Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli
  spaces
Symplectic homology, autonomous Hamiltonians,
and Morse-Bott moduli spaces
Frédéric Bourgeois Alexandru Oancea
Université Libre de Bruxelles Université Louis Pasteur
B-1050 Bruxelles, Belgium F-67084 Strasbourg, France
fbourgeo@ulb.ac.be oancea@math.u-strasbg.fr
11 March 2008
Abstract
We define Floer homology for a time-independent, or autonomous
Hamiltonian on a symplectic manifold with contact type boundary, under
the assumption that its 1-periodic orbits are transversally nondegenerate.
Our construction is based on Morse-Bott techniques for Floer trajectories.
Our main motivation is to understand the relationship between linearized
contact homology of a fillable contact manifold and symplectic homology
of its filling.
2000 Mathematics Subject Classification: 53D40.
Contents
1 Introduction 2
2 Symplectic homology 6
3 The Morse-Bott chain complex 11
4 Morse-Bott moduli spaces 21
4.1 Transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Compactness for Morse-Bott trajectories . . . . . . . . . . . . . . 26
4.3 Gluing for Morse-Bott moduli spaces . . . . . . . . . . . . . . . . 30
4.4 Coherent orientations . . . . . . . . . . . . . . . . . . . . . . . . 69
A Appendix: Asymptotic estimates 84
http://arxiv.org/abs/0704.1039v2
Symplectic homology for autonomous Hamiltonians 2
1 Introduction
One crucial hypothesis in the definition of Floer homology [12] of a Hamiltonian
H on a symplectic manifold (W,ω) is that the 1-periodic orbits of the Hamilto-
nian vector field XH are nondegenerate. Unless they are all constant – which
happens if the Hamiltonian is C2-small – this forces H to be time-dependent.
The purpose of this paper is to define Floer homology for a time-independent, or
autonomous Hamiltonian H :W → R under the assumption that its 1-periodic
orbits are transversally nondegenerate. This last condition is generic in the
space of autonomous Hamiltonians.
Although this generalization of Floer homology is interesting by itself, our
main motivation is to understand the relationship between linearized contact
homology of a fillable contact manifold (M, ξ) and symplectic homology of the
filling (W,ω). In this case there is a natural class of time-independent Hamilto-
nians on W whose nonconstant 1-periodic orbits correspond precisely to closed
Reeb orbits on M = ∂W , and for which the Floer trajectories can be related
to holomorphic cylinders in the symplectization M × R [3]. The goal of the
present paper is to relate the Floer trajectories of a specific time-dependent
perturbation to the Floer trajectories of the unperturbed Hamiltonian. Thus
Floer homology for time-independent Hamiltonians serves as a bridge between
symplectic homology and linearized contact homology. Moreover, the moduli
spaces of Floer trajectories for autonomous Hamiltonians are related to the
moduli spaces defining S1-equivariant symplectic homology [4, 3].
The Morse-Bott analysis in this paper is, to the best of our knowledge, new
to the literature, being based on ideas contained in the first author’s Ph.D.
dissertation [1] within the context of contact homology. Although our situation
is that of critical manifolds of dimension one, the complexity of the analytical
setup is the same as that of the higher dimensional case.
We must mention at this point Frauenfelder’s inspired approach [16, Ap-
pendix A] in which he defines a complex for a Morse-Bott function on a finite
dimensional manifold via “flow lines with cascades” – these being our Floer
trajectories with gradient fragments – and in which, without proving the cor-
respondence with gradient trajectories for some perturbed Morse function, he
directly shows deformation invariance of the resulting chain complex.
We now describe the structure of the paper. We give in the introduction only
a loose statement of our main Correspondence Theorem 3.7 and we recall in
Section 2 the construction of symplectic homology. Although this is well-known
to specialists we still need to establish notations, and we seize the occasion to
set up a general framework using the Novikov ring and nontrivial homotopy
classes of periodic orbits.
Section 3 describes the Morse-Bott complex and formally states the Corre-
spondence Theorem 3.7. The latter is complemented by Proposition 3.9 which
describes how the coherent orientation signs for the Morse-Bott complex are
related to the ones for the Floer complex.
Section 4 contains the proofs of the previous transversality, compactness,
Symplectic homology for autonomous Hamiltonians 3
gluing and orientation statements. Finally, the Appendix contains the state-
ments concerned with the asymptotic behaviour of the various types of Floer
trajectories that we use. These asymptotic estimates enter crucially in the proof
of the compactness statements, as well as in the definition of the Fredholm setup
for gluing.
We end the introduction with an informal presentation of our results. Let
H : W → R be an autonomous Hamiltonian defined on a symplectic manifold
(W,ω). We assume that H is a Morse function and that the nonconstant 1-
periodic orbits ofH are transversally nondegenerate. The set P(H) of 1-periodic
orbits of H is the set of critical points of the Hamiltonian action functional and
consists of isolated elements γep corresponding to critical points p̃ ∈ Crit(H),
and of nonisolated elements coming in families Sγ which are Morse-Bott non-
degenerate circles. These correspond to reparametrizations of some given orbit
γ ∈ P(H), with γ : S1 = R/Z →W .
For each circle Sγ we choose a perfect Morse function fγ : Sγ → R with
exactly one maximum Max and one minimum min . We denote by γmin , γMax
the orbits in Sγ corresponding to the minimum and the maximum of fγ respec-
tively. We choose a chart S1 × R2n−1 ∋ (τ, p) and a smooth cut-off function
ργ : S
1 × R2n−1 → R in the neighbourhood of each γ(S1) ⊂W , and we denote
by ℓγ ∈ Z+ the maximal positive integer such that γ(θ + 1/ℓγ) = γ(θ), θ ∈ S1.
Following [8], for δ > 0 small enough the time-dependent Hamiltonian
Hδ : S
1 ×W → R,
Hδ(θ, τ, p) := H − δ
ργ(τ, p)fγ(τ − ℓγθ)
has only nondegenerate 1-periodic orbits. Moreover, these are of the following
two types: they are either constant orbits γep corresponding to critical points
p̃ ∈ Crit(H), or they are nonconstant orbits of the form γp ∈ P(H) for p ∈
Crit(fγ). Thus, out of each circle Sγ of periodic orbits for H there are exactly
two orbits surviving for Hδ, namely γmin and γMax .
Let J be a generic time-dependent almost complex structure on W . Given
p ∈ Crit(fγ), q ∈ Crit(fγ) we denote by
M(γp, γq;Hδ, J)
the moduli space of Floer trajectories for the pair (Hδ, J) modulo reparametriza-
tion, with negative asymptote γp and positive asymptote γq. We also denote
M(Sγ , Sγ ;H, J)
the moduli space of Floer trajectories for the pair (H, J) modulo reparametriza-
tion, with negative asymptote in Sγ and positive asymptote in Sγ . Our goal is
to describe the moduli spaces of the first type in terms of moduli spaces of the
second type.
Symplectic homology for autonomous Hamiltonians 4
We denote by
M(p, q;H, {fγ}, J)
the moduli space of Floer trajectories for the pair (H, J) with intermediate
gradient fragments, consisting of tuples
[u] = (cm, [um], cm−1, [um−1], . . . , [u1], c0), m ≥ 0
such that:
(i) [ui] ∈ M(Sγi , Sγi−1 ;H, J), i = 1, . . . ,m with γm := γ, γ0 := γ;
(ii) cm is a semi-infinite gradient trajectory of fγ = fγm connecting γp to the
endpoint of um;
(iii) cj , j = 1, . . . ,m − 1 is a finite gradient trajectory of fγj connecting the
endpoints of uj+1 and uj ;
(iv) c0 is a semi-infinite gradient trajectory of fγ = fγ0 connecting the endpoint
of u1 to γq.
We give a pictogram of such an element [u] with m ≥ 1 in Figure 4 on page 45,
where one should read ci instead of vi. If m = 0 such an element [u] is simply
an infinite gradient trajectory of some fγ . Let us note that, just as the space
of Floer trajectories for a nondegenerate Hamiltonian can be compactified by
adding “broken” Floer trajectories, the space of Floer trajectories with interme-
diate gradient fragments can be compactified by adding “broken” such objects,
with an obvious meaning. We denote by M(p, q;H, {fγ}, J) these compactified
moduli spaces.
Our main result is the following comparison theorem.
Theorem. The following assertions hold.
(i) any sequence [vn] ∈ M(γp, γq;Hδn , J), δn → 0 converges to an element of
M(p, q;H, {fγ}, J);
(ii) any element of M(p, q;H, {fγ}, J) can be obtained as such a limit;
(iii) there is a bijective correspondence between elements of M(γp, γq;Hδ, J)
and elements of M(p, q;H, {fγ}, J) if the difference of index of the end-
points is equal to one, or equivalently if the moduli spaces have dimension
zero.
The rigorous forms for the statements (i), (ii), (iii) are given in Proposi-
tion 4.7, Proposition 4.22 and Theorem 3.7 respectively. Unsurprisingly, the
Fredholm setup for the previous theorem uses Sobolev norms with exponential
weights since we have degenerate asymptotics. Similarly, due to the convergence
estimates in the Appendix, there are such weights centered on the portions of the
Floer cylinders approaching gradient fragments. For each peak in the weight,
there is a special section supported around this peak which has constant norm
Symplectic homology for autonomous Hamiltonians 5
with respect to δ → 0. For each gradient fragment this section corresponds to
the reparametrization shift of the underlying gradient trajectory. As δ → 0, the
corresponding peak explodes and thus forbids all infinitesimal variations except
for the single degree of freedom coming from Morse theory.
To be useful for homological calculations the above theorem needs to be
complemented by a statement concerning signs. We describe in Section 4.4 how
to construct coherent orientations on the relevant spaces of Fredholm operators
and how to obtain signs ǫ(u) and ǫ(uδ) for elements [u] ∈ M(p, q;H, {fγ}, J)
and uδ ∈ M(γp, γq;Hδ, J) when the corresponding moduli spaces are zero-
dimensional. We recall in Remark 3.2 the definition of good orbits borrowed
from Symplectic Field Theory, where it plays a crucial role in all orientation
and signs problems. In the following statement we denote again by m ≥ 0 the
number of nonconstant Floer trajectories involved in u.
Proposition 3.9. Assume the moduli spaces under consideration have dimen-
sion zero. The bijective correspondence between elements uδ ∈ M(γp, γq;Hδ, J)
and [u] ∈ M(p, q;H, {fγ}, J) changes signs as follows:
(i) If m ≥ 1 we have
ǫ(u) = (−1)m−1ǫ(uδ);
(ii) If m = 0 we have u = uδ and ǫ(u) = ǫ(uδ), p is the minimum and q is the
maximum of the same function fγ , the moduli space M(p, q;H, {fγ}, J)
consists of the two gradient lines of fγ running from p to q, and their signs
are different if and only if the underlying orbit γ is good.
This result has two pleasant consequences. On the one hand we can construct
a “Morse-Bott” chain complex which computes symplectic homology by count-
ing with suitable signs rigid elements in the moduli spaces M(p, q;H, {fγ}, J).
On the other hand, this chain complex singles out algebraically the good orbits
and can be used to relate the symplectic homology of a manifold (W,ω) with
contact type boundary to the linearized contact homology of its boundary – the
latter being defined by a chain complex involving only good orbits. As already
mentioned at the beginning of this section, this is achieved in [3].
Acknowledgements. A.O. has benefited from a Swiss National Fund grant
under the supervision of Prof. Dietmar Salamon at ETH Zürich. Both authors
acknowledge financial support from the Fonds National de la Recherche Scien-
tifique, Belgium, the Forschungsinstitut für Mathematik, Zürich, the Institut de
Recherche Mathématique Avancée at Université Louis Pasteur, Strasbourg, as
well as from the Mathematisches Forschungsinstitut, Oberwolfach. The authors
would like to thank an anonymous referee for having carefully read through the
arguments in the paper and for having kindly suggested a solution to a gap in
the original proof of Proposition 4.9.
Symplectic homology for autonomous Hamiltonians 6
2 Symplectic homology
We define in this section the symplectic homology groups of a symplectically
aspherical manifold with contact type boundary. Our construction is modelled
on those of Cieliebak, Floer, Hofer and Viterbo [6, 7, 13, 28]. We consider
nontrivial homotopy classes of loops and we use the Novikov ring.
Let (W,ω) be a compact symplectic manifold with contact type boundary
M := ∂W . This means that there exists a vector field X defined in a neigh-
bourhood of M , transverse and pointing outwards along M , and such that
LXω = ω.
Such an X is called a Liouville vector field. The 1-form λ := (ιXω)|M is a
contact form on M . We denote by ξ the contact distribution defined by λ. The
Reeb vector field Rλ is uniquely defined by the conditions ker ω|M = 〈Rλ〉
and λ(Rλ) = 1. We denote by φλ the flow of Rλ. The action spectrum of
(M,λ) is defined by
Spec(M,λ) := {T ∈ R+ | there is a closed Rλ-orbit of period T }.
We assume throughout this paper the condition
f∗ω = 0 for all smooth f : T 2 →W. (1)
This guarantees that the energy of a Floer trajectory does not depend on its
homology class, but only on its endpoints (see below). Condition (1) plays an
important role in the Morse-Bott description of the symplectic homology groups.
Our main class of examples is provided by exact symplectic forms.
Let φ be the flow of X . We parametrize a neighbourhood U of M by
G :M × [−δ, 0] → U, (p, t) 7→ φt(p).
Then d(etλ) is a symplectic form on M ×R+ and G satisfies G∗ω = d(etλ). We
denote
Ŵ :=W
M × R+
and endow it with the symplectic form
ω̂ :=
ω, on W,
d(etλ), on M × R+.
Given a time-dependent HamiltonianH : S1×Ŵ → R, we define theHamil-
tonian vector field XθH by
ω̂(XθH , ·) = dHθ, θ ∈ S1 = R/Z,
where Hθ := H(θ, ·). We denote by φH the flow of XθH , defined by φ0H = Id and
φθH(x) = X
H(x)), θ ∈ R.
Symplectic homology for autonomous Hamiltonians 7
Let H be the set of admissible Hamiltonians, consisting of functions
H : S1 × Ŵ → R which satisfy
(i) H < 0 on W ;
(ii) H(θ, p, t) = αet + β for t large enough, with α /∈ Spec(M,λ);
(iii) every 1-periodic orbit γ : S1 → Ŵ of XθH is nondegenerate, i.e.
1l− dφ1H(γ(0))
6= 0.
We denote by P(H) the set of 1-periodic orbits of XθH and by Pa(H) the set of
1-periodic orbits in a given free homotopy class a in Ŵ .
Let J denote the set of admissible almost complex structures
J : S1 → End(TŴ ), J2 = −1l
which are compatible with ω̂ and have the following standard form for t large
enough: {
J(p,t)|ξ = J0,
J(p,t)
= Rλ.
Here J0 is any compatible complex structure on the symplectic bundle (ξ, dλ)
which is independent of θ and t.
Let us fix a reference loop la : S
1 → Ŵ for each free homotopy class a in
Ŵ such that [la] = a. If a is the trivial homotopy class we choose la to be a
constant loop. Recall that free homotopy classes of loops in Ŵ are in one-to-one
correspondence with conjugacy classes in π1(Ŵ ). As a consequence, the inverse
a−1 of a free homotopy class is well-defined. We require that la−1 coincides with
the loop la with the opposite orientation.
The Hamiltonian action functional acts on pairs (γ, [σ]) consisting of a
loop γ ∈ C∞(S1, Ŵ ) and the homology class (rel boundary) of a map σ : Σ → Ŵ
defined on a Riemann surface Σ with two boundary components ∂0Σ (with the
opposite boundary orientation) and ∂1Σ (with the boundary orientation), which
satisfies
σ|∂0Σ = l[γ], σ|∂1Σ = γ. (3)
Its values are defined by
AH(γ, [σ]) := −
σ∗ω̂ −
H(θ, γ(θ)) dθ. (4)
The differential dAH(γ, [σ]) : C∞(S1, γ∗TŴ ) → R is given by
dAH(γ, [σ])ζ :=
ω̂(γ̇ −XθH(γ), ζ) dθ.
Therefore the critical points of AH are pairs (γ, [σ]) such that γ ∈ P(H). We
fix from now on, for each γ ∈ P(H), a map σγ satisfying (3); then the set of all
pairs (γ, [σ]) can be identified with H2(W ;Z) for fixed γ.
Symplectic homology for autonomous Hamiltonians 8
Let us choose a symplectic trivialization
Φa : S
1 × R2n → l∗aTŴ
for each free homotopy class a in Ŵ . If a is the trivial homotopy class we
choose the trivialization to be constant. Moreover, we require that Φa−1(θ, ·) =
Φa(−θ, ·), θ ∈ S1 = R/Z. For each γ ∈ P(H) there exists a unique (up to
homotopy) trivialization
Φγ : Σ× R2n → σ∗γTŴ
such that Φγ = Φ[γ] on ∂0Σ× R2n. Let
Ψ : [0, 1] → Sp(2n), Ψ(θ) := Φ−1γ ◦ dφθH(γ(0)) ◦ Φγ . (5)
Because γ is nondegenerate we can define the Conley-Zehnder index µ(γ) by
µ(γ) := µ(γ, σγ) := −µCZ(Ψ), (6)
where µCZ(Ψ) is the Conley-Zehnder index of a path of symplectic matrices [23].
Remark 2.1. If, in the previous construction, we replace σγ with σγ#A for
some A ∈ H2(W ;Z), then the resulting index will be
µ(γ, σγ#A) = µ(γ, σγ)− 2〈c1(TW ), A〉. (7)
We define the Novikov ring Λω as the set of formal linear combinations
A∈H2(W ;Z)
A, λA ∈ Z such that
# {A |λA 6= 0, ω(A) ≤ c} <∞
for all c > 0. The multiplication in Λω is given by
λ ∗ λ′ :=
A,B∈H2(W ;Z)
We note that, if ω is exact, then Λω = Z[H2(W ;Z)]. We define a grading on
Λω by |eA| := −2〈c1(TW ), A〉. For each free homotopy class a in Ŵ and each
admissible Hamiltonian H we define the symplectic chain group SCa∗ (H) as
the free Λω-module generated by elements γ ∈ Pa(H). The grading is given by
|eAγ| := µ(γ)− 2〈c1(TW ), A〉.
We define the space of Floer trajectories M̂A(γ, γ;H, J) as the set of solu-
tions u : R× S1 → Ŵ of the equation
∂su+ Jθ(∂θu−XθH) = 0, (8)
such that
u(s, θ) = γ(θ), lim
u(s, θ) = γ(θ), lim
∂su = 0 (9)
uniformly in θ and
[σγ#u] = [σγ#A]. (10)
Symplectic homology for autonomous Hamiltonians 9
Remark 2.2. Under the nondegeneracy assumption on γ, γ condition (9) is
equivalent to the finiteness of the energy
E(u) := EJ,H(u) :=
|∂su|2θ + |∂tu−XθH |2θ
dsdθ. (11)
Because γ, γ are nondegenerate the linearized operator Du : W
1,p(R ×
S1, u∗TŴ ) → Lp(R× S1, u∗TŴ ), p > 2 given by
Duζ := ∇sζ + Jθ∇θζ + (∇ζJθ)∂θu−∇ζ
, u ∈ M̂A(γ, γ;H, J) (12)
is Fredholm with index
ind(Du) = µ(γ)− µ(γ) + 2〈c1(TW ), A〉. (13)
An almost complex structure J ∈ J is called regular for u ∈ M̂A(γ, γ;H, J) if
Du is surjective, and it is called regular if Du is surjective for all γ, γ ∈ P(H),
A ∈ H2(W ;Z) and u ∈ M̂A(γ, γ;H, J). It is proved in [15] that the space
Jreg(H) of regular almost complex structures is of the second category in J . For
every J ∈ Jreg(H) the space M̂A(γ, γ;H, J) is a smooth manifold of dimension
µ(γ)− µ(γ) + 2〈c1(TW ), A〉. From now on we fix some J ∈ Jreg(H).
If γ 6= γ or A 6= 0, the additive group R acts freely on M̂A(γ, γ;H, J) by
s0 · u(·, ·) := u(s0 + ·, ·). We define the moduli space of Floer trajectories
MA(γ, γ;H, J) := M̂A(γ, γ;H, J)/R.
