also satisfies the perturbed Cauchy–Riemann equation

\frac{∂u_{n}^{+}}{∂s} + J ( \frac{∂u_{n}^{+}}{∂t} − X_{ρ_{Rn +} H} (u_{n}^{+} )) = 0,

the line u_{n}^{+} (ℝ × {0}) is on the zero section and u_{n}^{+} (ℝ × {1}) is on the conormal bundle. The function ρ_{R_{n} +} is defined by

ρ_{R_{n} +} (s) = \{ \begin{matrix}0, & −2R_{n} + R_{0} ⩽ s ⩽ R_{0} \\ 1, & s ∈ (−∞, −2R_{n} + R_{0} − 1] ∪ [R_{0} + 1, +∞). \end{matrix}

The strip u_{n}^{±} converges locally uniformly with all derivatives to some u^{±} that satisfies the equation

\frac{∂u^{±}}{∂s} + J( \frac{∂u^{±}}{∂t} − X_{ρ_{R_{0}}^{±} H} (u^{±} )) = 0.

It is obvious that u^{−} (−∞, t) = x(t) and u^{+} (+∞, t) = y(t). At the +∞-end, the strip u^{−} converges to a point p ∈ N ⊂ o_{M} since u^{−} is holomorphic for s ⩾ −R_{0} and it has a finite energy. The positive strip u^{+} is holomorphic at −∞ and it converges to a point q ∈ N. Since u_{n}^{−} (R_{n} −R_{0}, t) = u_{n}^{+} (−R_{n} +R_{0}, t), we conclude that p = q. Thus, the pair (u^{−} , u^{+} ) belongs to ℳ̃(x, y; H).

Step B. For a given (u_{−} , u_{+} ) ∈ ℳ̃, we can find a sequence of elements (R, ω_{R}) ∈ ℳ̌ that Gromov-converges to (u_{−} , u_{+} ) as R → +∞ (see [4, Theorem 4.1.2], [9, Chapter 4.7] and [31, Theorem 7.1]).

The main technique is gluing, and goes as follows. The strips u_{−} and u_{+} are holomorphic around the point u_{−} (+∞) = u_{+} (−∞), and we can preglue them to obtain a map u_{R}. This is an approximate solution of the Cauchy–Riemann equation, u_{R} satisfies it everywhere except a small neighbourhood of u_{R}(0) = u_{−} (+∞) = u_{+} (−∞). Next, we construct a right inverse to the linearization D_{u_{R}} of the operator \overline{∂}. Using the implicit-function theorem, we find a genuine solution ω_{R} to this equation, that is in a neighborhood of an approximate solution.

Biran and Cornea in [4] glued two holomorphic discs with the boundary on one Lagrangian submanifold. Frauenfelder in [9] and Schmäschke in [31] worked with two cleanly intersecting (compact) submanifolds in a compact symplectic manifold. The cotangent bundle is not a compact manifold, but, with appropriate choice on an almost complex structure (see the definition of j^{c} below), the image of every holomorphic strip lies in a compact subset of T^{∗}M [23, Theorem 3.2]. So we can assume that everything happens in a compact subset of our symplectic manifold. We also need a special Riemannian metrics on T^{∗}M such that o_{M} and ν^{∗}N are totally geodesic submanifolds with respect to these metrics.

Following [24, 10, 33], we explain choices on almost complex structures and Riemannian metrics on T^{∗}M. Fix a Riemannian metric g on M. The associated Levi-Civita connection induces the canonical almost complex structure on T^{∗}M, which we denote by J_{g}. We define the subset j^{c} of the set of almost complex structure on T^{∗}M by

j^{c} = {J | J is compatible to ω, J = J_{g} outside a compact subset in T^{∗}M}.

Let J_{t} be a smooth path in j^{c}. Then there exists a smooth family of metrics g_{t} such that

(1) o_{M} is totally geodesic with respect to g_{0} and J_{0}(q)T_{q}o_{M} is the orthogonal complement of T_{q}o_{M} for every q ∈ o_{M},

(2) ν^{∗}N is totally geodesic with respect to g_{1} and J_{1}(q)T_{q}(ν^{∗}N) is the orthogonal complement of T_{q}(ν^{∗}N) near the intersection point of two holomorphic strips that we glue,

(3) g_{t}(J_{t}(q)u, J_{t}(q)v) = g_{t}(u, v) for q ∈ T^{∗}M and u, v ∈ T_{q}(T^{∗}M).