Its dimension is
dim MA(γ, γ;H, J) := µ(γ)− µ(γ) + 2〈c1(TW ), A〉 − 1.
If γ = γ and A = 0, the space M̂0(γ, γ;H, J) consists of a single point, cor-
reponding to a constant solution (i.e. independent of s). The R action is then
trivial and we define the moduli space by M0(γ, γ;H, J) := M̂0(γ, γ;H, J).
A straightforward application of the maximum principle [28] using the special
form of admissible Hamiltonians for large t shows that all solutions of equa-
tions (8) and (9) are contained in a compact set. Moreover, by condition (1),
there are no J-holomorphic spheres that can bubble off. Therefore the mod-
uli space MA(γ, γ;H, J) can be compactified [12] to a space MA(γ, γ;H, J)
consisting of all tuples
([uk], [uk−1], . . . , [u1]), [ui] ∈ MAi(γi, γi;H, J)
such that γ
= γ, γi = γi+1, γk = γ and
iAi = A. We call such a tuple
([uk], [uk−1], . . . , [u1]) a broken trajectory of level k. The topology of the
compactified moduli space is described by the following notion of convergence:
a sequence [uν ] ∈ MA(γ, γ;H, J) is said to converge to the broken trajectory
Symplectic homology for autonomous Hamiltonians 10
γ = γ
γ1 = γ2
γ2 = γ3
γ = γ3
Figure 1: Broken trajectory.
([uk], [uk−1], . . . , [u1]) if there exist sequences s
i ∈ R, 1 ≤ i ≤ k such that sνi ·uν
converges uniformly on compact sets to ui.
If the space M̂A(γ, γ;H, J) is nonempty then its dimension is strictly positive
due to the action of R. In this case, the broken trajectories involved in the
compactification have level at most dim M̂A(γ, γ;H, J). In particular, when
µ(γ) − µ(γ) + 2〈c1(TW ), A〉 = 1 the moduli space MA(γ, γ;H, J) is compact
and consists of a finite number of points. In this situation one can associate a
sign ǫ(u) to each element [u] of this moduli space [13] (see also Section 4.4). We
define the Floer differential
∂ : SCa∗ (H) → SCa∗−1(H)
∂γ :=
µ(γ)−µ(γ)+2〈c1(TW ),A〉=1
[u]∈MA(γ,γ;H,J)
ǫ(u)eAγ. (14)
According to Floer [12] we have ∂2 = 0. We define the symplectic homol-
ogy groups of the pair (H, J) by
SHa∗ (H, J) := H∗(SC
∗ (H), ∂).
Remark 2.3. In view of condition (1) the Novikov ring Λω can be replaced by
Z[H2(W ;Z)], or even by Z at the price of losing the grading. Indeed, the energy
of a Floer trajectory depends only on its endpoints, hence the moduli spaces
M(γ, γ;H, J) :=
(γ, γ;H, J) are compact. Therefore the sum (14) in-
volves only a finite number of classes A.
By a standard argument [12] the groups SHa∗ (H, J) do not depend on J ∈
Jreg(H). Nevertheless, they do depend onH and, in order to obtain an invariant
of (W,ω), we need an additional algebraic limit construction. We define an
Symplectic homology for autonomous Hamiltonians 11
admissible homotopy of Hamiltonians as a map H : R×S1×Ŵ → R with
the following properties:
(i) H(s, ·, ·) = H− ∈ H for s ≤ −1, H(s, ·, ·) = H+ ∈ H for s ≥ 1;
(ii) H < 0 on W and there exist t0 ≥ 0 and functions α, β : R → R such that,
for all t ≥ t0, we have
H(s, t, p) = α(s)et + β(s);
(iii) ∂sH ≥ 0.
An admissible homotopy of almost complex structures is a map J : R →
J such that J(s) = J− for s ≤ −1 and J(s) = J+ for s ≥ 1. Given an admissible
homotopy of Hamiltonians one defines regular admissible homotopies of almost
complex structures in the usual way, by linearizing the equation
∂su(s, θ) + J(s, θ, u(s, θ))(∂θu(s, θ)−XθH(s, θ, u(s, θ))) = 0, (15)
subject to the limit conditions
u(s, ·) = γ ∈ P(H−), lim
u(s, ·) = γ ∈ P(H+). (16)
Regular admissible homotopies of almost complex structures form again a set
of the second category in the space of admissible homotopies and the rigid
behaviour of H for t ≥ t0, together with the condition ∂sH ≥ 0, ensures again
that solutions of (15) and (16) stay in a compact set (see [22]). The usual count
of solutions of (15) and (16) induces the monotonicity morphism
σ : SHa∗ (H−) → SHa∗ (H+), (17)
which does not depend on the choice of admissible homotopy connecting H−
and H+. These morphisms form a direct system on the set {SHa∗ (H), H ∈ H}
and we define the symplectic homology groups of (W,ω) by
SHa∗ (W,ω) := lim→
SHa∗ (H).
According to [6, Lemma 3.7] and [28, Theorem 1.7] these groups do not depend
on the choice of the Liouville vector field X .
3 The Morse-Bott chain complex
In this section we apply the Morse-Bott formalism of [1] to the case of Hamil-
tonians H : Ŵ → R having circles of 1-periodic orbits.
We denote by Pλ the set of closed unparametrized Rλ-orbits in M . For each
free homotopy class of loops b in M we denote by Pbλ the set of all γ ∈ Pλ in the
homotopy class b. The inclusion i : M →֒ W induces a map (still denoted by
Symplectic homology for autonomous Hamiltonians 12
i) between the sets of free homotopy classes of loops in M and W respectively.
For each free homotopy class a in W we denote
−1(a)
b∈i−1(a)
We assume in this section that the closed Reeb orbits on M are transversally
nondegenerate in M . This means that, for every orbit γ of period T > 0, we
1l− dφTλ (γ(0))|ξ
6= 0.
This can always be achieved by an arbitrarily small perturbation of λ or, equiv-
alently, of X , and such perturbations do not change the symplectic homology
groups. If all orbits γ ∈ Pλ are transversally nondegenerate one can assign to
each of them a Conley-Zehnder index µCZ(γ) according to the following recipe.
We fix a reference loop lb : S
1 → M for each free homotopy class b in M
such that [lb] = b. If b is the trivial homotopy class we choose lb to be a constant
loop and we require that lb−1 coincides with lb with the opposite orientation.
We define symplectic trivializations
Φb : S
1 × R2n−2 → l∗bξ
as follows. For each class b we choose a homotopy hab : S
1 × [0, 1] → W from
la, a = i(b) to lb such that
ha−1b−1(τ, ·) = hab(−τ, ·). (18)
We extend the trivialization Φa : S
1 × R2n → l∗aTŴ over the homotopy hab to
get a trivialization Φ′b : S
1 × R2n → l∗bTŴ . This trivialization is homotopic to
another one, still denoted Φ′b, such that
Φ′b(S
1 × R2n−2 × {0} × {0}) = l∗bξ,
Φ′b(S
1 × {0} × R× {0}) = l∗b 〈
〉, (19)
Φ′b(S
1 × {0} × {0} × R) = l∗b 〈Rλ〉.
We define Φb := Φ
b|S1×R2n−2×{0}×{0}. If b is the trivial homotopy class we
choose hab to be a path of constant loops, so that Φb is constant.
We fix for each γ ∈ Pλ a map σγ : Σ → M , where Σ is a Riemann surface
with two boundary components ∂0Σ (with the opposite boundary orientation)
and ∂1Σ (with the boundary orientation), satisfying
σ|∂0Σ = l[γ], σ|∂1Σ = γ. (20)
For each γ ∈ Pλ there exists a unique (up to homotopy) trivialization
Φγ : Σ× R2n−2 → σ∗γξ
Symplectic homology for autonomous Hamiltonians 13
such that Φγ = Φ[γ] on ∂0Σ× R2n−2. Let
Ψ : [0, T ] → Sp(2n− 2), Ψ(τ) := Φ−1γ ◦ dφτλ(p) ◦ Φγ , p ∈ im γ. (21)
Because γ is nondegenerate we can define the Conley-Zehnder index µ(γ) by
µ(γ) := µ(γ, σγ) := µCZ(Ψ), (22)
where µCZ(Ψ) is the Conley-Zehnder index of a path of symplectic matrices [23].
Remark 3.1. If, in the previous construction, we replace σγ with σγ#A for
some A ∈ H2(M ;Z), then the resulting index will be
µ(γ, σγ#A) = µ(γ, σγ) + 2〈c1(ξ), A〉. (23)
Note that c1(ξ) = i
∗c1(TW ) because i
∗TW = ξ⊕〈 ∂
, Rλ〉. Moreover, the parity
of µ(γ) is well-defined independently of the trivialization of ξ along γ.
Remark 3.2. For each simple orbit γ ∈ Pλ we denote by γk, k ∈ Z+ its positive
iterates. The parity of the Conley-Zehnder index of all the odd, respectively even
iterates is the same. If these two parities differ we say that all even iterates γ2k,
k ∈ Z+ are bad orbits. It is proved in [27, Lemma 3.2.4] that the even iterates
of a simple orbit γ of period T are bad if and only if dφTλ (p)|ξ, p ∈ im γ has
an odd number of real negative eigenvalues strictly smaller than −1 (see also
Lemma 4.25). The orbits in Pλ which are not bad are called good orbits.
We define a new class H′ of admissible Hamiltonians consisting of elements
H : Ŵ → R such that
(i) H |W is a C2-small Morse function and H < 0 on W ;
(ii) H(p, t) = h(t) outside W , where h(t) is a strictly increasing function with
h(t) = αet + β, α, β ∈ R, α /∈ Spec(M,λ) for t bigger than some t0, and
such that h′′ − h′ > 0 on [0, t0[.
Note that the 1-periodic orbits of XH in W are constant and nondegenerate
by assumption (i). A direct computation shows that
Xh(p, t) = −e−th′(t)Rλ. (24)
The 1-periodic orbits of XH fall in two classes:
(1) critical points of H in W ;
(2) nonconstant 1-periodic orbits of Xh, located on levels M × {t}, t ∈]0, t0[,
which are in one-to-one correspondence with closed −Rλ-orbits of period
e−th′(t).
Symplectic homology for autonomous Hamiltonians 14
Recall that, for every critical point p̃ ∈ Crit(H), the corresponding constant
XH-orbit γep has Conley-Zehnder index
µ(γep) = ind(p̃;−H)− n, n =
dim W,
where ind(p̃;−H) is the Morse index of p̃ with respect to −H [25, Lemma 7.2].
Let α := limt→∞ e
−tH(p, t). We denote by P≤αλ the set of all γ ∈ Pλ such
γ∗λ ≤ α. BecauseH is independent of θ, every orbit γ ∈ P≤αλ gives rise to
a whole circle of nonconstant 1-periodic orbits γH of XH . We denote by Sγ the
set of such orbits and identify Sγ with its image under the natural embedding
Sγ → Ŵ given by γH 7→ γH(0). Note that all elements of Sγ differ by a shift in
the parametrization, and that the γH are noninjective if their minimal period
is smaller than 1.
Lemma 3.3. Let H ∈ H′. Every nonconstant 1-periodic orbit γH of H is
transversally nondegenerate in Ŵ .
Proof. We have to show that the only eigenvector of dφ1H(γH(0)) corresponding
to the eigenvalue 1 is γ̇H(0). To this effect we note that ξ is an invariant space
and that
dφ1H(γH(0))|ξ =
e−th′(t)
(γH(0))|ξ.
Because we have assumed that all Rλ-orbits are transversally nondegenerate in
M , it follows that dφ1H(γH(0))|ξ has no eigenvalue equal to one. On the other
hand we have
dφ1H(γH(0))
− e−t(h′′ − h′)Rλ.
The conclusion follows because h′′(t)− h′(t) > 0.
For each γ ∈ P≤αλ we choose a Morse function fγ : Sγ → R with exactly one
maximum M and one minimum m. We fix from now on an element H ∈ H′
and, for each γ ∈ Pλ corresponding to a nonconstant γH ∈ P(H), we denote
by ℓγ ∈ Z+ the maximal positive integer such that γH(θ + 1ℓγ ) = γH(θ) for
all θ ∈ S1. We choose a symplectic trivialization ψ := (ψ1, ψ2) : Uγ
V ⊂ S1 × R2n−1 between open neighbourhoods Uγ ⊂ Ŵ of γH(S1) and V of
S1 × {0}, such that ψ1(γ(θ)) = ℓγθ. Here S1 × R2n−1 is endowed with the
symplectic form ω0 :=
i=1 dqi ∧ dpi, q1 ∈ S1, (p1, q2, p2, . . . , qn, pn) ∈ R2n−1.
Let ρ : S1 × R2n−1 → [0, 1] be a smooth cutoff function supported in a small
neighbourhood of S1 × {0} such that ρ|S1×{0} ≡ 1. For δ > 0 and (θ, p, t) ∈
S1 × Uγ we define
Hδ(θ, p, t) := h(t) + δρ(ψ(p, t))fγ(ψ1(p, t)− ℓγθ). (25)
The Hamiltonian Hδ coincides with H outside the open sets S
1 × Uγ . This is
precisely the perturbation described in [8, Proposition 2.2]. It is shown therein
that, for δ sufficiently small, the set P(Hδ) consists of the following elements:
Symplectic homology for autonomous Hamiltonians 15
(1) constant orbits, which are the same as those of H ;
(2) nonconstant orbits, which are nondegenerate and form pairs (γmin , γMax ),
where γ ∈ P≤αλ and γmin , γMax coincide with the orbits in Sγ starting at
the minimum and the maximum of fγ respectively.
Lemma 3.4. The periodic orbits γmin , γMax ∈ P(Hδ) satisfy
µ(γmin) = µ(γ) + 1, µ(γMax ) = µ(γ). (26)
Proof. We denote by γH the 1-periodic orbit of XH corresponding to γ ∈ P≤αλ .
We define the Robbin-Salamon index of γH by
µRS(γH) := µ(Ψ),
where Ψ : [0, 1] → Sp(2n) is given by (5) and µ(Ψ) is the Robbin-Salamon
index of an arbitrary path of symplectic matrices [23, §4]. It is shown in [8,
Proposition 2.2] that
− µ(γmin) = µRS(γH)−
, −µ(γMax ) = µRS(γH) +
. (27)
Note that γH has the orientation of −Rλ. Define Ψ̃ : [0, 1] → Sp(2n) by
Ψ̃(θ) := Φ−1γ (−θ) ◦ dφ−θH (γ(0)) ◦Φγ(0),
where Φγ : R/Z × R2n → γ∗HTŴ is the trivialization involved in (5). Then
µRS(−γH) = −µRS(γH) = µ(Ψ̃).
Let Sp∗(2n) ⊂ Sp(2n) be the set of symplectic matrices with no eigenvalue
equal to 1 and recall that we have denoted by a free homotopy classes of loops
in W and by b free homotopy classes of loops in M . By our choice (18) and (19)
of trivializations of TŴ over the reference loops lb, b ∈ i−1(a) we deduce that
the path Ψ̃ is homotopic with endpoint in Sp∗(2n) to the path
[0, 1] → Sp(2n) : θ 7→ Ψλ (Tθ)⊕
Here T := e−t(h′′(t) − h′(t)) and Ψλ : [0, T ] → Sp(2n − 2) is defined by (21).
By the symplectic shear axiom for the Robbin-Salamon index [23, Theorem 4.1]
the index of the above path is µ(γ) + 1
. As a consequence
−µRS(γH) = µRS(−γH) = µ(γ) +
Together with (27) this yields the conclusion of the Lemma.
Let p ∈ Crit(fγ); then γp ∈ P(Hδ) for all δ ∈]0, δ0] if δ0 is small enough, and
Lemma 3.4 says that µ(γp) = µ(γ) + ind(p; fγ). If p̃ is a critical point of H in
Symplectic homology for autonomous Hamiltonians 16
W we denote by γ
∈ P(H) the corresponding constant orbit. Our goal is to
describe the boundary points as δ → 0 of
MA]0,δ0[(γp, γq;H, {fγ}, J) :=
0<δ<δ0
{δ} ×MA(γp, γq;Hδ, J), (28)
µ(γp)− µ(γq) + 2〈c1(TW ), A〉 = 1,
where
γ, γ ∈ P≤αλ , p ∈ Crit(fγ), q ∈ Crit(fγ), A ∈ H2(W ;Z), J ∈ J
γ ∈ P≤αλ , p ∈ Crit(fγ), q ∈ Crit(H), A ∈ H2(W ;Z), J ∈ J .
Our description is very similar to that of [1] within the setting of contact
homology. We fix J ∈ J , γ, γ ∈ P≤αλ and q̃ ∈ Crit(H). We define two Morse-
Bott spaces of Floer trajectories M̂A(Sγ , Sγ ;H, J) and M̂A(Sγ , q̃;H, J) as
follows.
For γ, γ ∈ P≤αλ we denote by M̂A(Sγ , Sγ ;H, J) the set of solutions u :
R× S1 → Ŵ of the Floer equation (8) subject to the asymptotic conditions
u(s, θ) = γH(θ), lim
u(s, θ) = γ
(θ), lim
∂su = 0 (29)
uniformly in θ, with
γH ∈ Sγ , γH ∈ Sγ (30)
[σγ#u] = [σγ#A].
It is implicit in the above definition that the orbits γH and γH may vary for
different elements of M̂A(Sγ , Sγ ;H, J).
For γ ∈ P≤αλ and q̃ ∈ Crit(H) we denote by M̂A(Sγ , q̃;H, J) the set of
solutions u : R× S1 → Ŵ of the Floer equation (8) subject to the asymptotic
conditions
u(s, θ) = γH(θ), lim
u(s, θ) = q̃, lim
∂su = 0 (31)
uniformly in θ, with
γH ∈ Sγ (32)
[σγ#u] = A.
Again, the orbit γH may vary for different elements of M̂A(Sγ , q̃;H, J).
Symplectic homology for autonomous Hamiltonians 17
If γ 6= γ or A 6= 0, the additive group R acts freely on M̂A(Sγ , Sγ ;H, J)
and M̂A(Sγ , q̃;H, J) by s0 · u(·, ·) := u(s0 + ·, ·). We define the Morse-Bott
moduli spaces of Floer trajectories by
MA(Sγ , Sγ ;H, J) := M̂A(Sγ , Sγ ;H, J)/R
MA(Sγ , q̃;H, J) := M̂A(Sγ , q̃;H, J)/R.
If γ = γ and A = 0, the space M̂0(Sγ , Sγ ;H, J) is diffeomorphic to Sγ , con-
sists of constant cylinders (i.e. independent of s) and the R action is trivial.
In this case, we define the Morse-Bott moduli spaces by M0(Sγ , Sγ ;H, J) :=
M̂0(Sγ , Sγ ;H, J). We have natural evaluation maps
ev : MA(Sγ , Sγ ;H, J) → Sγ , ev : MA(Sγ , Sγ ;H, J) → Sγ
ev : MA(Sγ , q̃;H, J) → Sγ
defined by
ev([u]) := lim
u(s, ·), ev([u]) := lim
u(s, ·).
In the statement of the next result we denote by J ′ the set of almost complex
structures J ∈ J which are independent of θ ∈ S1.
Proposition 3.5. (i) Given H ∈ H′, let J ′(H) ⊂ J ′ be the (nonempty and
open) set of almost complex structures J such that, for any x ∈ Ŵ located
on a simple 1-periodic orbit of XH , we have
[XH , JXH ](x) 6= 0 and [XH , JXH ](x) /∈ 〈XH , JXH〉. (33)
There exists a set of second category J ′reg(H) ⊂ J ′(H) consisting of almost
complex structures J that are regular for all u ∈ M̂A(Sγ , Sγ ;H, J) or u ∈
M̂A(Sγ , q̃;H, J) with γ or γ being a simple orbit, and such that Jξ = ξ,
= Rλ outside a fixed open neighbourhood of the nonconstant periodic
orbits of XH .
(ii) Given H ∈ H′, there exists a set of second category Jreg(H) ⊂ J con-
sisting of regular almost complex structures J which, outside a fixed open
neighbourhood of the nonconstant periodic orbits of XH , are independent
of θ and satisfy Jξ = ξ, J ∂
= Rλ.