We can define a metric g_{0} such that

(1) o_{M} is totally geodesic with respect to g_{0} and J_{0}(q)T_{q}o_{M} is the orthogonal

complement of T_{q}o_{M} for every q ∈ o_{M},
(2) g_{0}(J_{0}(q)u, J_{0}(q)v) = g_{0}(u, v) for q ∈ T^{∗}M and u, v ∈ T_{q}(T^{∗}M),

see [10] for details. In the same way we can define a metric g_{1} that satisfies the same properties for the submanifold ν^{∗}N. In [10] the author assumes that the Lagrangian submanifold is compact. The conormal bundle ν^{∗}N is not a compact manifold, in general. But it is enough to find a metric such that ν^{∗}N is a totally geodesic submanifold near N ⊂ o_{M}, not the whole ν^{∗}N. The linear combination g_{t}(u, v) = \overline{g}_{t}(u, v) + \overline{g}_{t}(J_{t}u, J_{t}v), where \overline{g}_{t}(u, v) = (1 − t)g_{0}(u, v) + tg_{1}(u, v), gives an appropriate family of metrics (see also [33]). All the other technical details of gluing are the same as in [31].

Now, we return to the homomorphism ĩ. Using the one-dimensional component of ℳ̌(x, y; H) and the description of its boundary (3.1), we conclude that ĩ (i.e., ψ ∘ ϕ) is chain homotopic to the map

k : x ↦ ∑_{μ_{N} (y)=μ_{N} (x)} n(x, y; H) y.

If there is a nonconstant holomorphic strip that connects Hamiltonian orbits x and y, then μ_{N} (x) > μ_{N} (y). It follows that

n(x, y; H) = \{ \begin{matrix}1, & x = y \\ 0, & x ≠ y, \end{matrix}

i.e., the map k induces the identity in homology HF_{∗}(H). □

4. Commutative diagram

Proof of Theorem 1.2. This theorem states that S^{αβ} ∘ Ψ^{α} = Ψ^{β} ∘ T^{αβ} . The composition on the left-hand side is generated bay the map σ^{αβ} ∘ ψ^{α}, and the right-hand side is generated by ψ^{β} ∘ τ^{αβ} on the chain level. The idea is to prove that these maps on the chain level are homotopic to each other.

We separate proof in two steps. In Step 1 and Step 2 we define new maps χ and ξ that are homotopic to σ^{αβ} ∘ ψ^{α} and ψ^{β} ∘ τ^{αβ} , respectively. In the conclusion of the proof we show that χ and ξ are chain homotopic maps.

Step 1. From definitions it follows

(σ^{αβ} ∘ ψ^{α})(p^{α}) = ∑_{x^{α} ,x^{β}} n(p^{α} , f^{α} , g^{α}; x^{α} , H^{α} , J^{α}) n(x^{α} , x^{β}; H^{αβ} ) x^{β}.

This means that σ^{αβ} ∘ ψ^{α} counts the number of points of the set

⋃_{x^{α}} ℳ(p^{α} , f^{α} , g^{α}; x^{α} , H^{α} , J^{α}) × ℳ(x^{α} , x^{β}; H^{αβ} ),

where ℳ(x^{α} , x^{β}; H^{αβ} ) denotes the set of solutions of (1.2). Summation is taken over x^{α} , x^{β} such that m_{f^{α}} (p^{α}) = μ_{N} (x^{α}) + \frac{1}{2} dimN = μ_{N} (x^{β}) + \frac{1}{2} dimN. Let us define a family of homotopies between Hamiltonians H^{α} and H^{β} by

H_{T,s}^{αβ} = \{ \begin{matrix}H^{α} , & s ⩽ T \\ H^{β}, & s ⩾ T + 1, \end{matrix}

and a family of homotopies between almost complex structures J^{α} and J^{β} by

J_{T,s}^{αβ} = \{ \begin{matrix}J^{α} , & s ⩽ T \\ J^{β}, & s ⩾ T + 1. \end{matrix}

We consider the moduli space ℳ̆(p^{α} , f^{α}; x^{β}, H_{T,s}^{αβ}) defined as the set of the triples (T, γ, u) such that

T ⩾ T_{0}, γ : (−∞, 0] → N, u : ℝ × [0, 1] → T^{∗}M,

\frac{dγ}{ds} = −∇_{g^{α}} f^{α} (γ), \frac{∂u}{∂s} + J_{T,s}^{αβ}( \frac{∂u}{∂t} − X_{ρ_{R}^{+} H_{T,s}^{αβ}} (u)) = 0,

E(u) < +∞, γ(−∞) = p^{α} , u(s, 0) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N, s ∈ ℝ,

u(+∞, t) = x^{β}(t), γ(0) = u(−∞).

Using the same idea as in the proof of Theorem 1.1, from gluing and compactness arguments, it follows that boundary of ℳ̆ can be described as

∂ℳ̆(p^{α} , f^{α}; x^{β}, H_{T,s}^{αβ}) = ℳ(p^{α} , f^{α}; x^{β}, H_{T_{0} ,s}^{αβ})

∪ ⋃_{x^{α}} ℳ(p^{α} , f^{α} , g^{α}; x^{α} , H^{α} , J^{α}) × ℳ(x^{α} , x^{β}; H^{αβ} )

∪ ⋃_{q^{α}} ℳ(p^{α} , q^{α}; f^{α}) × ℳ̆(q^{α} , f^{α}; x^{β}, H_{T,s}^{αβ})

∪ ⋃_{y^{β}} ℳ̆(p^{α} , f^{α}; y^{β}, H_{T,s}^{αβ}) × ℳ(y^{β}, x^{β}; H^{β}).

The first element in a previous union is already described in (2.4) (for fixed homotopy H_{s}^{αβ} = H_{T_{0} ,s}^{αβ}). We define a map χ that counts the number of elements in ℳ(p^{α} , f^{α}; x^{β}, H_{s}^{αβ} ) by χ(p^{α}) = ∑ _{x^{β}} n(p^{α} , f^{α}; x^{β}, H_{s}^{αβ} ) x^{β}. From the description of topological boundary of ℳ̆, we conclude that χ and σ^{αβ} ∘ ψ^{α} are chain homotopic maps.

Step 2. The other composition satisfies the equation

ψ^{β} ∘ τ^{αβ} (p^{α}) = ∑_{p^{β} ,x^{β}} n(p^{α} , p^{β}; f^{αβ} ) n(p^{β}, f^{β}, g^{β}; x^{β}, H^{β}, J^{β}) x^{β}.

Now, ψ^{β} ∘ τ^{αβ} counts the number of points of the set

⋃_{p^{β}} ℳ(p^{α} , p^{β}; f^{αβ} ) × ℳ(p^{β}, f^{β}, g^{β}; x^{β}, H^{β}, J^{β}),

where ℳ(p^{α} , p^{β}; f^{αβ} ) is the set of solutions of (1.4). Here, we take a sum over p^{β}, x^{β} such that m_{f^{α}} (p^{α}) = m_{f^{β}} (p^{β}) = μ_{N} (x^{β}) + \frac{1}{2} dimN. We define a map ξ

Figure 9. Riemannian surface Σ

that counts the number of points in ℳ(p^{α} , f_{s}^{αβ} ; x^{β}, H^{β}). It follows, similarly as in Step 1, that ξ and ψ^{β} ∘ τ^{αβ} are chain homotopic maps.

Using the moduli space ℳ̂(p^{α} , f_{s,δ}^{αβ} ; x^{β}, H_{s,δ}^{αβ} ) defined in (2.3), we prove that χ and ξ are chain homotopic maps. Let us define a chain homomorphism

j : CM_{k−1} (f^{α}) → CF_{k}(H^{β}), j(p^{α}) = ∑_{μ_{N} (x^{β} )+dim N/2=k} n̂(p^{α} , f_{s,δ}^{αβ} ; x^{β}, H_{s,δ}^{αβ} ) x^{β},

where n̂(p^{α} , f_{s,δ}^{αβ} ; x^{β}, H_{s,δ}^{αβ} ) is the number of elements of the zero-dimensional manifold ℳ̂(p^{α} , f_{s,δ}^{αβ} ; x^{β}, H_{s,δ}^{αβ} ). From Proposition 2.7, it follows that

ξ − χ + j ∘ ∂_{M} + ∂_{F} ∘ j = 0.  □

5. Product in homology

In this section we prove Theorem 1.3. First we explain a construction of the product

⋆ : HF_{∗}(o_{M} , o_{M} : H_{1}) ⊗ HF_{∗}(o_{M} , ν^{∗}N : H_{2}) → HF_{∗}(o_{M} , ν^{∗}N : H_{3}).

Then we prove subadditivity of spectral invariants with respect to this product.

Proof of Theorem 1.3. We define a Riemannian surface with boundary Σ as the disjoint union ℝ × [−1, 0] ⊔ ℝ × [0, 1] with identification (s, 0^{−} ) ∼ (s, 0^{+} ) for s ⩾ 0 (see Figure 9). The surface Σ is conformally equivalent to a closed disc with three boundary punctures. Complex structure on Σ ∖ {(0, 0)} is induced by the inclusion (s, t) ↦ s + it, in ℂ. Complex structure at (0, 0) is given by the square root.

Denote by Σ_{1}^{−} , Σ_{2}^{−} , Σ^{+} the two “incoming" and one “outgoing" ends, such that

Σ_{1}^{−} , Σ_{2}^{−} ≈ [0, 1] × (−∞, 0], Σ^{+} ≈ [0, 1] × [0, +∞).