In each of the previous cases the relevant moduli spaces MA(Sγ , Sγ ;H, J),
MA(Sγ , q̃;H, J) are smooth manifolds of dimension
dim MA(Sγ , Sγ ;H, J) = µ(γ)− µ(γ) + 2〈c1(TW ), A〉,
dim MA(Sγ , q̃;H, J) = µ(γ)− µ(γeq) + 2〈c1(TW ), A〉,
and the evaluation maps ev, ev are smooth.
Symplectic homology for autonomous Hamiltonians 18
The proof of this statement is given in Section 4. Unless the contrary is
explicitly mentioned, all statements in this section hold both for J ∈ Jreg(H)
or J ∈ J ′reg(H), provided one considers moduli spaces with at least one simple
asymptotic orbit in the latter case.
Let now J ∈ Jreg(H) and fix for each γ ∈ P≤αλ a metric on Sγ such that
Rλ has length one. Let Freg(H, J) be the set of regular Morse functions,
consisting of families {fγ}, γ ∈ P≤αλ of perfect Morse functions fγ : Sγ → R
such that all the maps ev are transverse to the unstable manifolds Wu(p),
p ∈ Crit(fγ), all the maps ev are transverse to the stable manifolds W s(p),
p ∈ Crit(fγ) and all pairs
(ev, ev) : MA(Sγ , Sγ ;H, J) → Sγ × Sγ ,
(ev, ev) : MA1(Sγ , Sγ1 ;H, J) ev ×ev MA2(Sγ1 , Sγ ;H, J) → Sγ × Sγ (34)
are transverse to productsWu(p)×W s(q), p ∈ Crit(fγ), q ∈ Crit(fγ). Here and
in the sequel the unstable and stable manifolds are understood with respect to
∇fγ . Denote by C∞p (Sγ ,R) the set of perfect Morse functions on Sγ .
Lemma 3.6. The set Freg(H, J) is of the second Baire category in the space∏
C∞p (Sγ ,R).
Proof. The first two transversality conditions on ev, ev are satisfied if and only
if the maximum of each function fγ is a regular value of all the evaluation maps
ev having Sγ as target space, and if the minimum of each fγ is a regular value
of all the evaluation maps ev mapping to Sγ . The third transversality condition
requires in addition that each pair (M,m) ∈ Sγ × Sγ , with M the maximum of
fγ and m the minimum of fγ , is a regular value of (ev, ev).
By Sard’s theorem the minimum and maximum of each fγ can be chosen
inside a set of second category in Sγ . The conclusion follows.
Let now J ∈ Jreg(H) and {fγ} ∈ Freg(H, J). For p ∈ Crit(fγ) we denote
the Morse index by
ind(p) := dim Wu(p;∇fγ).
Let γ, γ ∈ P≤αλ and p ∈ Crit(fγ), q ∈ Crit(fγ). For m ≥ 0 we denote by
MAm(p, q;H, {fγ}, J) (35)
the union for γ1, . . . , γm−1 ∈ P≤αλ and A1+ . . .+Am = A of the fibered products
Wu(p)×ev (MA1(Sγ , Sγ1)×R+)ϕfγ1◦ev×ev (M
A2(Sγ1 , Sγ2)×R+)ϕfγ2◦ev×ev
. . . ϕfγm−1 ◦ev
×ev MAm(Sγm−1 , Sγ)ev×W s(q),
Symplectic homology for autonomous Hamiltonians 19
with the convention γ0 = γ. This is well defined as a smooth manifold of
dimension
dim MAm(p, q;H, {fγ}, J)
= ind(p)− 1 + (dim MA1(Sγ , Sγ1) + 1)− 1
+(dim MA2(Sγ1 , Sγ2) + 1)− 1 + ...
+dim MAm(Sγm−1 , Sγ)− 1 + (1− ind(q))
= µ(γ) + ind(p)− µ(γ)− ind(q) + 2〈c1(TW ), A1 + ...+Am〉 − 1
= µ(γp)− µ(γq) + 2〈c1(TW ), A〉 − 1.
The last equality follows from Lemma 3.4. Note that MA0 (p, q;H, {fγ}, J) is
naturally a submanifold of MA(Sγ , Sγ ;H, J). We denote
MA(p, q;H, {fγ}, J) :=
MAm(p, q;H, {fγ}, J)
and we call this the moduli space of Morse-Bott broken trajectories,
whereasMAm(p, q;H, {fγ}, J) is called the moduli space of Morse-Bott bro-
ken trajectories with m sublevels (see also Definition 4.1 and Figure 4).
Similarly, given γ ∈ P≤αλ , p ∈ Crit(fγ), q̃ ∈ Crit(H), we define mod-
uli spaces of Morse-Bott broken trajectories MAm(p, q̃;H, {fγ}, J), m ≥ 0 and
MA(p, q̃;H, {fγ}, J) by replacing the last term MAm(Sγm−1 , Sγ)ev×W s(q) in
the definition (35) with MAm(Sγm−1 , q̃;H, J). This is again well defined as a
smooth manifold of dimension
dim MA(p, q̃;H, {fγ}, J) = µ(γp)− µ(γeq) + 2〈c1(TW ), A〉 − 1.
Again, MA0 (p, q̃;H, {fγ}, J) is naturally a submanifold of MA(Sγ , q̃;H, J).
The significance of the above moduli spaces of broken Morse-Bott trajec-
tories is explained by the following theorem, which describes the boundary of
]0,δ0[
(γp, γq;H, {fγ}, J) in (28) as δ → 0.
Theorem 3.7 (Correspondence Theorem). Let H ∈ H′ be fixed and let
α := limt→∞ e
−tH(p, t) be the maximal slope of H. Let J ∈ Jreg(H) and
{fγ} ∈ Freg(H, J). There exists
δ1 := δ1(H, J) ∈ ]0, δ0[
such that, for any
γ, γ ∈ P≤αλ , p ∈ Crit(fγ), q ∈ Crit(fγ),
γ ∈ P≤αλ , p ∈ Crit(fγ), q ∈ Crit(H),
and any A ∈ H2(W ;Z) with
µ(γp)− µ(γq) + 2〈c1(TW ), A〉 = 1,
the following hold:
Symplectic homology for autonomous Hamiltonians 20
(i) J is regular for MA(γp, γq;Hδ, J) for all δ ∈]0, δ1[;
(ii) the space MA
]0,δ1[
(γp, γq;H, {fγ}, J) is a 1-dimensional manifold having a
finite number of components which are graphs over ]0, δ1[, i.e. the natural
projection MA
]0,δ1[
(γp, γq;H, {fγ}, J) →]0, δ1[ is a submersion;
(iii) there is a bijective correspondence between points
[u] ∈ MA(p, q;H, {fγ}, J)
and connected components of MA]0,δ1[(γp, γq;H, {fγ}, J).
The proof of this statement, including a discussion of gluing and compactness
for Morse-Bott moduli spaces, is given in Section 4.
We assume in the remainder of this section that the conclusions of Theo-
rem 3.7 are satisfied. For each [u] ∈ MA(p, q;H, {fγ}, J) the sign ǫ(uδ) is con-
stant on the corresponding connected component C[u] for continuity reasons.
We define a sign ǭ(u) by
ǭ(u) := ǫ(uδ), δ ∈]0, δ1[, (δ, [uδ]) ∈ C[u]. (36)
We define the Morse-Bott chain groups by
BCa∗ (H) :=
i−1(a),≤α
Λω〈γmin , γMax 〉, a 6= 0, (37)
BC0∗(H) :=
ep∈Crit(H)
Λω〈p̃〉 ⊕
i−1(0),≤α
Λω〈γmin , γMax 〉. (38)
where α := limt→∞ e
−tH(p, t) and P i
−1(0),≤α
λ = P
λ ∩ P
i−1(0)
λ . The grading is
defined by
|eAp̃| := ind(p̃;−H)− n− 2〈c1(TW ), A〉,
|eAγmin | := µ(γ) + 1− 2〈c1(TW ), A〉,
|eAγMax | := µ(γ)− 2〈c1(TW ), A〉.
We define the Morse-Bott differential
∂ : BCa∗ (H) → BCa∗−1(H)
∂p̃ :=
eq∈Crit(H)
|ep|−|eq|=1
[u]∈M0(ep,eq;H,{fγ},J)
ǭ(u)q̃, (39)
∂γp :=
eq∈Crit(H)
|γp|−|e
eq|=1
[u]∈MA(γp,eq;H,{fγ},J)
ǭ(u)eAq̃ (40)
,q∈Crit(fγ)
|γp|−|e
[u]∈MA(γp,γ
;H,{fγ},J)
ǭ(u)eAγ
, p ∈ Crit(fγ).
Symplectic homology for autonomous Hamiltonians 21
The sums (39) and (40) clearly involve only periodic orbits in the same free
homotopy class as that of p̃ or γp respectively.
Remark 3.8. Since H is C2-small, the moduli spaces MA(p̃, q̃;Hδ, J), p̃, q̃ ∈
Crit(H) of expected dimension ind(p̃;−H)−ind(q̃;−H)+2〈c1(TW ), A)〉−1 = 0
are independent of δ and consist exclusively of gradient trajectories of H in
W [17, Theorem 6.1](see also [25, Theorem 7.3]). As a consequence, these
moduli spaces are empty whenever A 6= 0.
We have, following directly from the definitions, an obvious isomorphism of
free Λω-modules
SCa∗ (Hδ) ≃ BCa∗ (H), δ ∈]0, δ1[.
It follows now from Theorem 3.7 and the definition (36) of signs in the Morse-
Bott complex that the corresponding differentials, defined by (14) and (39-40),
also coincide. Here we use the fact that the Hamiltonian action functional
decreases along Floer trajectories, hence the differential (14) applied to elements
p̃ ∈ Crit(H) does not involve nonconstant elements of P(Hδ) and reduces to (39)
by Remark 3.8. As a consequence, we have
H∗(BC
∗ (H), ∂) = SH
∗ (Hδ, J).
We shall construct in Section 4.4 a system of coherent orientations on the
Morse-Bott moduli spaces
MA(Sγ , Sγ ;H, J), MA(Sγ , q̃;H, J)
whenever γ, γ ∈ P≤αλ are good orbits. This in turn determines signs ǫ(u) via an
orientation rule for fiber products (see (87)).
Proposition 3.9. Assume dim MA(p, q;H, {fγ}, J) = 0. The bijective corre-
spondence between elements [u] ∈ MAm(p, q;H, {fγ}, J), m ≥ 1 and elements of
[uδ] ∈ MA(γp, γq;Hδ, J) given by Theorem 3.7 changes the signs by the rule
ǫ(u) = (−1)m−1ǫ(uδ).
Moreover, if m = 0 then u = uδ and ǫ(u) = ǫ(uδ), p is a minimum and q
is a maximum, the moduli space MA0 (p, q;H, {fγ}, J) consists of the two gra-
dient lines running from p to q and their signs are different if and only if the
underlying orbit is good.
In view of (36) and (39–40), this identification of signs between ǫ(u) and ǭ(u)
allows to define the Morse-Bott differential exclusively in terms of Morse-Bott
data.
4 Morse-Bott moduli spaces
The structure of this section is as follows. We give in §4.1 the proof of Proposi-
tion 3.5, whereas Theorem 3.7 is proved in §4.2–§4.3, which treat compactness
and gluing and correspond to assertions (i-ii) and (iii) respectively. Finally §4.4
contains a full discussion of orientation issues and the proof of Proposition 3.9.
Symplectic homology for autonomous Hamiltonians 22
4.1 Transversality
Proof of Proposition 3.5. We first prove (ii). Let J ℓ ⊂ J be the space of
admissible almost complex structures of class Cℓ, ℓ ≥ 1, and let J ℓ(H) ⊂ J ℓ be
the set of almost complex structures J which, outside a fixed neighbourhood of
the nonconstant periodic orbits of XH , are independent of θ and satisfy Jξ = ξ,
= Rλ. By a standard trick of Taubes [15, Theorem 5.1] it is enough to show
that there exists an open and dense set J ℓreg(H) ⊂ J ℓ(H) consisting of regular
elements. We define the universal moduli spaces
MA(Sγ , Sγ ;H,J ℓ(H)) = {(u, J) | J ∈ J ℓ(H), u ∈ MA(Sγ , Sγ ;H, J)}
MA(Sγ , q̃;H,J ℓ(H)) = {(u, J) | J ∈ J ℓ(H), u ∈ MA(Sγ , q̃;H, J)}.
The main point is to show that these universal moduli spaces are Banach man-
ifolds. Then the sets J ℓreg(H) consist of the regular values of the natural pro-
jections from the universal moduli spaces to J ℓ(H). We only treat the case of
MA(Sγ , Sγ ;H,J ℓ(H)) since the second case is entirely similar, and we assume
without loss of generality that γ 6= γ. This universal moduli space is the zero
set of a distinguished section of a Banach vector bundle E → BA×J ℓ(H) which
we now define.
Let p > 2 and d > 0. Let BA = B1,p,d(Sγ , Sγ , A;H) be the space of proper
maps u : R× S1 → Ŵ which are locally in W 1,p and satisfy
(i) the map u converges uniformly in θ as s→ ±∞ to γ(·+ θ0), respectively
γ(· + θ0), for some θ0, θ0 ∈ S1, and represents the homology class A ∈
H2(W ;Z);
(ii) there exist tubular neighbourhoods U and U of γ and γ respectively, to-
gether with parametrizations Ψ : U → S1×R2n−1 and Ψ : U → S1×R2n−1
such that
Ψ ◦ γ(θ) = {θ} × {0}, Ψ ◦ γ(θ) = {θ} × {0},
Ψ ◦ γ(θ + θ0)−Ψ ◦ u(s, θ) ∈ W 1,p(]−∞,−s0], ed|s|ds dθ),
Ψ ◦ γ(θ + θ0)−Ψ ◦ u(s, θ) ∈ W 1,p([s0,∞[, ed|s|ds dθ),
for some s0 > 0 sufficiently large.
Then BA is a Banach manifold and, for d/p strictly smaller than the constant
r in Proposition A.1, it contains the moduli spaces MA(Sγ , Sγ ;H, J) for all
J ∈ J ℓ. Let E → BA ×J ℓ(H) be the Banach vector bundle with fiber E(u,J) =
Lp(R×S1, u∗TŴ ; ed|s|ds dθ). Let ∂̄H : BA×J ℓ(H) → E be the section defined
∂̄H(u, J) := ∂su+ Jθ(∂θu−XH). (41)
Symplectic homology for autonomous Hamiltonians 23
Then MA(Sγ , Sγ ;H,J ℓ(H)) = ∂̄−1H (0) and it remains to show that ∂̄H is
transverse to the zero section. This means that the vertical differential
D∂̄H(u, J) : TuBA × TJJ ℓ(H) → E(u,J)
is surjective for all (u, J) ∈ ∂̄−1H (0). We have
TuBA =W 1,p(R× S1, u∗TŴ ; ed|s|ds dθ)⊕ V ⊕ V ,
where V , V are the one-dimensional real vector spaces generated by two sections
of u∗TŴ of the form (1 − β(s, θ))XH(γ(θ)) and β(s, θ)XH(γ(θ)) respectively,
with β(s, θ) = β(s) a smooth cutoff function which vanishes for s ≤ 0 and is
equal to 1 for s ≥ 1. The space TJJ ℓ(H) consists of matrix valued functions
Y : S1 → End(TŴ ) of class Cℓ satisfying the conditions
JθYθ + YθJθ = 0, ω̂(Yθv, w) + ω̂(v, Yθw) = 0, ∀v, w ∈ TŴ , (42)
and such that, outside fixed neighbourhoods of the nonconstant periodic orbits
of XH , they are independent of θ and have the form
with respect
to the splitting ξ ⊕ Span(Rλ, ∂∂t ). The operator D∂̄H(u, J) can be written
D∂̄H(u, J) · (ζ, Y ) = Duζ + Yθ(u)(∂θu−XH(u)).
Du : W
1,p(R× S1, u∗TŴ ; ed|s|ds dθ)⊕ V ⊕ V → Lp(R× S1, u∗TŴ ; ed|s|ds dθ)
is the linearization of the Cauchy-Riemann operator associated to the pair (H, J)
and is explicitly given by formula (12). It is proved in [5, Proposition 4] that
Du is a Fredholm operator. It is at this point that the exponential weight
plays a crucial role, due to the degeneracy of the asymptotic orbits. As a
consequence the range of D∂̄H(u, J) is closed and we are left to prove that it
is also dense. Let q > 1 be such that 1/p + 1/q = 1. We show that every
η ∈ Lq(R× S1, u∗TŴ ; ed|s|ds dθ) satisfying
〈η,Duζ〉ed|s|ds dθ = 0,
〈η, Yθ(u)(∂θu−XH(u))〉ed|s|ds dθ = 0
for all ζ and Y vanishes. The first equation implies, by elliptic regularity, that η
is of class Cℓ and has the unique continuation property. Assume by contradiction
that η does not vanish. Then the set {(s, θ) : η(s, θ) 6= 0} is open and dense.
On the other hand, it is proved in [15, Theorem 4.3] that the set
R(u) := {(s, θ) : ∂su(s, θ) 6= 0, u(s, θ) 6= γ(θ), γ(θ), u(s, θ) /∈ u(R \ {s}, θ)}
of regular points of u is open and dense (although nondegeneracy of the asymp-
totic orbits is a standing assumption in [15], it does not play any role in the
Symplectic homology for autonomous Hamiltonians 24
proof of this result). Let z0 = (s0, θ0) be a point in R(u) with η(z0) 6= 0 and
u(z0) belonging to the fixed open neighbourhood of γ (such a point exists since
we have assumed γ 6= γ). One can choose a matrix Yθ0(u(z0)) satisfying (42)
such that
〈η(z0), Yθ0(u(z0))J(u(z0))∂su(z0)〉 6= 0.
Because z0 is a regular point we can choose a time-dependent cutoff function
ρ : S1 × Ŵ → [0, 1] supported near (θ0, u(z0)) such that Y := ρYθ0(u(z0))
satisfies ∫
〈η, Yθ(u)(∂θu−XH(u))〉ed|s|ds dθ 6= 0.
This contradicts (43) and shows thatD∂̄H(u, J) is surjective, hence the universal
moduli space MA(Sγ , Sγ ;H,J ℓ(H)) is a Banach manifold as claimed.
We now prove (i). The set J ′(H) is obviously open. The fact that it is
nonempty can be seen as follows. The space S1 ×R2 admits the “skating ring”
contact form α = sin θdx− cos θdy, (θ, x, y) ∈ S1×R2 for which ∂
∈ ξ = kerα.
If J denotes the almost complex structure on ξ satisfying J ∂
= cos θ ∂
sin θ ∂
, then [ ∂
, J ∂
] 6= 0 and [ ∂
, J ∂
] /∈ ξ = 〈 ∂
, J ∂
〉. This simple model
can be adapted to our situation as follows. We can symplectically trivialize a
neighbourhood of the simple orbit γ as S1 × R2n−1 ∋ (θ, t, q2, p2, . . . , qn, pn)
with the standard symplectic form dθ ∧ dt + dq2 ∧ dp2 + . . . + dqn ∧ dpn, so
that XH corresponds to
. Let J be a compatible almost complex structure
such that J ∂
+ cos θ ∂
+ sin θ ∂
. Since ∂
and ∂
commute we have
[XH , JXH ] = [
, cos θ ∂
+ sin θ ∂
] 6= 0 and [XH , JXH ] /∈ 〈XH , JXH〉, so
that J ∈ J ′(H).
Let J ′ℓ ⊂ J ′ be the space of admissible almost complex structures of class
Cℓ, ℓ ≥ 1 which are independent of θ ∈ S1, and let J ′ℓ(H) ⊂ J ′ℓ be the space
of almost complex structures J which, outside a fixed neighbourhood of the
nonconstant periodic orbits of XH , satisfy Jξ = ξ, J
= Rλ. It is enough
to show that there exists an open and dense set J ′ℓreg(H) ⊂ J ′ℓ(H) consisting
of elements which are regular for Floer trajectories with one nontrivial simple
asymptote.