By u_{j}^{−} := u|_{Σ_{j}^{−}} , j = 1, 2, and u^{+} := u|_{Σ^{+}} , we denote the restriction of the map defined on the surface Σ. Let ρ^{±} : ℝ → [0, 1] denote the smooth cut-off functions such that

ρ^{−} (s) = \{ \begin{matrix}1, & s ⩽ −2 \\ 0, & s ⩾ −1 \end{matrix} ρ^{+} (s) = ρ^{−} (−s).

For x_{1}^{−} ∈ CF_{∗}(o_{M} , o_{M} :H_{1}), x_{2}^{−} ∈ CF_{∗}(o_{M} , ν^{∗}N:H_{2}) and x^{+} ∈ CF_{∗}(o_{M} , ν^{∗}N:H_{3}) , we define the moduli space ℳ(x_{1}^{−} , x_{2}^{−} ; x^{+} ) as a set of maps u : Σ → T^{∗}M such that

∂_{s}u_{j}^{−} + J(∂_{t}u_{j}^{−} − X_{ρ^{−} H_{j}} (u_{j}^{−} )) = 0, j = 1, 2,

Figure 10. Set of trees ℳ^{tree} (p_{1}^{−} , p_{2}^{−} ; p^{+} )

∂_{s}u^{+} + J(∂_{t}u^{+} − X_{ρ^{+} H_{3}} (u^{+} )) = 0,

∂_{s}u + J∂_{t}u = 0 on Σ_{0} = Σ ∖ (Σ_{1} ∪ Σ_{2} ∪ Σ_{3}),

u(s, −1) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N, s ∈ ℝ,

u(s, 0^{−} ), u(s, 0^{+} ) ∈ o_{M} , s ⩽ 0,

u_{j}^{−} (−∞, t) = x_{j}^{−} (t), j = 1, 2, u^{+} (+∞, t) = x^{+} (t).

We use the notation \overline{∂}_{J,H} (u) = 0 for the perturbed Cauchy–Riemann equation that we consider in ℳ(x_{1}^{−} , x_{2}^{−} ; x^{+} ). Elements of a moduli space ℳ(x_{1}^{−} , x_{2}^{−} ; x^{+} ) are perturbed holomorphic discs u. The boundary of u is on the Lagrangian submanifold o_{M} ∪ ν^{∗}N with the clean self-intersection along N.

For generic choices of Hamiltonians and an almost complex structure, the manifold ℳ(x_{1}^{−} , x_{2}^{−} ; x^{+} ) is smooth and of dimension

μ_{M} (x_{1}^{−} ) + μ_{N} (x_{2}^{−} ) − μ_{N} (x^{+} ) − \frac{1}{2} dimM.

By x_{1}^{−} ⋆ x_{2}^{−} = ∑ _{x^{+}} ♯_{2}ℳ(x_{1}^{−} , x_{2}^{−} ; x^{+} ) x^{+} , we define a product on generators of Floer complexes. Here ♯_{2}ℳ(x_{1}^{−} , x_{2}^{−} ; x^{+} ) denotes the number (modulo 2) of elements of the zero-dimensional component of ℳ(x_{1}^{−} , x_{2}^{−} ; x^{+} ). (Similar type of product is defined in [6], where it was used in the comparison of spectral invariants in Lagrangian and Hamiltonian Floer theory.) We extend the product ⋆ by bilinearity on CF_{∗}(o_{M} , o_{M} : H_{1}) ⊗ CF_{∗}(o_{M} , ν^{∗}N : H_{2}). The operation ⋆ commutes with the corresponding boundary operators and induces product in homology

⋆ : HF_{k}(o_{M} , o_{M} : H_{1}) ⊗ HF_{l}(o_{M} , ν^{∗}N : H_{2}) → HF_{k+l−dim M} (o_{M} , ν^{∗}N : H_{3}).

The next step is to define an exterior intersection product on the Morse homology. Let us take Morse functions f_{1} : M → ℝ and f_{2}, f_{3} : N → ℝ. For p_{1}^{−} ∈ Crit(f_{1}), p_{2}^{−} ∈ Crit(f_{2}) and p^{+} ∈ Crit(f_{3}) we define the set of trees ℳ^{tree} (p_{1}^{−} , p_{2}^{−} ; p^{+} ) (see Figure 10). The moduli space ℳ^{tree} (p_{1}^{−} , p_{2}^{−} ; p^{+} ) contains the triples (γ_{1}^{−} , γ_{2}^{−} , γ^{+} ) such that

γ_{1}^{−} : (−∞, 0] → M, γ_{2}^{−} : (−∞, 0] → N, γ^{+} : [0, +∞) → N,

\frac{dγ_{1}^{−}}{ds} = −∇_{g_{1}} f_{1}(γ_{1}^{−} ), \frac{dγ_{2}^{−}}{ds} = −∇_{g_{2}} f_{2}(γ_{2}^{−} ), \frac{dγ^{+}}{ds} = −∇_{g_{3}} f_{3}(γ+ ),

γ_{1}^{−} (−∞) = p_{1}^{−} , γ_{2}^{−} (−∞) = p_{2}^{−} , γ^{+} (+∞) = p^{+} , γ_{1}^{−} (0) = γ_{2}^{−} (0) = γ^{+} (0).

For generic choices of Morse–Smale pairs (f_{j} , g_{j}), j ∈{1, 2, 3}, ℳ^{tree} (p_{1}^{−} , p_{2}^{−} ; p^{+} ) is a smooth manifold of dimension m_{f_{1}} (p_{1}^{−} )+m_{f_{2}} (p_{2}^{−} )−m_{f_{3}} (p^{+} )−dimM. On chain complexes we define

· : CF_{k}(M : f_{1}) ⊗ CF_{l}(N : f_{2}) → CF_{k+l−dim M} (N : f_{3}),

by

p_{1}^{−} · p_{2}^{−} = ∑_{p^{+}} ♯_{2}ℳ^{tree} (p_{1}^{−} , p_{2}^{−} ; p^{+} ) p^{+} .

It is a chain map that defines the exterior intersection product

· : HF_{k}(M : f_{1}) ⊗ HF_{l}(N : f_{2}) → HF_{k+l−dim M} (N : f_{3}).

Once we have defined the exterior intersection product, we can prove that PSS preserves the algebraic structure, i.e., it maps · to ⋆. The idea is to show that · and ϕ(ψ ⋆ ψ) are chain homotopic maps. As in the previous situation, p_{1}^{−} , p_{2}^{−} , p^{+} are critical points. We know that

ϕ(ψ(p_{1}^{−} ) ⋆ ψ(p_{2}^{−} )) = ∑_{x_{1}^{−} ,x_{2}^{−} ,x^{+} ,p^{+}} ♯_{2}ℳ(p_{1}^{−} , f_{1}; x_{1}^{−} , H_{1}) ♯_{2}ℳ(p_{2}^{−} , f_{2}; x_{2}^{−} , H_{2})
♯_{2}ℳ(x_{1}^{−} , x_{2}^{−} ; x^{+} ) ♯_{2}ℳ(x^{+} , H_{3}; p^{+} , f_{3}) p^{+} .

Following [16], we define two auxiliary manifolds. The first of them, the manifold ℳ_{R}^{prod} (p_{1}^{−} , p_{2}^{−} , p^{+} ; f⃗, H⃗), is the set of all (γ_{1}^{−} , γ_{2}^{−} , γ^{+} , u) that satisfy

γ_{1}^{−} : (−∞, 0] → M, γ_{1}^{−} (−∞) = p_{1}^{−} , γ_{2}^{−} : (−∞, 0] → N, γ_{2}^{−} (−∞) = p_{2}^{−} ,

γ^{+} : [0, +∞) → N, γ^{+} (+∞) = p^{+} , u : Σ → T^{∗}M,

\frac{dγ_{1}^{−}}{ds} = −∇_{g_{1}} f_{1}(γ_{1}^{−} ), \frac{dγ_{2}^{−}}{ds} = −∇_{g_{2}} f_{2}(γ_{2}^{−} ), \frac{dγ^{+}}{ds} = −∇_{g_{3}} f_{3}(γ+ ),

∂_{s}u_{j}^{−} + J(∂_{t}u_{j}^{−} − X_{κ_{R}^{−} H_{j}} (u_{j}^{−} )) = 0, j = 1, 2, ∂_{s}u^{+} + J(∂_{t}u^{+} − X_{κ_{R}^{+} H_{3}} (u^{+} )) = 0,

∂_{s}u + J∂_{t}u = 0 on Σ_{0}, E(u) < +∞,

u(s, −1) ∈ o_{M} , u(s, 1) ∈ ν^{∗}N, s ∈ ℝ, u(s, 0^{−} ), u(s, 0^{+} ) ∈ o_{M} , s ⩽ 0,

u_{j}^{−} (−∞) = γ_{j}^{−} (0), j = 1, 2, u^{+} (+∞) = γ^{+} (0),

where R > 2. A function κ_{R}^{−} : (−∞, 0] → [0, 1] is defined by

κ_{R}^{−}(s) = \{ \begin{matrix}1, & −R ⩽ s ⩽ −2 \\ 0, & s ⩽ −R − 1, s ⩾ −1, \end{matrix}

and κ_{R}^{+} : [0, +∞) → [0, 1], κ_{R}^{+}(s) = κ_{R}^{−}(−s). We include (f⃗, H⃗) in the notation for manifold ℳ_{R}^{prod} in order to emphasize that we have different functions, f_{j}, and Hamiltonians, H_{j}, at appropriate ends. Another moduli space is

ℳ^{prod} (p_{1}^{−} , p_{2}^{−} , p^{+} ; f⃗, H⃗) =
{(R, γ_{1}^{−} , γ_{2}^{−} , γ^{+} , u) | R > R_{0}, (γ_{1}^{−} , γ_{2}^{−} , γ^{+} , u) ∈ ℳ_{R}^{prod} (p_{1}^{−} , p_{2}^{−} , p^{+} ; f⃗, H⃗)}.