We have J ′ℓ ⊂ J ℓ and the main point is to show that the corresponding
universal moduli spaces MA(Sγ , Sγ ;H,J ′ℓ(H)) ⊂ MA(Sγ , Sγ ;H,J ℓ(H)) and
MA(Sγ , q̃;H,J ′ℓ(H)) ⊂ MA(Sγ , q̃;H,J ℓ(H)) are Banach manifolds. We again
treat only MA(Sγ , Sγ ;H,J ′ℓ(H)) and assume without loss of generality that
γ is a simple orbit and γ 6= γ. This universal moduli space is the zero set of
the section of the restricted bundle E → BA × J ′ℓ(H) defined by (41), and we
have to show that the vertical differential D∂̄H(u, J) : TuBA × TJJ ′ℓ(H) →
E(u,J) is surjective. Arguing by contradiction, we get an element η ∈ Lq(R ×
S1, u∗TŴ ; ed|s|ds dθ) of class Cℓ which does not vanish on an open and dense
subset of R×S1 and satisfies
〈η, Y (u)(∂θu−XH(u))〉ed|s|ds dθ = 0 for any
Y ∈ TJJ ′ℓ(H). The main difference with respect to (ii) is that TJJ ′ℓ(H) ⊂
TJJ ℓ(H) consists of elements Y ∈ End(TŴ ) which are independent of θ.
Symplectic homology for autonomous Hamiltonians 25
I(u) :=
(s, θ) : ∂su(s, θ) 6= 0, u−1(u(s, θ)) = {(s, θ)}
be the set of injective points, and denote IR(u) := I(u)∩ ]R,∞[×S1, R > 0.
The main observation is that our special choice of J ∈ J ′(H) implies that
IR(u) is open and dense in ]R,∞[×S1 for R large enough. This is proved
exactly as in [15, §7], and the main steps are the following. Since γ is simple,
every u as above is simple, i.e. for every integer m > 1 there exists a point
(s, θ) ∈ R × S1 = R × R/Z such that u(s, θ + 1
) 6= u(s, θ). Let U be a
neighbourhood of γ in which [XH , JXH ] 6= 0 and [XH , JXH ] /∈ 〈XH , JXH〉. We
call a point (s, θ) regular if ∂su, ∂θu, XH(u), JXH(u) are linearly independent
at (s, θ), and we denote by R(u) the set of regular points. Then [15, Lemma 7.6]
holds and [15, Lemma 7.7] shows that the set {(s, θ) ∈ R(u) : u(s, θ) ∈ U} is
open and dense in u−1(U). Note that we crucially use here our hypothesis
J ∈ J ′(H), which plays the role of the hypothesis J ∈ Jad(M,ω,X) in [15].
Finally [15, Lemma 7.8] shows that the set of points which are regular and
injective is open and dense in u−1(U), and in particular IR(u) is open and
dense in ]R,∞[×S1 for R large enough.
We can then choose a point z0 = (s0, θ0) ∈ IR(u) such that η(z0) 6= 0 and
a matrix Y (u(z0)) satisfying (42) and 〈η(z0), Y (u(z0))J(u(z0))∂su(z0)〉 6= 0.
Since z0 is an injective point we can choose a cutoff function ρ : Ŵ → R
supported near u(z0) such that Y := ρY (u(z0)) satisfies
〈η, Y (u)(∂θu −
XH(u))〉ed|s|ds dθ 6= 0. This contradiction shows that D∂̄H(u, J) is surjective
and therefore MA(Sγ , Sγ ;H,J ′ℓ(H)) is a Banach manifold as claimed.
The dimension of the moduli space MA(Sγ , Sγ ;H, J), J ∈ Jreg(H) is equal
to ind(Du) − 1. The restriction of the operator Du to the subspace W 1,p(R ×
S1, u∗TŴ ; ed|s|ds dθ) is conjugated to a Cauchy-Riemann operator
Du :W 1,p(R× S1, u∗TŴ ; ds dθ) → Lp(R× S1, u∗TŴ ; ds dθ)
via multiplication by e
|s|. If the asymptotics of Du were nondegenerate, the
Fredholm index of Du would be given by [26]
µRS(γ)− µRS(γ) + 2〈c1(TW ), A〉.
Due to the one-dimensional degeneracy of γ and γ, the actual index of Du is
obtained by a calculation analogous to [5, Proposition 4] (see also Lemma 3.4) :
(µRS(γ)−
)− (µRS(γ) +
) + 2〈c1(TW ), A〉. (44)
We have proved in Lemma 3.4 that µRS(γ) = µ(γ) +
, hence
ind(Du) = ind(Du) + 2 = µ(γ)− µ(γ) + 2〈c1(TW ), A〉+ 1.
Finally note that the evaluation maps ev, ev are well-defined and smooth on
BA. Hence their restrictions to the moduli spaces are smooth as well.
Symplectic homology for autonomous Hamiltonians 26
4.2 Compactness for Morse-Bott trajectories
Definition 4.1. Let H, {fγ} and J be fixed as above, and let p ∈ Crit(fγ),
q ∈ Crit(fγ). The space M̂A(p, q;H, {fγ}, J) of parametrized Morse-Bott
broken trajectories consists of tuples
u = (cm, um, cm−1, um−1, . . . , u1, c0)
such that
(i) ui ∈ M̂Ai(Sγi , Sγi−1 ;H, J), i = 1, . . . ,m with γm = γ, γ0 = γ and A1 +
. . .+Am = A;
(ii) c0 : [−1,+∞[→ Sγ0 , ci : [−Ti/2, Ti/2] → Sγi , i = 1, . . . ,m − 1 and
cm :]−∞, 1] → Sγm satisfy ċi = ∇fγi ◦ ci, i = 0, . . . ,m;
(iii) ev(ui) = ev(ci), ev(ui) = ev(ci−1), i = 1, . . . ,m and c0(+∞) = q,
cm(−∞) = p.
The space MA(p, q;H, {fγ}, J) of unparametrized Morse-Bott broken tra-
jectories consists of equivalence classes
[u] = (cm, [um], cm−1, [um−1], . . . , [u1], c0)
such that u ∈ M̂A(p, q;H, {fγ}, J).
Definition 4.2. Let
uk = (cmk,k, umk,k, cmk−1,k, . . . , u1,k, c0,k) ∈ M̂A(pk, qk;H, {fγ}, J)
with k = 1, . . . , ℓ, and satisfying qk = pk−1 for k = 2, . . . , ℓ. We denote p := pℓ,
q := q1. A sequence vn ∈ M̂A(γp, γq;Hδn , J) with δn → 0, n → ∞ is said to
converge to u := (uℓ, . . . ,u1) if there exist shifts (s
i,k) ∈ R, i = 1, . . . ,mk such
vn(·+ sni,k , ·) → ui,k, n→ ∞
uniformly on compact sets in R× S1. We write in this case vn → u.
A sequence [ṽn] ∈ MA(γp, γq;Hδn , J) with δn → 0, n → ∞ is said to
converge to [u] ∈ MA(p, q;H, {fγ}, J) if there exist representatives vn and
v such that vn → v (this condition is obviously independent on the choice of
representatives). We write in this case [ṽn] → [u].
We call u a broken Floer trajectory with gradient fragments. We
call each of the uk’s a Floer trajectory with gradient fragments. Each uk
is a level of u and each ui,k is a sublevel of uk.
Definition 4.3. An element
u = (cm, um, cm−1, . . . , u1, c0) ∈ M̂A(p, q;H, {fγ}, J)
with m ≥ 1 is stable if each ui, i = 1, . . . ,m is a nonconstant Floer trajectory
and if each ci, i = 1, . . . ,m − 1 defined on an interval of nonzero length is
Symplectic homology for autonomous Hamiltonians 27
nonconstant. An element u = (c0) ∈ MA(p, q;H, {fγ}, J) is stable if p 6= q.
A broken Floer trajectory with gradient fragments u = (uℓ, . . . ,u1) is stable if
each uk, k = 1, . . . , ℓ is stable.
Remark 4.4. A convergent sequence vn of nonconstant Floer trajectories has
a stable limit u which is unique up to shifts on the ci,k and ui,k.
The proofs of the next two lemmas use the asymptotic estimates proved
in the Appendix. The relevant notation is introduced at the beginning of the
Appendix, and we briefly recall it here for the reader’s convenience. For each
γ ∈ P(H) we choose coordinates (ϑ, z) ∈ S1 × R2n−1 parametrizing a tubular
neighbourhood of γ, such that ϑ ◦ γ(θ) = θ and z ◦ γ(θ) = 0. Given a smooth
function fγ : Sγ → R, we denote by ϕ
s the gradient flow of fγ with respect to
the natural metric on S1.
In a neighbourhood of γ ∈ P(H) the Floer equation ∂su+ J∂θu− JXH = 0
becomes ∂sZ+J∂θZ+J
−JXH = 0, where Z(s, θ) := (ϑ◦u(s, θ)−θ, z◦u(s, θ)).
Since XH =
on {z = 0} this can be rewritten as ∂sZ + J∂θZ + Sz = 0 for
some matrix-valued function S = S(ϑ, z). The matrix S∞(θ) := S(θ, 0) is
symmetric. Let A∞ : H
k(S1,R2n) → Hk−1(S1,R2n) be the operator defined by
A∞Z := J
Z + S∞(θ)z. The kernel of A∞ has dimension one and is spanned
by the constant vector e1 := (1, 0, . . . , 0). We denote by Q∞ the orthogonal
projection onto (ker A∞)
⊥ and we set P∞ := 1l−Q∞.
Lemma 4.5. Let vn ∈ M̂A(γp, γq;Hδn , J) with δn → 0, n→ ∞ and s
1 < s
shifts such that vn(·+ sn1 , ·) → u1, vn(·+ sn2 , ·) → u2 uniformly on compact sets,
with u1 ∈ M̂A1(Sγ1 , Sγ ;H, J) and u2 ∈ M̂A2(Sγ , Sγ2 ;H, J). Any two sequences
of shifts sn1 < s
+ < s
− < s
2 satisfying s
+ − sn1 → ∞, sn2 − sn− → ∞ and
+ − sn1 ) → 0, δn(sn2 − sn−) → 0, (45)
have the property that
vn(·+ sn+, ·) → ev(u1), vn(·+ sn−, ·) → ev(u2)
uniformly on compact sets.
Proof. We claim that there exists K > 0 such that vn([s
1 +K, s
2 −K] × S1)
is contained in a given small neighbourhood of Sγ . If that was not the case, we
could find a sequenceKn → ∞ and a sequence (sn, θn) ∈ [sn1 +Kn, sn2 −Kn]×S1
such that dist(vn(sn, θn), Sγ) is bounded away from zero. Up to a subsequence,
vn(· + sn, ·) converges to some Floer trajectory v which must be nonconstant.
On the other hand, for any s ∈ R and for any K > 0 we have, for n large
enough,
vn(s+ s
2 −K, ·)
≤AHδn
vn(s+ sn, ·)
≤AHδn
vn(s+ s
1 +K, ·)
and in the limit AH(u2(s − K, ·)) ≤ AH(v(s, ·)) ≤ AH(u1(s + K, ·)). We let
K go to infinity and obtain AH(γ) ≤ AH(v(s, ·)) ≤ AH(γ). This holds for all
Symplectic homology for autonomous Hamiltonians 28
s ∈ R and therefore the cylinder v is constant over some element of P(H), a
contradiction which proves the claim.
By (98) in the proof of Proposition A.3 applied to vn on [s
1 + K, s
2 −K]
we get
|Q∞vn(s, θ)| ≤ Cmax(‖Q∞vn(sn1 +K)‖, ‖Q∞vn(sn2 −K)‖). (46)
Let γ+ be the limit in Sγ of vn(s
+, ·), and let In(ǫ) := [sn+(ǫ), sn+] ⊂ [sn1 +
K, sn+] be the maximal subinterval containing s
+ such that P∞vn(s), s ∈ In(ǫ)
is at distance at least ǫ from the critical points of fγ , except maybe γ+ (if the
latter is a critical point). By the second part of Proposition A.3 applied to vn
on In(ǫ) we obtain
|ϑ ◦ vn(s, θ)− θ−ϕ
δs (θ0)| ≤ Cmax(‖Q∞vn(s
+(ǫ))‖, ‖Q∞vn(sn+)‖)eMδn(s
Since δn(s
+ − sn1 ) → 0 and taking into account (46) we get
|ϑ ◦ vn(s, θ)− ϑ ◦ γ+(θ)| ≤ C1 max(‖Q∞vn(sn1 +K)‖, ‖Q∞vn(sn2 −K)‖). (47)
For K large enough the right hand term becomes so small that the distance
between P∞vn(s), s ∈ In(ǫ) and the critical points of fγ , except possibly γ+, is
strictly bigger than ǫ, hence In(ǫ) = [s
1 +K, s
+] by maximality (this holds for
K large enough). Applying (47) to s = sn1 +K we obtain
|ϑ ◦ vn(sn1 +K, θ)− ϑ ◦ γ+(θ)| ≤ C1 max(‖Q∞vn(sn1 +K)‖, ‖Q∞vn(sn2 −K)‖).
Passing to the limit in the above inequality we obtain
|ϑ ◦ u1(K, θ)− ϑ ◦ γ+(θ)| ≤ C1 max(‖Q∞u1(K)‖, ‖Q∞u2(−K)‖).
Letting K → ∞ we obtain ev(u1) = γ+. That this implies uniform convergence
on compact sets to the constant cylinder over ev(u1) can be seen in two ways:
either one notices that the above estimates hold uniformly when sn+ is replaced
with sn++K+, whereK+ is a bounded constant, or one uses the fact that a Floer
trajectory passing through a periodic orbit is necessarily a constant cylinder, by
unique continuation applied to the infinite jet at that orbit [21, Theorem 2.3.2].
A similar argument proves the assertion involving ev(u2).
Lemma 4.6. Let vn ∈ M̂A(p, q;Hδn , J) with δn → 0, n→ ∞. Assume we are
given two sequences of shifts sn1 < s
2 such that vn(· + sni , ·), i = 1, 2 converge
uniformly on compact sets to constant cylinders uγi over orbits γi belonging to
the same family Sγ . Then there exists a (possibly broken) gradient trajectory
of fγ starting at γ1 and ending at γ2. Moreover, the length of the gradient
trajectory is T = limn→∞ δn(s
2 − sn1 ).
Proof. We claim that, for n large enough, vn([s
1 , s
2 ]×S1) is entirely contained
in an arbitrarily small neighbourhood of γ. By contradiction, if this fails we
can reparametrize the sequence vn so that it converges to a nonconstant Floer
Symplectic homology for autonomous Hamiltonians 29
trajectory v ∈ MB(γ′, γ′;H, J) for some class B ∈ H2(M ;Z) and some γ′, γ′ ∈
P(H) such that their actions satisfy AH(γ) ≤ AH(γ′) < AH(γ′) ≤ AH(γ),
which is impossible.
We first assume that γ1 is not a critical point of fγ . Let ǫ > 0 be fixed and
denote by In(ǫ) = [s
1 , s
2 (ǫ)] ⊂ [sn1 , sn2 ] the maximal subinterval containing sn1
such that the distance between P∞vn(s), s ∈ In(ǫ) and Crit(fγ) is at least ǫ.
We can apply Proposition A.3 to vn and In(ǫ). In particular, for some sequence
θn ∈ S1 we have
(s,θ)∈In(ǫ)×S1
|ϑ ◦ vn(s, θ)− θ − ϕδnfγs (θn)| = 0.
Since vn(s
1 , ·) converges to γ1, we also have
(s,θ)∈In(ǫ)×S1
|ϑ ◦ vn(s, θ)− θ − ϕ
δn(s−s
(γ1)| = 0. (48)
Modulo passing to a subsequence we know that vn(s
2 (ǫ), ·) converges, which
together with (48) implies that δn(s
2 (ǫ)−sn1 ) converges to T (ǫ) ∈ R+. This holds
for each ǫ > 0 and, since sn2 (ǫ) < s
′) if ǫ > ǫ′, the limit limǫ→0 T (ǫ) = T ∈ R+
exists. Then ϕ
s (γ1), s ∈ [0, T ] is a gradient trajectory starting at γ1.
If T is finite then this trajectory, and therefore vn(In(ǫ)×S1) stay at a fixed
distance from Crit(fγ) for n large enough. Hence In(ǫ) = In for ǫ sufficiently
small and we are done. If T is infinite and the limit lims→∞ ϕ
s (γ1) is equal
to γ2, we are also done. Otherwise we are in the next case, with shifts s̃
limǫ→0 s
2 (ǫ) and s̃
2 := s
We now assume that γ1 is a critical point of fγ and γ1 6= γ2. Given ǫ > 0 we
denote by In(ǫ) = [s
1 , s
2 (ǫ)] ⊂ [sn1 , sn2 ] the maximal subinterval containing sn1
such that the distance between P∞vn(s), s ∈ In(ǫ) and Crit(fγ)\{γ1} is at least
ǫ. For ǫ > 0 small enough the loops P∞vn(s
2 (ǫ)) are at a distance bigger than ǫ
from γ1 and, up to a subsequence, vn(s
2 (ǫ), ·) converges to some γ̃2 ∈ Sγ which
is not a critical point of fγ . The same argument as in the previous case applied
“backwards” to the shifts sn1 < s
2 (ǫ) produces a negative gradient trajectory
running from γ̃2 to some critical point γ̃1. By definition of In(ǫ), we must have
γ̃1 = γ1 and we thus obtain a gradient trajectory from γ1 to γ̃2. We are now in
the first case with shifts s̃n1 := s
2 (ǫ) and s̃
2 := s
We successively apply the above two cases in order to produce a broken gra-
dient trajectory from γ1 to γ2. This is a finite process since a broken trajectory
has a finite number of nonconstant fragments.
Proposition 4.7. Let vn ∈ M̂A(p, q;Hδn , J) with δn → 0, n → ∞. There
exists a broken Floer trajectory with gradient fragments u and a subsequence
(still denoted by vn) such that vn → u.
Proof. The energy E(vn) := EJ,Hδn (vn) defined in (11) satisfies
E(vn) = −
Hδn(θ, γp(θ)) dθ +
Hδn(θ, γq(θ)) dθ.
Symplectic homology for autonomous Hamiltonians 30
Since Hδn → H we infer that E(vn) is uniformly bounded.
Floer’s compactness theorem [12, Proposition 3c] applies to our situation and
provides a collection of Floer trajectories ui, i = 1, . . . ,m for the pair (H, J)
together with holomorphic spheres attached to them, as well as shifts (sni ) such
that vn(·+ sni , ·) converges to ui and its associated holomorphic spheres in the
sense of nodal curves. Condition (1) implies symplectic asphericity 〈ω, π2(Ŵ )〉 =
0, therefore holomorphic spheres in (Ŵ , J) are constant and the shifted vn
converge to ui uniformly on compact sets.
Because the action spectrum of ∂W was assumed to be discrete and injective
the trajectories ui connect with each other, in the sense that ev(ui) and ev(ui+1)
belong to the same family of trajectories Sγi , i = 1, . . . ,m−1. Moreover, ev(um)
belongs to Sγ and ev(u1) belongs to Sγ .
By Lemma 4.5 there exist shifts sni,± such that vn(· + sni,+, ·) converges to
the constant cylinder over ev(ui), and vn(· + sni,−, ·) converges to the constant
cylinder over ev(ui). Applying Lemma 4.6 with shifts s
i,+ < s
i−1,−, i = 2, . . . ,m
and n large enough, we obtain broken gradient trajectories ci−1 starting at
ev(ui) and ending at ev(ui−1). Let now s
+ be shifts such that vn(·+sn−, ·) →
p and vn(· + sn+, ·) → q. Applying Lemma 4.6 with shifts sn− < snm,− and with
shifts sn1,+ < s
+ we obtain broken gradient trajectories cm starting at p and
ending at ev(um) and c0 starting at ev(u1) and ending at q. Since all Sγ are
circles of periodic orbits, the broken gradient trajectories ci, i = 0, . . . ,m consist
each of a single fragment.
The construction of a stable broken Floer trajectory with gradient fragments
out of the data ci, ui is straightforward and goes as follows. The collection of
points of the form ev(ui+1), ev(ui) which are critical points of fγi determine a
partition
(cmℓ,ℓ, umℓ,ℓ, cmℓ−1,ℓ, . . . , c1,ℓ, u1,ℓ, c0,ℓ), . . . , (cm1,1, um1,1, . . . , u1,1, c0,1)
of the ordered tuple (cm, um, . . . , c1, u1, c0). Note that the cmk,k and c0,k may
either be missing or be constant and exactly one of c0,k and cmk−1,k−1 is missing.