The boundary of one-dimensional component of ℳ^{prod} is of the form

∂ℳ^{prod} (p_{1}^{−} , p_{2}^{−} , p^{+} ; f⃗, H⃗) = ℳ_{R_{0}}^{prod} (p_{1}^{−} , p_{2}^{−} , p^{+} ; f⃗, H⃗)

∪ ⋃_{q_{1}^{−} ∈CF_{∗} (f_{1} )} ℳ(p_{1}^{−} , q_{1}^{−} ; f_{1}) × ℳ^{prod} (q_{1}^{−} , p_{2}^{−} , p^{+} ; f⃗, H⃗)

∪ ⋃_{q_{2}^{−} ∈CF_{∗} (f_{2} )} ℳ(p_{2}^{−} , q_{2}^{−} ; f_{2}) × ℳ^{prod} (p_{1}^{−} , q_{2}^{−} , p^{+} ; f⃗, H⃗)

∪ ⋃_{q^{+} ∈CF_{∗} (f_{3} )} ℳ^{prod} (p_{1}^{−} , p_{2}^{−} , q^{+} ; f⃗, H⃗) × ℳ(q^{+} , p^{+} ; f_{3})

∪ ⋃_{x_{1}^{−} ,x_{2}^{−} ,x^{+}} ℳ(p_{1}^{−} , f_{1}; x_{1}^{−} , H_{1}) × ℳ(p_{2}^{−} , f_{2}; x_{2}^{−} , H_{2})

× ℳ(x_{1}^{−} , x_{2}^{−} ; x^{+} ) × ℳ(x^{+} , H_{3}; p^{+} , f_{3}).

We conclude that, at the homology level, ϕ(ψ(p_{1}^{−} ) ⋆ ψ(p_{2}^{−} )) equals homomorphism that counts the number of elements of ℳ_{R_{0}}^{prod} (p_{1}^{−} , p_{2}^{−} , p^{+} ; f⃗, H⃗). The latter is independent of the choice of Hamiltonians. We used the same idea to show that the homomorphism L, in the proof of Theorem 1.1, is independent of Hamiltonian. Therefore, we can take Hamiltonians to be zero. Homolomorphic pants with boundary on o_{M} ∪ ν^{∗}N are constant, thus

ℳ_{R_{0}}^{prod} (p_{1}^{−} , p_{2}^{−} , p^{+} ; f⃗, H⃗ = 0) = ℳ^{tree} (p_{1}^{−} , p_{2}^{−} ; p^{+} ).

It follows that α · β = Φ(Ψ(α) ⋆ Ψ(β)), for α ∈ HM_{∗}(M) and β ∈ HM_{∗}(N).

Now we prove the inequality between spectral invariants stated in Theorem 1.3. Since a concatenation does not have to be a smooth function, we can find a Hamiltonian H′ that is regular, smooth and close enough to the concatenation H_{1}♯H_{2}, i.e., ∥H′− H_{1}♯H_{2}∥_{C^{0}} < ε. The first step is to prove that the product ⋆ defines a product

CF_{∗}^{λ}(H_{1}) × CF_{∗}^{μ}(H_{2}) → CF_{∗}^{λ+μ+ε} (H′),

on filtered complexes, for every ε > 0 that is small enough. Let us take a smooth family of Hamiltonians K : ℝ × [−1, 1] × T^{∗}M → ℝ such that

K(s, t, ·) = \{ \begin{matrix}H_{1}(t + 1, ·), & s ⩽ −1, −1 ⩽ t ⩽ 0 \\ H_{2}(t, ·), & s ⩽ −1, 0 ⩽ t ⩽ 1 \\ \frac{1}{2}H′(\frac{t+1}{2} , ·), & s ⩾ 1. \end{matrix}

We can choose K such that ∥ \frac{∂K}{∂s} ∥ ⩽ ε, s ∈ [−1, 1], and \frac{∂K}{∂s} = 0, elsewhere. Let us take x_{1}^{−} ∈ CF_{∗}^{λ}(H_{1}) and x_{2}^{−} ∈ CF_{∗}^{μ}(H_{2}). Assume that there exists an element u ∈ ℳ(x_{1}^{−} , x_{2}^{−} ; x^{+} ) for some x^{+} ∈ CF_{∗}(H′) (u is a solution of the equation ¯∂_{K,J} (u) = 0). Then it holds

(5.1)  0 ⩽ ∫_{Σ} ∥ \frac{∂u}{∂s} ∥_{J}^{2} ds dt = ∫_{Σ} ω( \frac{∂u}{∂s}, J \frac{∂u}{∂s}) ds dt

= ∫_{Σ} ω( \frac{∂u}{∂s}, \frac{∂u}{∂t} − X_{K} (u)) ds dt

= ∫_{Σ} u^{∗}ω − ∫_{Σ} dK( \frac{∂u}{∂s}) ds dt.