In such a situation we set cmk,k or c0,k to be a constant trajectory at the relevant
critical point, defined on a semi-infinite interval.
4.3 Gluing for Morse-Bott moduli spaces
We prove in this subsection the assertions (i-ii) of Theorem 3.7. The following
notation was introduced in the previous subsection. For γ ∈ P(H) we choose
coordinates (ϑ, z) ∈ S1 × R2n−1 parametrizing a tubular neighbourhood of γ,
such that ϑ ◦ γ(θ) = θ and z ◦ γ(θ) = 0. Given a smooth function fγ : Sγ → R,
we denote by ϕ
s the gradient flow of fγ with respect to the natural metric on
S1. The orthogonal projection onto the 1-dimensional kernel of the asymptotic
operator at γ ∈ P(H) is denoted by P∞, and we denote Q∞ := 1l− P∞.
Let p > 2, d > 0 and δ > 0. Let BAδ = B
1,p,d
δ (γp, γq, A;H, {fγ}) be the space
of proper maps u : R× S1 → Ŵ which are locally in W 1,p and satisfy
Symplectic homology for autonomous Hamiltonians 31
(i) the map u converges uniformly in θ as s → ±∞ to γ
, respectively γp,
and represents the homology class A ∈ H2(W ;Z);
(ii) there exist tubular neighbourhoods U and U of γ and γ respectively,
parametrized by (ϑ, z) ∈ S1 × R2n−1 such that
ϑ ◦ u(s, θ)− θ − ϕδfγs (θ0) ∈ W 1,p(]−∞,−s0]× S1,R; ed|s|ds dθ),
z ◦ u(s, θ) ∈ W 1,p(]−∞,−s0]× S1,R2n−1; ed|s|ds dθ),
ϑ ◦ u(s, θ)− θ − ϕδfγs (θ0) ∈ W 1,p([s0,∞[×S1,R; ed|s|ds dθ),
z ◦ u(s, θ) ∈ W 1,p([s0,∞[×S1,R2n−1; ed|s|ds dθ),
for some s0 > 0 sufficiently large and some θ0, θ0 ∈ S1 satisfying
ϕfγs (θ0) = p, lim
ϕfγs (θ0) = q. (49)
Then BAδ is a Banach manifold and, for d > 0 sufficiently small, it contains
the moduli spaces MA(γp, γq;Hδ, J) for all J ∈ J (see Proposition A.2 in the
Appendix). Let E → BAδ be the Banach vector bundle with fiber E(u,J) =
Lp(R× S1, u∗TŴ ; ed|s|ds dθ). Let ∂̄Hδ,J : BAδ → E be the section defined by
∂̄Hδ,J(u) := ∂su+ Jθ(∂θu−XHδ ).
Then MA(γp, γq;Hδ, J) = ∂̄
(0). From now on we fix J ∈ Jreg(H). In
order to prove (i) in Theorem 3.7 we need to show that the vertical differential
Du : TuBAδ → Eu defined by (12) is surjective for all u ∈ MA(γp, γq;Hδ, J)
when δ > 0 is sufficiently small and the expected dimension of the moduli space
is zero. We have
TuBAδ =W 1,p(R× S1, u∗TŴ ; ed|s|ds dθ)⊕ V u ⊕ V u,
where V u, V u are real vector spaces of dimension
dim V u = ind(p), dim V u = 1− ind(q).
When their dimension is nonzero V u and V u are respectively generated by two
sections of u∗TŴ of the form
(1− β(s, θ))∇fγ(ϑ ◦ u(s, 0)) and β(s, θ)∇fγ(ϑ ◦ u(s, 0)),
with β(s, θ) = β(s) a smooth cutoff function which vanishes for s ≤ 0 and is
equal to 1 for s ≥ 1. The fact that V u and V u have varying dimensions is a
consequence of condition (49).
We shall prove surjectivity of Du by showing that the elements of the moduli
spaceMA(γp, γq;Hδ, J) can be approximated, for δ > 0 small enough, by gluing
the elements of MA(Sγ , Sγ ;H, J) with fragments of gradient trajectories of the
Morse functions fγ .
Symplectic homology for autonomous Hamiltonians 32
Given a, b ∈ R, a < b we define intervals
I(a, b) =
[a, b] if a, b ∈ R,
]−∞, b] if a = −∞, b ∈ R,
[a,∞[ if a ∈ R, b = ∞.
For b − a > 4 and |ǫ| < 1, we let ha,b,ǫ : R → I(a, b) ⊂ R be a collection of
smooth increasing functions such that ha,b,ǫ(s) = a if s ≤ a − ǫ/2, ha,b,ǫ(s) =
b if s ≥ b + ǫ/2 and ha,b,ǫ(s) := s if a − ǫ/2 + 1 < s < b + ǫ/2 − 1. We
can of course make the family {ha,b,ǫ} depend smoothly on a, b and ǫ. We
define ka,b,ǫ(s) :=
|σ=0h′a−σ,b+σ,ǫ(s). The support of ka,b,ǫ is contained in
[a − ǫ/2, a− ǫ/2 + 1] ∪ [b + ǫ/2 − 1, b + ǫ/2]. We may assume without loss of
generality that h′a,b,ǫ and ka,b,ǫ are uniformly bounded.
Convention. If ǫ = 0 we shall omit it from all subsequent decorations, and we
set ǫ = 0 if a = −∞ or b = +∞.
a, b ∈ R a ∈ R, b = +∞
ka,b,ǫ
ha,b,ǫ
h′a,b,ǫ
ka,b,ǫ
ha,b,ǫ
h′a,b,ǫ
Figure 2: The reparametrization function ha,b,ǫ and its derivatives.
Let γ ∈ Pλ and c : I(a, b) → Sγ ⊂ Ŵ be a fragment of gradient trajectory
for the function fγ , i.e. ċ = ∇fγ ◦ c. We define the corresponding gradient
cylinder
uδ,γ,a,b,ǫ : R× S1 → Sγ ⊂ Ŵ
by the equation
ϑ ◦ uδ,γ,a,b,ǫ(s, θ) = ϑ ◦ c(δh a
(s)) + θ. (50)
Then lim
ϑ ◦ uδ,γ,a,b,ǫ(s, θ) = ϑ ◦ c(a) + θ and lim
ϑ ◦ uδ,γ,a,b,ǫ(s, θ) = ϑ ◦
c(b) + θ.
Symplectic homology for autonomous Hamiltonians 33
For γ ∈ Pλ we define Banach manifolds B1,p,dδ (Sγ , Sγ ; fγ), B
1,p,d
δ (p, Sγ ; fγ),
p ∈ Crit(fγ) and B1,p,dδ (Sγ , q; fγ), q ∈ Crit(fγ) consisting of maps u : R× S1 →
Ŵ which are locally of class W 1,p, whose asymptotics are translates of γ, which
represent the zero homology class and which satisfy the following asymptotic
conditions.
(i) for B1,p,dδ (Sγ , Sγ ; fγ): there exists a neighbourhood U of Sγ together with
a parametrization (ϑ, z) : U → S1 × R2n−1 such that
ϑ ◦ u(s, θ)− θ − θ0 ∈ W 1,p(]−∞,−s0]× S1,R; ed|s|ds dθ),
z ◦ u(s, θ) ∈ W 1,p(]−∞,−s0]× S1,R2n−1; ed|s|ds dθ),
ϑ ◦ u(s, θ)− θ − θ0 ∈ W 1,p([s0,∞[×S1,R; ed|s|ds dθ),
z ◦ u(s, θ) ∈ W 1,p([s0,∞[×S1,R2n−1; ed|s|ds dθ),
for some θ0, θ0 ∈ S1 and some s0 > 0. Moreover, there exists T > 0 such
T (θ0) = θ0; (51)
(ii) for B1,p,dδ (p, Sγ ; fγ): there exists a neighbourhood U of Sγ parametrized
by (ϑ, z) ∈ S1 × R2n−1 such that
ϑ ◦ u(s, θ)− θ − ϕδfγs (θ0) ∈ W 1,p(]−∞,−s0]× S1,R; ed|s|ds dθ),
z ◦ u(s, θ) ∈ W 1,p(]−∞,−s0]× S1,R2n−1; ed|s|ds dθ),
ϑ ◦ u(s, θ)− θ − θ0 ∈ W 1,p([s0,∞[×S1,R; ed|s|ds dθ),
z ◦ u(s, θ) ∈ W 1,p([s0,∞[×S1,R2n−1; ed|s|ds dθ),
for some θ0, θ0 ∈ S1 such that lims→−∞ ϕ
s (θ0) = lims→−∞ ϕ
s (θ0) = p
and some s0 > 0;
(iii) for B1,p,dδ (Sγ , q; fγ): there exists a neighbourhood U of Sγ parametrized
by (ϑ, z) ∈ S1 × R2n−1 such that
ϑ ◦ u(s, θ)− θ − θ0 ∈ W 1,p(]−∞,−s0]× S1,R; ed|s|ds dθ),
z ◦ u(s, θ) ∈ W 1,p(]−∞,−s0]× S1,R2n−1; ed|s|ds dθ),
ϑ ◦ u(s, θ)− θ − ϕδfγs (θ0) ∈ W 1,p([s0,∞[×S1,R; ed|s|ds dθ),
z ◦ u(s, θ) ∈ W 1,p([s0,∞[×S1,R2n−1; ed|s|ds dθ),
for some θ0, θ0 ∈ S1 such that lims→∞ ϕ
s (θ0) = lims→∞ ϕ
s (θ0) = q and
some s0 > 0.
We will designate one of the above three spaces by B′δ. We define evaluation
maps ev and ev on B′δ by
ev(u) = lim
u(s, ·), ev(u) = lim
u(s, ·).
Symplectic homology for autonomous Hamiltonians 34
Any map u = uδ,γ,a,b,ǫ belongs to a suitable space B′δ, depending on a, b being
finite or not. The tangent space TuB′δ has a natural decomposition
TuB′δ =W 1,p,d(R× S1, u∗TŴ )⊕ V
u ⊕ V
u, (52)
where V
u are real vector spaces of dimensions
1 if a ∈ R,
ind(p) if a = −∞, dim V
1 if b ∈ R,
1− ind(q) if b = +∞. (53)
When the dimensions are respectively nonzero the generators of V
u are
sections given as follows.
(i) for B1,p,dδ (Sγ , Sγ ; fγ) the sections are
(1− β(s, θ))XH(γ(θ + θ0)) and β(s, θ)XH(γ(θ + θ0));
(ii) for B1,p,dδ (p, Sγ ; fγ) the sections are
(1− β(s, θ))∇fγ(ϑ ◦ u(s, 0)) and β(s, θ)XH(γ(θ + θ0));
(iii) for B1,p,dδ (Sγ , q; fγ) the sections are
(1− β(s, θ))XH(γ(θ + θ0)) and β(s, θ)∇fγ(ϑ ◦ u(s, 0)).
We recall that β(s, θ) = β(s) is a smooth cutoff function which vanishes for
s ≤ 0 and is equal to 1 for s ≥ 1. The norm on TuB′δ is chosen such that the
norm of the above generators of V
u is equal to 1. Let E → B′δ be the
Banach vector bundle with fiber
Eu = Lp(R× S1, u∗TŴ ; ed|s|ds dθ).
We are interested in the family of sections ∂̄a,b,ǫ := ∂̄H′
a,b,ǫ
,J : B′δ → E , with
H ′a,b,ǫ = H + h
(s)(Hδ −H)
= H + δh′a
(s)ρfγ(ℓγϑ− ℓγθ). (54)
Here we use the definition (25) of Hδ. This is a three-parameter family in case
(i) and a two-parameter family in cases (ii) and (iii). Its main feature is that
∂̄a,b,ǫ(uδ,γ,a,b,ǫ) = 0.
We note that neither of the operators ∂̄H,J and ∂̄Hδ,J defines a section B′δ → E
if a or b is infinite. The vertical differential Du := D
a,b,ǫ
u : TuB′δ → Eu of each
of the sections ∂̄a,b,ǫ is given by formula (12) and is a Fredholm operator whose
index has the following values (see also (44)).
Symplectic homology for autonomous Hamiltonians 35
(i) for B1,p,dδ (Sγ , Sγ ; fγ)
ind(Du) = (µRS(γ)−
)− (µRS(γ) +
) + 2 = 1,
(ii) for B1,p,dδ (p, Sγ ; fγ)
ind(Du) = (µRS(γ)−
)− (µRS(γ) +
) + ind(p) + 1 = ind(p),
(iii) for B1,p,dδ (Sγ , q; fγ)
ind(Du) = (µRS(γ)−
)− (µRS(γ) +
) + 1 + 1− ind(q) = 1− ind(q).
In formulas (ii) and (iii) the asymptotics of the operator obtained by conjugation
with e
|s| do not depend on ind(p), ind(q) because, for δ small, the exponential
weight d
overrides the contribution of the perturbation Hδ −H .
Proposition 4.8. Let u = uδ,γ,a,b,ǫ ∈ B′δ. The operator
Du :W
1,p(R× S1, u∗TŴ ; ed|s|)⊕ V ′u ⊕ V ′u → Lp(R× S1, u∗TŴ ; ed|s|dsdθ)
is surjective for δ > 0 small enough.
Proof. In order to compute Du we choose ∇ to be the Levi-Civita connection
corresponding to a (split) metric given by (dλ + dt ∧ λ)(·, J ·). It is a general
fact that the operator Du can be written in a unitary trivialization of u
∗TŴ as
(Duζ)(s, θ) = ∂sζ + J0∂θζ + S(s, θ)ζ(s, θ),
where J0 is the standard complex structure on R
2n and S is asymptotically
symmetric as s → ±∞. We can choose the trivialization so that XH and ∂/∂t
correspond to constant vectors in R2n. We denote S := lims→−∞ S(s, ·). In this
situation the matrix S has the following properties:
(i) ‖S(s, θ) − S(ϑ ◦ u(s, θ) − ϑ ◦ c(a))‖ is bounded by a constant multiple of
δ. This is because, for s ∈ R, the restriction of u to [s − 1, s+ 1] × S1 is
δ-close to the constant cylinder over the orbit u(s, ·) ∈ Sγ .
(ii) the action of S(s, θ) on the (constant) vector of R2n corresponding to XH
is multiplication by
δk(s) := δh′a
(s)f ′′γ (ϑ(u(s, θ))− θ),
and this expression goes to zero with δ.
(iii) the matrix S(s, θ) sends the subspace corresponding to ξ to itself and
sends ∂/∂t on a multiple of the form (E + δF (s, θ))∂/∂t, with E > 0 and
F a bounded function of (s, θ). This follows from (12), (24) and the fact
that ∇∂/∂tRλ = 0 and ∇vRλ ∈ ξ, v ∈ ξ;
Symplectic homology for autonomous Hamiltonians 36
(iv) there is a constant C > 0 such that ‖S′(s, θ)‖ ≤ Cδ for all s ∈ R and
θ ∈ S1. This follows from (50) due to the presence of the factor δ in front
of the reparametrization function h a
We characterize now the kernel of Du. We first show that each ζ ∈ ker Du is
a multiple of the (constant) vector corresponding to XH , or that its component
ζ⊥ on the orthogonal complement vanishes. Let F (s) denote the self-adjoint
operator J0∂θ + S(s, θ), so that Du = ∂s + F (s). If ζ ∈ ker Du we have
(∂s − F (s))(∂s + F (s))ζ = 0, i.e.
∂2sζ − F (s)2ζ + S′(s)ζ = 0.
By taking the scalar product in L2(S1,R2n) with ζ⊥ and using property (ii) for
S we get
〈ζ⊥, ∂2sζ⊥〉 − ‖F (s)ζ⊥‖2 + 〈ζ⊥, S′(s)ζ⊥〉 = 0.
The Morse-Bott assumption and property (i) guarantee that ‖F (s)ζ⊥‖L2 ≥
c‖ζ⊥‖L2 for some c > 0. We obtain
∂2s‖ζ⊥‖2L2 ≥ 2〈ζ⊥, ∂2sζ⊥〉L2 ≥ 2(c2 − Cδ)‖ζ⊥‖2L2 ≥ c2‖ζ⊥‖2L2
if δ > 0 is sufficiently small. In particular ‖ζ⊥‖2L2 can have no local maximum
on R. Since ‖ζ⊥‖2
→ 0 as s→ ±∞ we deduce that ζ⊥ ≡ 0.
We now show that all elements of ker Du are independent of θ. Let ζ ∈
ker Du. Because ζ
⊥ = 0 we have ∂sζ + J0∂θζ + δk(s)ζ = 0, with ∂sζ + δk(s)ζ
and ∂θζ pointwise colinear with XH . Hence ∂sζ + δk(s)ζ = 0 and ∂θζ = 0.
This shows that the elements of ker Du also belong to the kernel of the
linearized Morse operator
ζ 7→ ∂sζ + δh′a
(s)f ′′γ (ϑ ◦ c(δh a
This is a differential equation on R for which the Cauchy problem has a unique
solution. Hence the space of solutions is one-dimensional in C∞(R,R) and, in
order to determine the dimension of ker Du, we just have to check whether the
solutions belong or not to its domain.
If a and b are finite the solutions are constant near ±∞, hence belong to the
domain of Du and dim ker Du = 1. If a = −∞ (and b is finite) we distinguish
two cases: either p is a maximum, in which case f ′′γ (p) < 0, the solutions are
unbounded near −∞ and ker Du = 0, or p is a minimum, in which case f ′′γ (p) >
0, the solutions coincide near −∞ with the elements of V ′u and dim ker Du = 1.
Hence dim ker Du = ind(p). A similar argument shows that dim ker Du =
1− ind(q) if b = +∞ (and a is finite). In all cases we have
dim ker Du = ind(Du),
so that Du is surjective.
Symplectic homology for autonomous Hamiltonians 37
Up to a translation, the defining interval I(a, b) of a gradient cylinder can be
considered to be [−T/2, T/2], T > 0 in case (i), or ]−∞, 1], [−1,∞[ in cases (ii)
and (iii) respectively. We shall thus assume in the sequel that the parameters
a, b take the values
a = −T/2, b = T/2 for T > 0, or a = −∞, b = 1, or a = −1, b = +∞.
We consider a tuple (γ, a, b, ǫ) and the gradient cylinder u := uδ := uδ,γ,a,b,ǫ
for δ small enough. Let (sδ) be a family of parameters such that sδ ≤ s∗δ and
→ 1 as δ → 0, where
s∗δ :=
(T + ǫ)/2δ, if a = −T/2, b = T/2,
1/δ, otherwise.
In particular we have sδ → ∞ as δ → 0. Our goal now is to define modified
norms ‖ · ‖1,δ and ‖ · ‖δ on the domain and target of the operators Du = Duδ
such that they admit uniformly bounded right inverses with respect to δ → 0.
Let wδ : R → R+ be the weight function defined by
wδ(s) =
ed||s|−sδ|, if a and b are finite,
ed|s−sδ|, if a = −∞ and b is finite,
ed|s+sδ|, if a is finite and b = ∞.
0 sδ−sδ
Figure 3: Weight function ||s| − sδ| for a, b finite (logarithmic scale).
The new norm ‖ · ‖δ on the target of Du is the Lp-norm with weight wδ, and
we emphasize it by writing the target as
Lp(R× S1, u∗TŴ ;wδ(s)dsdθ).
Let V
u,δ, V
u,δ be vector spaces of the same dimension as V
u, given
by (53), and which, when their dimension is nonzero, are spanned by the fol-
lowing sections.
(i) if a, b are both finite the sections are
(1− β(s+ sδ, θ))XH(γ(θ + θ0)) and β(s− sδ, θ)XH(γ(θ + θ0));
Symplectic homology for autonomous Hamiltonians 38
(ii) if a = −∞ and b is finite the sections are
(1− β(s− sδ, θ))∇fγ(ϑ ◦ u(s, 0)) and β(s− sδ, θ)XH(γ(θ + θ0));
(iii) if a is finite and b = +∞ the sections are
(1− β(s+ sδ, θ))XH(γ(θ + θ0)) and β(s+ sδ, θ)∇fγ(ϑ ◦ u(s, 0)).