Using Stoke’s formula we obtain ∫_{Σ} u^{∗}ω = − ∫ x_{1}^{−∗} λ − ∫ x_{2}^{−∗} λ + ∫ x^{+∗} λ. Using the equality

∫_{Σ} \frac{∂}{∂s} (K ∘ u) ds dt = ∫_{Σ} dK( \frac{∂u}{∂s}) ds dt + ∫_{Σ} \frac{∂K}{∂s} (u) ds dt,

and Stoke’s formula again, we get the estimate

− ∫_{Σ} dK( \frac{∂u}{∂s}) ds dt ⩽ ∫_{0}^{1} H_{1}(x_{1}^{−} (t), t) dt
+ ∫_{0}^{1} H_{2}(x_{2}^{−} (t), t) dt − ∫_{0}^{1} H′(x^{+} (t), t) dt + 4ε.

Thus 𝒜_{H′} (x^{+} ) ⩽ 𝒜_{H_{1}} (x_{1}^{−} ) + 𝒜_{H_{2}} (x_{2}^{−} ) + 4ε. From the definition of the operation ·, it easily follows

l(α · β; o_{M} , ν^{∗}N : H′) ⩽ l(α; o_{M} , o_{M} : H_{1}) + l(β; o_{M} , ν^{∗}N : H_{2}) + 4ε.

We know [23] that spectral invariants are continuous with respect to Hamiltonians. If we pass to the limit as ε → 0, then we get the triangle inequality

l(α · β; o_{M} , ν^{∗}N : H_{1}♯H_{2}) ⩽ l(α; o_{M} , o_{M} : H_{1}) + l(β; o_{M} , ν^{∗}N : H_{2}).  □

Proof of Theorem 1.4. For x∈CF_{∗}(o_{M} , ν^{∗}N:H) and y ∈CF_{∗}(o_{M} , o_{M} :H), we define an auxiliary moduli space ℳ^{j} (x, y; H) to be the set of all the maps u: ℝ × [0, 1] → T^{∗}M such that

∂_{s}u + J(∂_{t}u − X_{H} (u)) = 0, u(s, 0) ∈ o_{M} , s ∈ ℝ, u(s, 1) ∈ ν^{∗}N, s ⩽ 0,

u(s, 1) ∈ o_{M} , s ⩾ 0, u(−∞, t) = x(t), u(+∞, t) = y(t).

Strips of this type, with a jump on the boundary, were discussed in [1]. On the generators of CF_{∗}(o_{M} , ν^{∗}N : H) we define m to be m(x) = ∑ _{y} ♯_{2}ℳ^{j} (x, y; H) y. The boundary of the one-dimensional component of ℳ^{j} (x, y; H) is

∂ℳ_{[1]}^{j}(x, y; H) =  ⋃_{x′∈CF_{∗} (o_{M} ,ν^{∗} N:H)} ℳ(x, x′; H) × ℳ^{j} (x′, x; H)

∪ ⋃_{y′∈CF_{∗} (o_{M} ,o_{M} :H)} ℳ^{j} (x, y′; H) × ℳ(y′, y; H).

Thus, m induces a map on homology level (denoted by m, again)

m : HF_{∗}(o_{M} , ν^{∗}N : H) → HF_{∗}(o_{M} , o_{M} : H).

We can explicitly describe the induced morphism on the Morse side

Φ ∘ m ∘ Ψ : HM_{∗}(N) → HM_{∗}(M).

In order to do this, we need to correlate somehow the Morse functions on N and M. Let us take a Morse function f : N → ℝ. Following [32], we can find a Morse function F : M → ℝ extending f, F |_{N} = f, in such a way that there are no trajectories for the negative gradient flow of F leaving N (see [32, Proposition 4.16 and Corollary 4.17]).

On chain complexes, ϕ ∘ m ∘ ψ is

ϕ(m(ψ(p))) = ∑_{x,y,q} ♯_{2}ℳ(p, f, g; x, H, J)♯_{2}ℳ^{j} (x, y; H)♯_{2}ℳ(y, H, J; q, F, g) q,

where the summation is taken over

x ∈ CF_{k}(o_{M} , ν^{∗}N : H), y ∈ CF_{k}(o_{M} , o_{M} : H), q ∈ CM_{k}(M : F).

Note that g is a metric on M. We will use the same idea as in the proof of Theorem 1.1, when we defined ℳ_{R}(p, q, f; H) and \overline{ℳ}(p, q, f; H). So, we define two auxiliary manifolds. One of them, denoted by ℳ_{R}^{aux} (p, f; q, F; H), is defined as the set of triples (γ_{−} , u, γ_{+} ) such that

γ_{−} : (−∞, 0] → N, dotγ_{−} = −∇f(γ_{−} ), γ_{+} : [0, +∞) → M, γ̇_{+} = −∇F (γ_{+} ),

u : ℝ × [0, 1] → T^{∗}M, ∂_{s}u + J(∂_{t}u − X_{σ_{R} H} (u)) = 0,
u(s, 0) ∈ o_{M} , s ∈ ℝ, u(s, 1) ∈ ν^{∗}N, s ⩽ 0, u(s, 1) ∈ o_{M} , s ⩾ 0,
γ_{−} (−∞) = p, γ_{+} (+∞) = q, u(−∞) = γ_{−} (0), u(+∞) = γ_{+} (0).

The dimension of ℳ_{R}^{aux} (p, f; q, F; H) is m_{f} (p) − m_{F} (q). The other manifold is

ℳ^{aux} (p, f; q, F; H) = {(R, γ_{−} , u, γ_{+} ) | (γ_{−} , u, γ_{+} ) ∈ ℳ_{R}^{aux} (p, f; q, F; H), R > R_{0}},

of dimension m_{f} (p) − m_{F} (q) + 1. For p ∈ CM_{k}(N : f) and q ∈ CM_{k}(M : F), the boundary of the one-dimensional manifold ℳ^{aux} (p, f; q, F; H) is

∂ℳ^{aux} (p, f; q, F; H) = ℳ_{R_{0}}^{aux} (p, f; q, F; H)

∪ ⋃_{r∈CM_{k−1} (N:f)} ℳ(p, r; f) × ℳ^{aux} (r, f; q, F; H)

∪ ⋃_{s∈CM_{k+1} (M:F)} ℳ^{aux} (p, f; s, F; H) × ℳ(s, q; F)

∪ ⋃_{x∈CF_{k} ,y∈CF_{k}} ℳ(p, f, g; x, H, J) × ℳ^{j} (x, y; H) × ℳ(y, H, J; q, F, g).

Thus, ϕ ∘ m ∘ ψ is chain homotopic to the map η : CM_{k}(N : f) → CM_{k}(M : F), defined by

η(p) = ∑_{q} ♯_{2}ℳ_{R_{0}}^{aux} (p, f; q, F; H) q.

This map is an analogue of the map l defined in the proof of Theorem 1.1. In the same way, η is going to be chain homotopic to the map η_{0} that counts combined object (γ_{−} , u, γ_{+} ) where u is a holomorphic disc (perturbed by zero Hamiltonian) with the boundary on o_{M} ∪ ν^{∗}N. We have already showed that all such discs are constant, thus η_{0} counts the number of gradient trajectories of F (since F = f on N) that connect p ∈ N with some q ∈ CM_{k}(M : F). We assume that there are no negative gradient trajectories of F leaving N. Since p and q are of the same Morse index, a gradient trajectory connecting p and q does not exist when p ≠ q. We conclude that η_{0} = i : CM_{k}(N : f) → CM_{k}(M : F), is the inclusion of chain complexes. Once again, we construct F as an extension of f. Thus the inclusion i of chain complexes makes sense in this situation. In Morse homology, Φ ∘ m ∘ Ψ and i induce the same map.

We are only left to prove inequality (1.6) among spectral invariants. Using the same idea as in (5.1), one can prove that the action functional 𝒜_{H} decreases along the holomorphic strip u ∈ ℳ^{j} (x, y; H). It means that m induces the homomorphism

m : HF_{∗}^{λ}(o_{M} , ν^{∗}N : H) → HF_{∗}^{λ}(o_{M} , o_{M} : H).

on filtered homology. So, if Ψ(α) is realized as an element from HF_{∗}^{λ}(o_{M} , ν^{∗}N : H), then an element m(Ψ(α)) = Ψ(Φ(m(Ψ(α)))), is realized as an element from HF_{∗}^{λ}(o_{M} , o_{M} : H). The inequality (1.6) directly follows. □

Acknowledgments. The author thanks Jelena Katić, Darko Milinković and Katrin Wehrheim for useful discussions during the preparation of this paper. The author also thanks the anonymous referee for many valuable suggestions and corrections.

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Faculty of Mathematics (Received 30 10 2014) University of Belgrade (Revised 15 07 2015, 23 01 2017, and 12 06 2017) Belgrade Serbia

    jovanadj@matf.bg.ac.rs