In case a = −∞, b finite or a finite, b = +∞ we define the new norm ‖ · ‖1,δ
on the domain of Du by splitting it as
domDu =W
1,p(R× S1, u∗TŴ ;wδ(s)dsdθ) ⊕ V
u,δ ⊕ V
and setting the norm of the above generators of V
u,δ, V
u,δ to be equal to 1.
In case a, b are finite we split the domain of Du as above and further modify
the weighted norm on the W 1,p-space. We recall from the proof of Proposi-
tion 4.8 that kerDu is 1-dimensional and is spanned by a section ζδ which is
constant for |s| ≥ s∗δ . We normalize ζδ by requiring that its value at 0 be equal
to the constant vector corresponding to XH . Let 〈·, ·〉 be the scalar product in
L2(S1). For an element ζ ∈ W 1,p(R× S1, u∗TŴ ;wδ(s)dsdθ) we denote
κδ :=
〈ζ(0, ·), ζδ(0, ·)〉
〈ζδ(0, ·), ζδ(0, ·)〉
We denote
χδ(s, θ) := β(s+ sδ)β(−s+ sδ)ζδ(s, θ)
and define the norm ‖ · ‖1,δ on W 1,p(R× S1, u∗TŴ ;wδ(s)dsdθ) by
‖ζ‖1,δ := ‖ζ − κδχδ‖1,p,δ + |κδ|.
Here ‖ · ‖1,p,δ is the weighted norm on W 1,p(R× S1, u∗TŴ ;wδ(s)dsdθ).
Proposition 4.9. Let u = uδ = uδ,γ,a,b,ǫ as above. There exists δ2 ∈]0, δ0] such
that the operator
Du : (domDu, ‖ · ‖1,δ) → (Lp(R× S1, u∗TŴ ;wδ(s)dsdθ), ‖ · ‖δ)
is surjective and has a uniformly bounded right inverse Qu = Quδ for δ ∈]0, δ2].
Proof. We choose a unitary trivialization of u∗TŴ as in the proof of Proposi-
tion 4.8, so that XH and ∂/∂t correspond to constant vectors in R
2n, and so
that the operator Du takes the form
(Duζ)(s, θ) = ∂sζ + J0∂θζ + S(s, θ)ζ(s, θ).
Here J0 is the standard complex structure on R
2n and S is asymptotically
symmetric as s→ ±∞. The matrix S(s, θ) can be written as S′(s, θ)⊕S′′(s, θ)
with respect to the splitting ξ ⊕ 〈∂/∂t,XH〉, so that the operator Du is also
Symplectic homology for autonomous Hamiltonians 39
split with respect to the decomposition ξ ⊕ L, where L := 〈∂/∂t,XH〉. It
is therefore enough to find uniformly bounded right inverses for each of the
surjective operators
D′u :W
1,p(R× S1, u∗ξ;wδ(s)dsdθ) → Lp(R× S1, u∗ξ;wδ(s)dsdθ),
D′′u :W
1,p(R× S1, u∗L; ‖ · ‖1,δ)⊕ V
u,δ ⊕ V ′u,δ → Lp(R× S1, u∗L;wδ(s)dsdθ).
Here we use the fact that the norm ‖·‖1,δ coincides with the weightedW 1,p-norm
on sections with values in the subbundle u∗ξ. Note that D′u is an isomorphism
since it has index 0, whereas ind(D′′u) = ind(Du) is either 0 or 1.
We treat D′′u and consider first the case of a semi-infinite gradient trajec-
tory. The two possible cases are entirely similar, and we assume without loss of
generality that a = −∞, b = 1. Let
S′′0 :=
so that limδ→0 S
′′(s, θ) = S′′0 uniformly in (s, θ). Consider the operator
D′′0,δ :W
1,p(R×S1, u∗L;wδ(s)dsdθ)⊕V
u,δ⊕V
u,δ → Lp(R×S1, u∗L;wδ(s)dsdθ)
defined by D′′0,δ := ∂s + J0∂θ + S
0 . As in the proof of Proposition 4.8 one sees
that D′′0,δ is surjective, and we claim that it admits a right inverse Q
0,δ that
is uniformly bounded with respect to δ. Indeed, let Q′′0 be a right inverse of
D′′0 := D
0,δ=1 and consider the shift operators
(Tδζ)(s) := ζ(s+ sδ)
acting from dom(D′′0,δ) → dom(D′′0 ) and from Lp(wδ(s)dsdθ) → Lp(ed|s|dsdθ).
It follows from the definitions of ‖ · ‖1,δ and ‖ · ‖δ that the operators Tδ are
isometries, and we have D′′0,δ = T
0Tδ since D
0 is independent of s ∈ R.
Hence Q′′0,δ = T
0Tδ is a right inverse for D
0,δ such that ‖Q′′0,δ‖ = ‖Q′′0‖,
and the claim is proved.
Now, if δ is small enough we have ‖S′′(s, θ) − S′′0 ‖ ≤ 1/2‖Q′′0‖, s ∈ R and
therefore ‖D′′u −D′′0,δ‖ ≤ 1/2‖Q′′0‖. This implies that
‖D′′uQ′′0,δ − Id‖ = ‖D′′uQ′′0,δ −D′′0,δQ′′0,δ‖ ≤
Thus D′′uQ
0,δ is invertible and the norm of its inverse is ≤ 2. Finally a right
inverse for D′′u is given by Q
0,δ(D
−1 and has norm ≤ 2‖Q′′0‖.
We now treat the case a = −T/2, b = T/2 for T > 0. Let u := uδ,γ,−(T+ǫ)/2,0,
u := uδ,γ,0,(T+ǫ)/2 and
:= D′′u :W
1,p(R× S1, u∗L; ed|s|dsdθ) ⊕ V ′u ⊕ V ′u → Lp(ed|s|dsdθ),
D′′ := D′′u :W
1,p(R× S1, u∗L; ed|s|dsdθ) ⊕ V ′u ⊕ V
u → Lp(ed|s|dsdθ).
Symplectic homology for autonomous Hamiltonians 40
The same argument as above, using the constant operator D′′0 , shows that D
and D′′ admit right inverses which are uniformly bounded with respect to δ →
0. Both operators have index 1 and it follows from the description of their
kernels given in the proof of Proposition 4.8 that their restrictions toW 1,p⊕V ′u,
respectively W 1,p ⊕V ′u are isomorphisms. We choose the right inverses Q
, Q′′
to be the inverses of their respective restrictions.
Let ζ ∈ kerD′′, ζ ∈ kerD′′ be two sections such that their values at +∞
and respectively −∞ are equal to the (constant) vector corresponding to XH
in the chosen trivialization of u∗TŴ . Let V ′, V
be the 1-dimensional vector
spaces spanned by βζ and (1 − β)ζ respectively. Setting the norm of these
generators to be equal to 1 defines a new norm on dom(D
) and dom(D′′),
which we emphasize by decomposing the latter as
dom(D
) =W 1,p(R× S1, u∗L; ed|s|dsdθ)⊕ V ′u ⊕ V ′,
dom(D′′) = D′′u :W
1,p(R× S1, u∗L; ed|s|dsdθ) ⊕ V ′ ⊕ V ′u.
It follows from our special choice of the right inverses Q
, Q′′ that the latter
are also uniformly bounded with respect to this new norm as δ → 0.
Let D′′ := D
′′ be the operator obtained by gluing D
cut at sδ and
D′′ cut at −sδ, with the ‖ · ‖1,δ-norm on its domain and the ‖ · ‖δ-norm on its
target. It follows as in [5, Proposition 5] that the right inverses Q
, Q′′ give rise
to a uniformly bounded right inverse Q′′ for D′′ as δ → 0. On the other hand,
we have ‖D′′u − D′′‖ → 0 as δ → 0, and we obtain a uniformly bounded right
inverse for D′′u by the previous formula Q
u := Q
′′(D′′uQ
′′)−1. We note that,
upon gluing, the exponential weights at ±∞ for D′′, D′′ give rise to the peak
in the weight function wδ for D
′′, and the fibered sum operation on V ′, V
which the norm is fixed, is responsible for the appearance of the distinguished
cutoff section ζδ leading to the modified norm ‖ · ‖1,δ.
We now treat D′u and start by making a few general remarks. For each
s0 ∈ R the operator
D′(s0) := ∂s+J0∂θ+S(s0, θ) :W
1,p(R×S1, u∗ξ; dsdθ) → Lp(R×S1, u∗ξ; dsdθ)
is δ-close to the R-invariant operator with nondegenerate asymptotics corre-
sponding to the constant cylinder over the orbit u(s0, ·). Hence, for δ > 0
small enough, both operators are isomorphisms [24, Lemma 2.4]. Moreover,
this property also holds in the presence of weights ed|s|, eds or e−ds. For
the weight ed|s| we argue as follows. The operator is still Fredholm between
the W 1,p and Lp spaces with weights, of the same index 0. Since the corre-
sponding W 1,p space is contained in W 1,p(R × S1, u∗ξ; dsdθ) we infer that the
operator is injective, hence an isomorphism. For the weight eds we argue as
follows. Multiplication by e
s determines linear isomorphisms M : W 1,p(R ×
S1, u∗ξ; edsdsdθ) →W 1,p(R×S1, u∗ξ; dsdθ) andM : Lp(R×S1, u∗ξ; edsdsdθ) →
Symplectic homology for autonomous Hamiltonians 41
Lp(R× S1, u∗ξ; dsdθ). The operator M−1D′(s0)M is an isomorphism and, for
ζ ∈ W 1,p(R× S1, u∗ξ; edsdsdθ), we have
M−1D′(s0)Mζ = D
′(s0)ζ +
Since d > 0 is small as in Proposition A.2 and p > 2, the operatorM−1D′(s0)M
is R-invariant and has nondegenerate asymptotics, hence is an isomorphism [24,
Lemma 2.4]. An analogous reasoning using the multiplication by e−
s proves
the claim for the weight e−ds.
We now prove that D′u admits a uniformly bounded right inverse in the case
a = −T/2, b = T/2, sδ = (T+ǫ)/2δ. We recall the notation u := uδ,γ,−(T+ǫ)/2,0,
u := uδ,γ,0,(T+ǫ)/2 and set
:= D′u :W
1,p(R× S1, u∗ξ; ed|s|dsdθ) → Lp(R× S1, u∗ξ; ed|s|dsdθ),
D′ := D′u :W
1,p(R× S1, u∗ξ; ed|s|dsdθ) → Lp(R× S1, u∗ξ; ed|s|dsdθ).
We claim that each of the operatorsD
, D′ is an isomorphism with uniformly
bounded right inverse as δ → 0. We give the proof forD′ since the proof forD′ is
entirely analogous. We choose a finite number of points −∞ = s−m < s−m+1 <
· · · < s−1 < 0 = s0 < s1 < · · · < sm+1 = +∞ such that ‖S(s, θ) − S(s′, θ)‖ ≤
1/4C for all θ ∈ S1 and s, s′ ∈ [si, si+1], i = −m, . . . ,m, with C > 0 a constant
to be chosen below. Let
bi−1 := ai := c
−1(u(si, 0)), i = −m, . . . ,m+ 1.
We consider the operators
D′i := D
uδ,γ,ai−1,bi−1
, i = −m+ 1, . . . ,−1,
D′0 := D
uδ,γ,a−1,b0
D′i := D
uδ,γ,ai,bi
, i = 1, . . . ,m.
For each i = −m + 1, . . . ,m we denote by ui = ui,δ the gradient cylinder
corresponding to the operator D′i. The domain and range of the operators D
are as follows:
D′i :W
1,p(R× S1, u∗i ξ; e−dsdsdθ) → Lp(R× S1, u∗i ξ; e−dsdsdθ), i < 0,
D′0 :W
1,p(R× S1, u∗0ξ; ed|s|dsdθ) → Lp(R× S1, u∗0ξ; ed|s|dsdθ),
D′i :W
1,p(R× S1, u∗i ξ; edsdsdθ) → Lp(R× S1, u∗i ξ; edsdsdθ), i > 0.
We have seen that D′(s0) is an isomorphism for all s0 ∈ R if one uses any of
the weights ed|s|, eds, e−ds. Since S(s0, ·) belongs to a compact set of loops of
matrices we infer that the norm of the inverse Q′(s0) := D
′(s0)
−1 is uniformly
bounded with respect to s0 ∈ R for each of these three weights. We choose
C := maxweight∈{ed|s|,eds,e−ds} maxs0∈R ‖Q′(s0)‖.
Symplectic homology for autonomous Hamiltonians 42
The same argument as forD′′u shows that the inverse of eachD
i is bounded by
2C independently of δ. We glue together the operators D′i into D̃
′ by cutting at
ai/δ and bi/δ. Then D̃
′ is still surjective and the norm of its inverse is bounded
by 2CC̃2m−1, with C̃ a universal constant (see [24, Proposition 3.9]). Note that
our choice of weights for the operators D′i is such that the resulting weight for
the domain and target of D̃′ is still ed|s|. On the other hand we have
‖D̃′ −D′‖ → 0, δ → 0.
This is because the two operators coincide outside 2m− 1 intervals of length 2,
where the variation of S tends to zero as δ → 0. As a consequence the inverse
is also uniformly bounded when δ is small enough.
We now glue the operator D
cut at sδ with the operator D
′ cut at −sδ,
and denote the resulting operator by D′. The argument in [5, Proposition 5]
shows that D′ admits a uniformly bounded right inverse Q′, provided one uses
the weight wδ(s) on its domain and target. On the other hand
‖D′u −D′‖ → 0, δ → 0
since the two operators differ on a segment of length 2 where the variation of S
goes to zero. We infer that D′u also admits a uniformly bounded right inverse.
The cases when a = −∞, b = 1 or a = −1, b = ∞ follow now easily by
combining the proof of the existence of uniformly bounded right inverses for the
operators D
with the previous use of a shift operator (Tδζ)(s) = ζ(s± sδ).
Remark 4.10. Note that, if a = −T/2, b = T/2, Our construction of a right
inverse for D′′ in the proof of Proposition 4.9 is such that its norm is uniformly
bounded as δ → 0 even if one uses the “non-compensated” norm ‖·‖1,p,δ instead
of ‖ · ‖1,δ. However, our choice of the norm ‖ · ‖1,δ will be essential in the proof
of Proposition 4.16.
In order to describe the pregluing construction it is convenient to work with
a single section over B′δ rather than with a family of sections. We recall that,
up to a translation, the defining interval I(a, b) of a gradient cylinder can be
considered to be [−T/2, T/2], T > 0 in case (i), or ] − ∞, 1], [−1,∞[ in cases
(ii) and (iii) respectively. We are therefore led to consider the section
∂̄ : B′δ → E (56)
defined by ∂̄ := ∂̄−∞,1 and ∂̄ := ∂̄−1,∞ in cases (ii) and (iii), and by
∂̄(u) = ∂̄ǫ(u) := ∂̄−Tu/2,Tu/2,ǫ(u)
in case (i). Here Tu > 0 is the time needed to flow along the gradient of fγ from
the negative limit to the positive limit of u (see (51)).
Remark 4.11. In case (i) the section ∂̄ can be described as follows. The one
parameter family of sections ∂̄T := ∂̄−T/2,T/2,ǫ gives rise to a section denoted
Symplectic homology for autonomous Hamiltonians 43
{∂̄T } of the pull-back bundle pr∗1E → B′δ × R+. There is a codimension one
embedding ι : B′δ → B′δ × R+ given by ι(u) = (u, Tu), the composition pr1 ◦ ι is
the identity and we have
∂̄ = {∂̄T }|im ι.
The situation is summarized in the following commutative diagram.
pr∗1E //
� ι //
B′δ × R+
pr1 //
{∂̄T }
Given u ∈ B′δ we denote by D′u : TuB′δ → Eu the vertical differential of ∂̄. In
cases (ii) and (iii) we have seen that D′u is a Fredholm operator of index ind(p)
and 1− ind(q) respectively. In case (i) the vertical differential can be computed
explicitly as follows. The vertical differential of {∂̄T }, denoted by D{∂̄T }, is
D{∂̄T }(u, T ) · (ζ, τ) = D−T/2,T/2,ǫu ζ − τ(JXHδ−H)
h′−T/2δ,T/2δ,ǫ/δ(s)
= D−T/2,T/2,ǫu ζ −
(JXHδ−H)k−T/2δ,T/2δ,ǫ/δ(s).
Let us write a section ζ ∈ TuB′δ as ζ = ζ0 + aζ + bζ, with ζ0 ∈ W 1,p,d, a, b ∈ R
and ζ, ζ being the distinguished generators of V
u respectively. The vertical
differential D′u acts by
D′uζ = D{∂̄T}(u, Tu) · (ζ, dTu · ζ)
= D−Tu/2,Tu/2,ǫu ζ −
dTu · ζ
(JXHδ−H)k−Tu/2δ,Tu/2δ,ǫ/δ(s).
One can explicitly compute
dTu · ζ = dTu · (aζ + bζ) =
ċ(−Tu/2)b− ċ(Tu/2)a
ċ(−Tu/2) · ċ(Tu/2)
where c : R → Sγ is the gradient trajectory satisfying c(−Tu/2) = θ0, c(Tu/2) =
θ0 and ċ is the derivative with respect to the XH -parametrization of Sγ .
Proposition 4.12. Let T > 0 and u = uδ,γ,−T/2,T/2,ǫ. The index of D
equal to 1, its kernel has dimension 2 and a complement of imD′u is spanned by
a section supported in
[−(T + ǫ)/2δ,−(T + ǫ)/2δ + 1]× S1
[(T + ǫ)/2δ − 1, (T + ǫ)/2δ]× S1.
Morever, D′u admits a right inverse defined on its image which is uniformly
bounded with respect to δ → 0.
Symplectic homology for autonomous Hamiltonians 44
Proof. The first order differential operators D′u and D
−Tu/2,Tu/2,ǫ
u differ by a
term of order zero, hence their indices are equal and indD′u = 1.
The operator D{∂̄T }(u, Tu) is surjective and has index 2. As a consequence
dimkerD′u ≤ dim kerD{∂̄T }(u, Tu) = 2. Let c : R → Sγ be the gradient curve
defining u = uδ,γ,−T/2,T/2,ǫ. For σ close to zero we define c
σ(s) := c(σ + s)
and denote by uσ1 := u
δ,γ,−T/2,T/2,ǫ
the gradient cylinder defined by cσ. Then
∂̄(uσ1 ) = ∂̄T (u
1 ) = 0, hence ζ
1 := d
|σ=0uσ1 ∈ kerD′u. We also define uσ2 :=
uδ,γ,−(T+σ)/2,(T+σ)/2,ǫ to be the gradient cylinder associated to c. Then ∂̄(u
2 ) =
∂̄T+σ(u
2 ) = 0, hence ζ
2 := d
|σ=0uσ2 ∈ kerD′u. Since ζ1 and ζ2 are linearly
independent, we infer that dimkerD′u = 2.
We claim that the section η := 1
(JXHδ−H)k−Tu/2δ,Tu/2δ,ǫ/δ(s) spans a com-
plement of imD′u. This follows from (i) in Lemma 4.13 below with ℓ := dTu,
φ := D
−Tu/2,Tu/2,ǫ
u , φ̃ := D
u, y := η and xy := ζ
2. That D′u admits a uniformly
bounded right inverse defined on its image follows from (ii) in Lemma 4.13 and
the fact that D
−Tu/2,Tu/2,ǫ
u has a uniformly bounded right inverse by Proposi-
tion 4.9.
Lemma 4.13. Let φ : E → F be a surjective map of Banach vector spaces,
ℓ : E → R be a nonzero linear functional, y = φ(xy) ∈ F be fixed and φ̃ : E → F
be defined by
φ̃(x) = φ(x) − ℓ(x)y.
We assume that kerφ ⊂ ker ℓ. Then im φ̃ = φ(ker ℓ) if and only if ℓ(xy) = 1, in
which case the following hold.
(i) The element y spans a complement of im φ̃.
(ii) If Q : F → E is a right inverse for φ, then Q|φ(ker ℓ) is a right inverse for
φ̃ defined on its image.
Proof. We first note that im φ̃ ⊇ φ(ker ℓ). Let us now assume that im φ̃ =
φ(ker ℓ). For x /∈ ker ℓ we obtain φ(x)− l(x)φ(xy) ∈ φ(ker ℓ), hence x− l(x)xy ∈
ker ℓ, implying ℓ(x) − ℓ(x)ℓ(xy) = 0 and ℓ(xy) = 1. Conversely, if ℓ(xy) = 1 we
obtain x− ℓ(x)xy ∈ ker ℓ for any x ∈ E, hence φ̃(x) = φ(x− ℓ(x)xy) ∈ φ(ker ℓ).
The element y does not belong to φ(ker ℓ) because y = φ(xy) with ℓ(xy) = 1
and the preimage xy is well-defined up to an element of kerφ ⊂ ker ℓ. This
proves the equivalence in the statement of the Lemma, as well as (i).
To prove (ii) we need to show that Q(φ(ker ℓ)) ⊂ ker ℓ. We prove the stronger
statement imQ ∩ ker ℓ = Q(φ(ker ℓ)). The inclusion imQ ∩ ker ℓ ⊂ Q(φ(ker ℓ))
follows from the observation that, given x = Qz with ℓ(x) = 0, we have z =
φ(Qz) = φ(x) ∈ φ(ker ℓ). On the other hand note that Qφ is the projection to
imQ along kerφ. Since kerφ ⊂ ker ℓ, it follows that Qφ(ker ℓ) ⊂ imQ∩ker ℓ.
We describe now the pre-gluing construction for elements of the Morse-Bott
moduli spaces and gradient cylinders of the form uδ,γ,a,b,ǫ. We define the space
B̃δ := B̃1,p,dδ (γp, Sγm−1 , . . . , Sγ1 , γq, A;H, {fγ})
Symplectic homology for autonomous Hamiltonians 45
consisting of tuples w̃ := (u1, . . . , um, v0, . . . , vm) satisfying the following condi-
tions.
(i) ui ∈ B1,p,d(Sγi , Sγi−1 , Ai;H), i = 1, . . . ,m, with Sγ0 := Sγ , Sγm := Sγ ,
Sγi 6= Sγi−1 , i = 1, . . . ,m and A1 + . . .+Am = A;
(ii) v0 ∈ B1,p,dδ (Sγ , q; fγ), vi ∈ B
1,p,d
δ (Sγi , Sγi ; fγi) for i = 1, . . . ,m − 1, and
vm ∈ B1,p,dδ (p, Sγ ; fγ);
(iii) ev(vi−1) = ev(ui) and ev(vi) = ev(ui) for i = 1, . . . ,m;
(iv) ev(v0) belongs to the stable manifold of q, and ev(vm) belongs to the
unstable manifold of p.
By the definition of the spaces B1,p,dδ (Sγi , Sγi ; fγi) we have ev(vi) 6= ev(vi) for
i = 1, . . . ,m − 1. We denote by Ti > 0 the unique positive real number such
that ϕ
(ev(vi)) = ev(vi), where ϕ
s is the gradient flow of fγ .
Figure 4: Broken Morse-Bott trajectory w̃.
Let us choose a tubular neighbourhood Uγ ⊂ Ŵ for each γ ∈ P(H),
parametrized by (ϑ, z) ∈ S1 × R2n−1. Given any subset
K ⊂ B̃1,p,dδ (γp, Sγm−1 , . . . , Sγ1 , γq, A;H, {fγ})
for which there exists s0 > 0 such that, for |s| ≥ s0, the components of any
w̃ ∈ K belong to the respective tubular neighbourhoods of their asymptotics,
Symplectic homology for autonomous Hamiltonians 46
we construct, for δ > 0 small enough and ǫi ∈ R, i = 1, . . . ,m− 1 small enough
in absolute value, a pre-gluing map
Gδ,ǫ : K → B1,p,dδ (γp, γq, A;H, {fγ}), ǫ := (ǫ1, . . . , ǫm−1).
Let β : R → [0, 1] be a smooth increasing cutoff function vanishing on ]−∞, 0]
and identically equal to 1 on [1,∞[. Define the gluing profile R = R(δ) by
. (57)
We define for i = 1, . . . ,m the maps ûi : [−R,R]× S1 → Ŵ by
ûi(s, θ) :=


z(s, θ) = β(s+R)z ◦ ui(s, θ),
ϑ(s, θ) = θ + β(s+R)(ϑ ◦ ui(s, θ)− θ),
s ∈ [−R,−R+ 1],
ui(s, θ), s ∈ [−R+ 1, R− 1],{
z(s, θ) = β(−s+R)z ◦ ui(s, θ),
ϑ(s, θ) = θ + β(−s+R)(ϑ ◦ ui(s, θ)− θ),
s ∈ [R− 1, R].
We define for i = 1, . . . ,m− 1 the maps
v̂i : [−(Ti + ǫi)/2δ, (Ti + ǫi)/2δ]× S1 → Ŵ
by the analogous formulas in which we replace R by Ti+ǫi
. We also define
v̂0 : [−1/δ,+∞[×S1 → Ŵ
v̂0(s, θ) :=
z(s, θ) = β(s+ 1
)z ◦ v0(s, θ),
ϑ(s, θ) = θ+β(s+ 1
)(ϑ◦v0(s, θ)−θ),
s ∈ [− 1
+ 1],
v0(s, θ), s ∈ [− 1δ + 1,+∞[,
as well as
v̂m :]−∞, 1/δ]× S1 → Ŵ
by the analogous formula with s replaced by −s and v0 replaced by vm. Finally,
we define
Gδ,ǫ(w̃)
as the catenation v̂m, ûm, v̂m−1, . . . , û1, v̂0. The catenation of these maps is
performed in the above order and with (obvious) shifts
0 = svm < sum < svm−1 < . . . < su1 < sv0
in the domain defined by
suj = svj + ℓj, (58)
svj−1 = suj + ℓj−1
Symplectic homology for autonomous Hamiltonians 47
for j = 1, . . . ,m. Here we denote
ℓi := R + (Ti + εi)/2δ (59)
for i = 0, . . . ,m, with the convention Tm = T0 = 2 and εm = ε0 = 0. We have
in particular
v̂i(s, θ) = Gδ,ǫ(w̃)(s+ svi , θ), (s, θ) ∈ dom(v̂i), i = 0, . . . ,m,
ûj(s, θ) = Gδ,ǫ(w̃)(s+ suj , θ), (s, θ) ∈ dom(ûj), j = 1, . . . ,m.
Given u = (cm, um, . . . , u1, c0) ∈ M̂A(p, q;H, {fγ}, J), we denote by
Gδ,ǫ(u)
the element Gδ,ǫ(w̃) ∈ Bδ, where w̃ := (vm, um, . . . , u1, v0) and vi := uδ,γi,ai,bi,ǫi ,
i = 0, . . . ,m is the gradient cylinder corresponding to the gradient trajectory
ci : I(ai, bi) → Sγi .
The section ∂̄Hδ,J(Gδ,ǫ(w̃)) belongs to the space
Lp(R× S1, Gδ,ǫ(w̃)∗TŴ ; gδ,ǫ(s)dsdθ),
where the continuous function gδ,ǫ(s) is the catenation of the following functions:
(i) gδ,ui(s) := e
d|s| on the domain [−R,R] of ûi;
(ii) gδ,ǫi,vi(s) = e
d||s|−si,δ| on the domain [−(Ti + ǫi)/2δ, (Ti + ǫi)/2δ] of v̂i,
where si,δ =
Ti+ǫi
−R ≤ s∗i,δ =
Ti+ǫi
, i = 1, . . . ,m− 1;
(iii) gδ,v0(s) := e
d|s+s0,δ| on the domain [−1/δ,+∞[ of v̂0, where s0,δ = 1/δ −
R ≤ s∗0,δ = 1/δ;
(iv) gδ,vm(s) := e
d|s−sm,δ| on the domain ]−∞, 1/δ] of v̂m, with sm,δ = 1/δ −
R ≤ s∗m,δ = 1/δ.
We denote the norm on the above Lp space with weight gδ,ǫ by ‖ · ‖δ, omitting
in the notation the dependence on the numbers Ti + ǫi, i = 1, . . . ,m − 1. We
define a norm ‖ · ‖1,δ on the space
W 1,p(R× S1, Gδ,ǫ(w̃)∗TŴ ; gδ,ǫ(s)dsdθ)
as follows. For j = 1 . . . ,m let
〈ζ(suj −R, ·), XH〉
〈XH , XH〉
, κj =
〈ζ(suj +R, ·), XH〉
〈XH , XH〉
, (60)
where 〈·, ·〉 is the inner product in L2(S1). Here suj − R and suj + R are the
coordinates of the catenation circles between ûj and v̂j , respectively ûj and
v̂j−1. For i = 1, . . . ,m− 1 let
〈ζ(svi , ·), ζi,δ(0, ·)〉
〈ζi,δ(0, ·), ζi,δ(0, ·)〉
Symplectic homology for autonomous Hamiltonians 48
suj +R
sujsvj suj − 2R suj −R svj−1suj + 2R
Figure 5: The definition of ‖ · ‖1,δ.
where the section ζi,δ generates the kernel of the operator Dvi as in Proposition
4.8. The norm ‖ · ‖1,δ is then defined by
‖ζ‖1,δ :=
∥∥ζ −
κjβ(−s+ suj )β(s − suj + 2R)XH (61)
− κjβ(s− suj )β(−s+ suj + 2R)XH
κiβ(s− svi + ℓi − 2R)β(−s+ svi + ℓi − 2R)ζi,δ(· − svi , ·)
W 1,p(gδ,ǫ)
|κj |+ |κj |
|κi|.
Here ℓj is defined by (59), β : R → [0, 1] is the smooth cutoff function which
vanishes on ]−∞, 0] and is equal to 1 on [1,∞[, and ‖ · ‖W 1,p(gδ,ǫ) is the W 1,p-
norm with weight gδ,ǫ on W
1,p(ed|s|dsdθ). The graph of the function
β(−s+ suj )β(s− suj + 2R) + β(s− suj )β(−s+ suj + 2R)
+β(s− svj + ℓj − 2R)β(−s+ svj + ℓj − 2R)
+β(s− svj−1 + ℓj−1 − 2R)β(−s+ svj−1 + ℓj−1 − 2R)
is depicted in Figure 5.
Remark 4.14. The definition of ‖ · ‖1,δ is such that the norm of the gluing
map G constructed in the proof of Proposition 4.18 below is uniformly bounded
with respect to δ → 0.
Proposition 4.15. Let w̃ ∈ B̃δ and ǫ(δ) := (ǫ1(δ), . . . , ǫm−1(δ)) be such that
(i) ǫi(δ) → 0, δ → 0 for i = 1, . . . ,m− 1;
(ii) ui ∈ MAi(Sγi , Sγi−1 ;H, J), i = 1, . . . ,m;
Symplectic homology for autonomous Hamiltonians 49
(iii) the components vi are of the form uδ,γi,ai,bi,ǫi , with bi = −ai = Ti/2 for
i = 1, . . . ,m − 1, b0 = +∞, a0 = −1, ǫ0 = 0 and bm = 1, am = −∞,
ǫm = 0.
‖∂̄Hδ,J(Gδ,ǫ(w̃))‖δ = 0.
Proof. We must check that ‖∂̄Hδ,J(Gδ,ǫ(w̃))|I×S1‖δ → 0 as δ → 0 when I ⊂ R
is an interval of the following type.
(i) I = [−R + 1, R − 1] is contained in the domain of ûi. Then ∂̄Hδ,J(ûi) =
−J(XHδ − XH) ◦ ûi. The norm of this map is pointwise bounded by a
constant multiple of δ. Hence its δ-norm is bounded by a constant multiple
of δedR → 0, δ → 0;
(ii) I = [−R,−R + 1] or I = [R − 1, R] is contained in the domain of ûi.
We have ∂̄Hδ,J(ûi) = ∂̄H,J(ûi) − J(XHδ − XH) ◦ ûi. The second term is
bounded as in (i). The term ∂̄H,J(ûi) is pointwise bounded by the norms
of z ◦ ûi, ϑ ◦ ûi − θ and of their derivatives. By Proposition A.1 their
δ-norm is bounded by a constant multiple of e(d−r)R → 0, δ → 0;
(iii) I = [−(Ti + ǫi)/2δ + 1, (Ti + ǫi)/2δ − 1] for i = 1, . . .m − 1, or I =
[−1/δ + 1,+∞[ or I =] −∞, 1/δ − 1] and is contained in the domain of
some v̂i. Since ∂̄H′
−Ti/2,Ti/2,ǫi
,J(v̂i) = 0 and H
−Ti/2,Ti/2,ǫi
= Hδ for s ∈ I,
we already have ‖∂̄Hδ,J(Gδ,ǫ(w̃))|I×S1‖δ = 0;
(iv) I = [−(Ti+ ǫi)/2δ,−(Ti+ ǫi)/2δ+1] or I = [(Ti+ ǫi)/2δ− 1, (Ti+ ǫi)/2δ]
for i = 1, . . .m − 1, or I = [−1/δ,−1/δ + 1], or I = [1/δ − 1, 1/δ] and is
contained in the domain of some v̂i. Then ∂̄Hδ,J(v̂i) involves only ϑ◦ v̂i−θ,
its derivative with respect to s and δ∇fγi . By formula (50) the norm
of these expressions is pointwise bounded by a constant multiple of δ,
therefore their δ-norms are bounded by δedR → 0 as δ → 0.
Proposition 4.16. Let [ṽn] ∈ MA(γp, γq;Hδn , J) with δn → 0, n → ∞ and
let [u] ∈ MA(p, q;H, {fγ}, J) be a broken Floer trajectory of level ℓ = 1 whose
intermediate gradient fragments c1, . . . , cm−1 are nonconstant. Then [ṽn] → [u],
n→ ∞ if and only if there exist
• representatives vn ∈ [ṽn], v ∈ [u],
• real parameters ǫn = (ǫn1 , . . . , ǫnm−1) with ǫni → 0, n→ ∞,
• vector fields ζn ∈ TGδn,ǫn (v)Bδ with ζn = (ζ
n, ζn, ζn), such that
‖ζn‖1,δn := ‖ζ0n‖1,δn + ‖ζn‖+ ‖ζn‖ → 0, n→ ∞,
Symplectic homology for autonomous Hamiltonians 50
satisfying
vn := expGδn,ǫn(v)
(ζn).
Proof. We first prove the converse implication, namely that convergence in norm
implies geometric convergence. We define shifts (sni ), i = 1, . . . ,m inductively
snm := 1/δn +Rn, s
i := s
i+1 + 2Rn + (Ti + ǫ
i )/δn.
We claim that vn(·+sni , ·) → ui, n→ ∞ uniformly on compact sets. Let R0 > 0
be fixed. By assumption
‖ζ0n(·+ sni , ·)|[−R0,R0]×S1‖1,δn → 0, n→ ∞.
By the Sobolev embedding theorem this implies
‖ζ0n(·+ sni , ·)|[−R0,R0]×S1‖C0 → 0, n→ ∞.
Since
Gδn,ǫn(v)(· + sni , ·)|[−R0,R0]×S1 = ui|[−R0,R0]×S1
for n sufficiently large, the conclusion follows.
We now prove the direct implication. Let us pick a representative
v = (cm, um, cm−1, . . . , u1, c0) ∈ [u]
and let Ti, i = 0, . . . ,m be the lengths of the intervals of definition of ci, with the
convention T0 = Tm = +∞. We also choose arbitrary representatives vn ∈ [ṽn].
By assumption there exist shifts (sni ) such that vn(· + sni , ·) converges to ui
uniformly on compact sets. We define
ǫni := δn(s
i − sni+1 − 2Rn)− Ti, i = 1, . . . ,m− 1. (62)
By Lemma 4.6 we have ǫni → 0, n→ ∞. We define partitions of the real line
−∞ = anm ≤ bnm ≤ anm−1 ≤ . . . ≤ an0 ≤ bn0 = +∞
by bnm := 1/δn and
ani−1 := b
i + 2Rn, b
i−1 := a
i−1 + (Ti−1 + ǫ
i−1)/δn, i = 1, . . . ,m.
We define a sequence of shifts (sn) by
sn := snm − 1/δn −Rn
and we still denote by vn the shifted sequence vn(·+ sn, ·).
We first show the existence of a unique vector field ζn satisfying vn =
expGδn,ǫn(v)
(ζn). For that it is enough to prove
s∈In,θ∈S1
dist(vn(s, θ), Gδn,ǫn(v)(s, θ)) = 0, (63)
where In is an interval of the following form:
Symplectic homology for autonomous Hamiltonians 51
(i) [bni , a
i−1], i = 1, . . . ,m;
(ii) [ani , b
i ], i = 1, . . . ,m− 1;
(iii) [bnm −K/δn, bnm] or [an0 , an0 +K/δn], for any K > 0.
The asymptotic behaviour of vn and Gδn,ǫn(v) ensures that ζn is an element of
the relevant W 1,p-space.
We prove case (i) by contradiction. Assume that there exists ǫ > 0 and a
sequence (s̃n, θ̃n) ∈ [bmi , ani−1]× S1 such that
dist(vn(s̃n, θ̃n), Gδn,ǫn(v)(s̃n, θ̃n)) ≥ ǫ.
Since (63) is satisfied if one replaces vn by ui(·−sni , ·) (by definition ofGδn,ǫn(v)),
we also have
dist(vn(s̃n, θ̃n), ui(s̃n − sni , θ̃n)) ≥ ǫ/2 (64)
for n large enough. By the assumption of uniform convergence on compact
sets vn(· + sni , ·) → ui(·, ·), up to passing to a subsequence we can assume that
s̃n − sni → ±∞. We treat the case s̃n − sni → ∞, the other case being similar.
Since s̃n ∈ [bni , ani−1] and δn(ani−1 − bni ) = 2δnRn → 0, we have δn(s̃n − sni ) → 0.
By Lemma 4.5 we infer that vn(·+ s̃n, ·) → ev(ui), which means
vn(s̃n, ·) = lim
ui(s̃n − sni , ·)
and this contradicts (64).
Note that the above proof shows that vn(· + ani−1, ·) → ev(ui) and vn(· +
bni , ·) → ev(ui), i = 1, . . . ,m uniformly on compact sets.
We now prove case (ii). Let us fix 1 ≤ i ≤ m − 1. An action argument as
the one in the proof of Lemma 4.6 shows that vn(In × S1) is entirely contained
in a small neighbourhood of Sγi . We apply Proposition A.3 to vn and In × S1
to obtain
(s,θ)∈In×S1
|z ◦ vn(s, θ)| = 0
(s,θ)∈In×S1
|ϑ ◦ vn(s, θ)− θ − ϕ
δn(s−a
(ev(ui+1))| = 0.
The same two equations hold, by definition, if one replaces vn by Gδn,ǫn(v), and
the conclusion follows.
We now prove (iii). We treat only the case In = [a
0 , a
0 +K/δn], the other
case being similar. An action argument as above shows that vn(·+an0 +K/δn, ·)
converges uniformly on compact sets to a constant cylinder over some orbit
γ ∈ Sγ0 . By Lemma 4.6 we know that γ = ϕ
K (ev(u1)), and in particular is
not a critical point of fγ0 . Now the conclusion follows in the same way as in
case (ii).
We now show that
‖ζn|In×S1‖1,δn = 0 (65)
Symplectic homology for autonomous Hamiltonians 52
in each of the cases (i)-(iii). We denote in the sequel
|ζ(s, θ)|1 := |ζ(s, θ)| + |∇sζ(s, θ)|+ |∇θζ(s, θ)|.
We first consider case (i). Let us fix K > 0 large enough. For n large enough
we can write
In = [s
i −Rn, sni −K] ∪ [sni −K, sni +K] ∪ [sni +K, sni +Rn].
We first note that
∫ sni +K
|ζ(s, θ)|p1 gδn,ǫn(s)dsdθ =
∫ sni +K
|ζ(s, θ)|p1 ed|s−s
i |dsdθ
≤ sup
s∈[sn
−K,sn
|ζ(s, θ)|p1 · e
Since vn(· + sni , ·) and Gδn,ǫn(v)(· + sni , ·) converge uniformly on compact sets
together with their derivatives to ui, the last term goes to zero as n→ ∞.
In order to estimate the integral on the interval [sni −Rn, sni −K] we apply
Proposition A.3 on [sni+1 +K, s
i −K] to vn to obtain
|z ◦ vn(s, θ)|1 ≤ C(K)
cosh(ρ(s− s
i+1+s
cosh(ρ(
≤ C1C(K)eρ(s−s
i +K)
|ϑ ◦ vn(s, θ)− θ − ϕ
δn(s−b
(pni )|1 ≤ C1C(K)eρ(s−s
i +K),
where | · |1 stands for the pointwise C1-norm, for some pni ∈ Sγi such that
pni → ev(ui), n → ∞. Similar estimates hold, by definition, if one replaces vn
by Gδn,ǫn(v) and p
i with ev(ui). Hence we obtain
|ζn(s, θ)− κni XH |1 ≤ C1C(K)eρ(s−s
i +K), (66)
where κni → 0 as n→ ∞ and
C(K) = Cmax(‖Q∞vn(sni+1 +K)‖, ‖Q∞vn(sni −K)‖,
‖Q∞ṽn(sni+1 + sn +K)‖, ‖Q∞ṽn(sni + sn −K)‖). (67)
We obtain
∫ sni −K
|ζn(s, θ)− κni XH |
1 gδn(s)dsdθ
∫ sni −K
|ζn(s, θ)− κni XH |
−d(s−sni )dsdθ
≤ C2C(K)pedK .
Symplectic homology for autonomous Hamiltonians 53
A similar estimate holds when the interval of integration is [sni +K, s
i + Rn],
with C(K) replaced with C′(K). Letting n→ ∞ we obtain
In×S1
|ζn(s, θ)− κni β(−s+ sni )XH − κni β(s− sni )XH |
1 gδn(s)dsdθ
≤ C2(C(K)p + C′(K)p)edK .
We let now K → ∞. Proposition A.3 implies that, for K > K ′, we have
C(K ′) ≤ C3C(K)e−ρ(K
′−K), hence C(K)pedK → 0 as K → ∞ because d < ρp.
The equality (65) follows.
We now consider case (ii). We fix K > 0 large enough and apply Proposi-
tion A.3 on the interval [sni+1+K, s
i −K] ⊃ [sni+1+Rn, sni −Rn] = In to obtain
as in case (i)
|ζn(s, θ)− κni ζi,δ(s, θ)|1 ≤ C(K)
cosh(ρ(s− s
i+1+s
cosh(ρ(
where C(K) is given by (67), ζi,δ(s −
sni+1+s
) generates the kernel of the lin-
earized operator corresponding to gradient trajectory ci as in Proposition 4.8
and κni ζi,δ(b
i , ·) = κni XH . In particular, we have κni → 0, n→ ∞. We get
∫ sni −Rn
|ζn(s, θ)− κni ζi,δ|
1 gδn(s)dsdθ ≤ C2C(K)
pe(d−ρp)(Rn−K).
The last term goes to zero as n → ∞. Equality (65) follows now as in case (i).
Case (iii) is entirely similar to case (ii).
In order to complete the proof of ‖ζn‖1,δn → 0, n→ ∞, it is enough to show
that ‖ζn|In×S1‖1,δn → 0 if In =] − ∞, bnm − K/δn] or In = [an0 + K/δn,+∞[,
for any K > 1. The two cases are entirely similar and we give the argument
only for In =]−∞, bnm −K/δn]. By Proposition A.2, for n sufficiently large we
have vn(s, θ) = expuδn,γm,−∞,1(s,θ)(ηn(s, θ)), with ηn = (η
n, ηn), η
n ∈W 1,p(In×
S1, u∗δn,γm,−∞,1TŴ ; e
r|s|ds dθ), ηn ∈ V
. Since vn(b
n , ·) → ev(um) we have
‖ηn‖∞ → 0. Since Gδn,ǫn(v) = uδn,γm,−∞,1 on In, we obtain ζn = ηn, so that
‖ζn‖ → 0. The fact that ‖ζ0n‖1,δn → 0 follows from the fact that d < r.
We explain now how to construct a right inverse for DGδ,ǫ( ew) which is uni-
formly bounded with respect to δ → 0. The space B̃δ is a Banach manifold
whose tangent space at w̃ is
T ewB̃δ = TvmB′δ dev ⊕dev TumB dev ⊕dev Tvm−1B′δ dev ⊕ . . .⊕dev Tv0B′δ. (68)
Recall that the fibered sum of two vector spaces W1, W2 with respect to linear
maps fi :Wi →W is the vector space
W1 f1 ⊕f2 W2 := {(w1, w2) ∈W1 ⊕W2 : f1(w1) = f2(w2)}.
Symplectic homology for autonomous Hamiltonians 54
If (W1, ‖ · ‖1), (W2, ‖ · ‖2) and W are normed vector spaces, and f1, f2 are
continuous linear maps, then W1 f1 ⊕f2 W2 is a closed subspace of W1⊕W2 and
inherits the norm ‖ · ‖1 + ‖ · ‖2 from W1 ⊕W2. In our case
dev : TvmB′δ =W 1,p,d ⊕ V
′ ⊕ V ′ → Tev(vm)Sγm
factors through the projection on V ′, and similarly for the other evaluation
maps. Therefore the above fibered sum only affects the summands V , V , V
V ′, so that T ewB̃δ is a subspace of codimension 2m in
TvmB′δ ⊕ TumB ⊕ Tvm−1B′δ ⊕ . . .⊕ Tv0B′δ.
As above, the norm on T ewB̃δ is induced from the ambient space. Recall that
the W 1,p-component has weight ed|s| for each TujB, weight ed||s|−si,δ| for each
TviB′δ, i = 1, . . . ,m − 1, weight ed|s+s0,δ | for i = 0 and weight ed|s−sm,δ| for
i = m, with si,δ as in the definition of gδ,ǫ.
The sections ∂̄H,J : B → E and ∂̄ : B′δ → E defined by (41) and (56) give rise
to a section over B̃δ. We denote its vertical differential by
D ew : T ewB̃δ → Lp,d(v∗mTŴ )⊕ Lp,d(u∗mTŴ )⊕ . . .⊕ Lp,d(v∗0TŴ ),
where
Lp,d(v∗i TŴ ) := L
p(R× S1, v∗i TŴ ; gδ,ǫi,vi(s)dsdθ),
Lp,d(u∗iTŴ ) := L
p(R× S1, u∗iTŴ ; gδ,ui(s)dsdθ).
Lemma 4.17. Let J ∈ Jreg(H) and {fγ} ∈ Freg(H, J). Let ǫ =(ǫ1, . . . , ǫm−1)
and let w̃ ∈ B̃δ be as in Proposition 4.15. The image of the operator D ew has codi-
mension m−1 and admits a complement spanned by sections ηi ∈ Lp,d(v∗i TŴ ),
i = 1, . . . ,m− 1 which are respectively supported in
[−(Ti + ǫi)/2δ,−(Ti + ǫi)/2δ + 1]× S1 ∪ [(Ti + ǫi)/2δ − 1, (Ti + ǫi)/2δ]× S1.
The operator D ew admits a right inverse Q ew defined on its image and whose
norm is uniformly bounded with respect to δ → 0.
Proof. We show that
imD ew = imD
⊕ imDum ⊕ imD′vm−1 ⊕ . . .⊕ imD
=: E. (69)
By definition we have imD ew ⊂ E. Let us now choose (xm, ym, . . . , x0) ∈ E and
x̃i and ỹj such that D
(x̃i) = xi, Duj (ỹj) = yj . We need to modify x̃i and ỹj
by elements lying in the kernels of the corresponding operators so that
dev(ỹj) = dev(x̃j), dev(ỹj) = dev(x̃j−1), j = 1, . . . ,m. (70)
Let us first assume m > 1. We have
TvmM′δ,1,−∞(Sγ , Sγ ;H, J)× TumMAm(Sγ , Sγm−1 ;H, J) = kerD′vm × kerDum
Symplectic homology for autonomous Hamiltonians 55
and, because {fγ} ∈ Freg(H, J), the map
(dev, dev) : kerD′vm × kerDum → Tev(vm)Sγ × Tev(um)Sγ
is transverse to the diagonal. We can therefore modify x̃m and ỹm so that
dev(ỹm) = dev(x̃m). Similarly the map
(dev, dev) : kerDu1 × kerD′v0 → Tev(u1)Sγ × Tev(v0)Sγ
is transverse to the diagonal and we can modify ỹ1, x̃0 in order to achieve
dev(ỹ1) = dev(x̃1). For i = 1, . . . ,m− 1 the maps
(dev, dev) : kerD′vi → Tev(vi)Sγi × Tev(vi)Sγi
are surjective and we can modify x̃i so that (70) is satisfied.
If m = 1 the regularity hypothesis on fγ ensures that the map
(dev, dev, dev, dev) : kerD′v1 × kerDu1 × kerD
→ Tev(v1)Sγ × Tev(u1)Sγ × Tev(u1)Sγ × Tev(v0)Sγ
is transverse to the product of the diagonals in the first two and in the last
two factors. We can therefore modify simultaneously x̃1, ỹ1, x̃0 in order to
achieve (70). Therefore (69) is proved. It then follows from Proposition 4.12
that the image of D ew has codimension m − 1 and is spanned by sections ηi ∈
Lp,d(v∗i TŴ ) supported in the desired intervals.
We now prove that D ew admits a uniformly bounded right inverse defined on
its image. We observe that D ew is the restriction to dom(D ew) of the direct sum of
operators D := D′vm ⊕Dum ⊕D
⊕· · ·⊕Du1 ⊕D′v0 . Let ζm, ζ0 be generators
of kerD′vm , kerD
and, for i = 1, . . . ,m− 1, let ζ1i , ζ2i be the basis of kerD′vi
constructed in Proposition 4.12. We denote by K the vector space spanned by
these 2m sections, viewed as elements of dom(D). Then dim K = 2m and K
is a complement of dom(D ew). Let P : dom(D) → dom(D ew) be the projection
parallel to K, let Quj , j = 1, . . . ,m be uniformly bounded right inverses forDuj ,
let Qvi , i = 0, . . . ,m be uniformly bounded right inverses for D
defined on
their images as in Proposition 4.12, and denote Q := Qvm ⊕Qum⊕Qvm−1 ⊕· · ·⊕
Qu1 ⊕Qu0 . Since K ⊂ kerD the operator P ◦Q : im(D) = im(D ew) → dom(D ew)
is a right inverse for D ew defined on its image, and we claim that its norm is
uniformly bounded for δ → 0. The norm of Q is uniformly bounded for δ → 0,
so that it is enough to prove that the norm of P is uniformly bounded for δ → 0.
The sections ζ0, ζm and ζ
i , ζ
i for i = 1, . . . ,m − 1 have the property that
their respective asymptotic values (obtained by applying dev and dev) are not
simultaneously zero. Moreover, the same is true for any linear combination of ζ1i
and ζ2i for i = 1, . . . ,m− 1. As a consequence, there exists a uniform constant
C > 0 such that, for any x = (xm, 0, xm−1, . . . , 0, x0) ∈ K, we have
‖x‖1,δ ≤ C
|dev(xm)|+ |dev(x0)|+
|dev(xi)|+ |dev(xi)|
. (71)
Symplectic homology for autonomous Hamiltonians 56
Given v ∈ dom(D) we have P (v) = v + w for some vector w ∈ K which is
uniquely determined by the asymptotic values of the components of v, and it
follows from (71) that
‖w‖1,δ ≤ C‖v‖1,δ.
We obtain
‖P (v)‖1,δ
‖v‖1,δ
‖v + w‖1,δ
‖v‖1,δ
≤ 1 + C,
so that the norm of P is uniformly bounded by 1+C. This proves the Lemma.
Proposition 4.18. Let J ∈ Jreg(H) and {fγ} ∈ Freg(H, J). Let w̃ ∈ B̃δ and
ǫ(δ) = (ǫ1(δ), . . . , ǫm−1(δ)) be as in Proposition 4.15. The operator
DGδ,ǫ( ew) : W
1,p(R× S1, Gδ,ǫ(w̃)∗TŴ ; gδ,ǫ(s)dsdθ)⊕ V
⊕ V ′v0
→ Lp(R× S1, Gδ,ǫ(w̃)∗TŴ ; gδ,ǫ(s)dsdθ)
is surjective and admits a right inverse Qδ = Qδ,ǫ, ew whose δ-norm is uniformly
bounded with respect to δ → 0.
Proof. Our proof is modelled on the proof of the gluing theorem for holomorphic
spheres by McDuff and Salamon [21, Ch. 10] . Let
vδm, u
m−1, . . . , u
be the extensions of v̂m, ûm, v̂m−1, . . . , û1, v̂0 to R × S1 defined by the same
formulas. Note that
uδj(s, θ) = uj(s, θ), s ∈ [−R+ 1, R− 1],
vδm(s, θ) = vm(s, θ), s /∈ [1/δ − 1, 1/δ],
vδ0(s, θ) = v0(s, θ), s /∈ [−1/δ,−1/δ+ 1]
and vδi (s, θ) = vi(s, θ) for s outside [−(Ti + ǫi)/2δ,−(Ti + ǫi)/2δ + 1] ∪ [(Ti +
ǫi)/2δ − 1, (Ti + ǫi)/2δ] and i = 1, . . . ,m− 1. The difference between vδi and vi
on the one hand, and that between uδj and uj on the other hand is exponentially
small as δ → 0. This implies that the operatorsDuδ
andD′
are surjective
for δ small enough and admit uniformly bounded right inverses, while the op-
erators D′
, i = 1, . . . ,m− 1 have a codimension one image with a supplement
spanned by a smooth section ηi supported in [−(Ti+ ǫi)/2δ,−(Ti+ ǫi)/2δ+1]×
S1 ∪ [(Ti + ǫi)/2δ − 1, (Ti + ǫi)/2δ]× S1, and admit uniformly bounded “right
inverses” defined on their image. It follows that the vertical differential D ewδ
satisfies the conclusions of Lemma 4.17, where w̃δ := (uδ1, . . . , u
0 , . . . , v
In particular, it admits a uniformly bounded right inverse defined on its image,
which we denote by Q ewδ (see [21, Lemma 10.6.1] for a similar statement in the
case of holomorphic spheres). This means that there exists a constant c0 > 0
such that
‖Q ewδx‖W 1,p,d ≤ c0‖x‖Lp,d
Symplectic homology for autonomous Hamiltonians 57
for all x ∈ imD ewδ and δ > 0.
We define an operator Tδ by the commutative diagram
T ewδ B̃δ
Lp,d(w̃δ∗TŴ )
dom(DGδ,ǫ( ew)) L
p(R× S1, Gδ,ǫ(w̃)∗TŴ ; gδ,ǫ(s)dsdθ)
where
Lp,d(w̃δ∗TŴ ) := Lp,d(vδm
TŴ )⊕ Lp,d(uδm
TŴ )⊕ . . .⊕ Lp,d(vδ0
TŴ ).
In the rest of the proof we shall omit the subscript ǫ from Gδ,ǫ and gδ,ǫ. An
element of Lp,d(w̃δ∗TŴ ) is denoted by
x = (xm, ym, . . . , x0).
The mixing map P , the splitting map S and the gluing map G are defined below,
and we shall prove that P, S,G are uniformly bounded with respect to δ → 0.
We shall also prove that Tδ is an approximate right inverse for DGδ( ew), i.e.
‖DGδ( ew)Tδη − η‖δ ≤
‖η‖δ (72)
for δ sufficiently small and η ∈ Lp(R × S1, Gδ(w̃)∗TŴ ; gδ(s)dsdθ). This im-
plies that DGδ( ew)Tδ is invertible (with the norm of its inverse bounded by 2),
and Tδ(DGδ( ew)Tδ)
−1 is a right inverse for DGδ( ew). Since P, S,G are uniformly
bounded, the norm of Tδ(DGδ( ew)Tδ)
−1 is bounded by a constant multiple of
‖Q ewδ‖, hence is uniformly bounded and the conclusion of the Proposition fol-
lows.
For every L > 0 we fix a smooth function
βL : R → [0, 1]
which vanishes for s ≤ 0, which is constant equal to 1 for s ≥ L and whose
derivative is bounded by 2/L. We moreover require that, for L large enough,
the function βL vanishes for s ≤ 1.
We define the mixing map P . Let
pi : L
p,d(w̃δ∗TŴ ) → imD′
, i = 0, . . . ,m
be the projection on Lp,d((vδi )
∗TŴ ) followed by the projection on imD′
allel to ηi. Recall the definition (59) of ℓi for i = 0, . . . ,m and let
qj : L
p,d(w̃δ∗TŴ ) → imDuδ
qj(x)(s, θ) := yj(s, θ)
+ β1(s− ℓj) ·
(1l− pj)(xj)
(s− ℓj, θ)
+ (1− β1(s− ℓj−1)) ·
(1l− pj−1)(xj−1)
(s− ℓj−1, θ)
Symplectic homology for autonomous Hamiltonians 58
for j = 1, . . . ,m. We define
P : Lp,d(w̃δ∗TŴ ) → imD ewδ
P := pm + qm + pm−1 + . . .+ q1 + p0.
The norm of P is uniformly bounded with respect to δ → 0 since the norm of
each pi is uniformly bounded by 1.
We define now the splitting map
S(η) := x = (xm, ym, . . . , x0).
We recall the definition (58) of the catenation shifts
0 = svm < sum < svm−1 < . . . < su1 < sv0 ,
and set
xm(s, θ) := β1(1/δ − s)η(s, θ),
x0(s, θ) := β1(1/δ + s)η(s+ sv0 , θ),
and, for i = 1, . . . ,m− 1, j = 1, . . . ,m,
yj(s, θ) :=
(1− β1(−R− s))η(s+ suj , θ), s ≤ 0,
(1− β1(−R+ s))η(s+ suj , θ), s ≥ 0,
xi(s, θ) :=
β1((Ti + ǫi)/2δ + s)η(s+ svi , θ), s ≤ 0,
β1((Ti + ǫi)/2δ − s)η(s+ svi , θ), s ≥ 0.
It follows from the definition that the norm of S is uniformly bounded by 1.
We define now the gluing map ζ := G(x̃), x̃ = (x̃m, ỹm, x̃m−1, . . . , x̃0) ∈
T ewδ B̃δ by “slowly interpolating” the components of x̃. For j = 1, . . . ,m, i =
1, . . . ,m− 1 we put
ζ(s, θ) :=
x̃m(s, θ), −∞ < s ≤ 1/δ −R/2,
ỹj(s− suj , θ), suj −R/2 ≤ s ≤ suj +R/2,
x̃i(s− svi , θ), svi − ℓi + 3R/2 ≤ s ≤ svi + ℓi − 3R/2,
x̃0(s− sv0 , θ), sv0 − 1/δ +R/2 ≤ s < +∞.
The above formula leaves out two types of intervals, on which the actual inter-
polation takes place (see Figure 6).
• If svj + ℓj − 3R/2 ≤ s ≤ suj −R/2 (interval of length R), we define
ζ(s, θ) := x̃j(+∞, θ)
+ (1− βR
(s− svj − ℓj +R))
x̃j(s− svj , θ)− x̃j(+∞, θ)
+ (1− βR
(−s+ suj −R))
ỹj(s− suj , θ)− ỹj(−∞, θ)
Symplectic homology for autonomous Hamiltonians 59
R 0 −R−R
x̃j−1 0 −Tj−1+ǫj−12δ
−Tj−1+ǫj−1
Tj+ǫj
Tj+ǫj
Figure 6: The gluing map G.
• If suj +R/2 ≤ s ≤ svj−1 − ℓj−1 + 3R/2 (interval of length R), we define
ζ(s, θ) := x̃j−1(−∞, θ)
+ (1−βR
(−s+svj−1−ℓj−1 +R))
x̃j−1(s− svj−1 , θ)− x̃j−1(−∞, θ)
+(1− βR
(s− suj −R))
ỹj(s− suj , θ)− ỹj(+∞, θ)
The section ζ is indeed of class W 1,p because
ỹj(−∞, θ) = x̃j(+∞, θ), ỹj(+∞, θ) = x̃j−1(−∞, θ).
That the norm of G is uniformly bounded with respect to δ → 0 follows directly
from the definition (68) of the norm on T ewδ B̃δ, as well as from the definition (61)
of the norm ‖ · ‖1,δ on dom(DGδ,ǫ( ew)) (see also Remark 4.14).
Let us now prove the estimate (72). On each of the intervals appearing
in (73) we have (DGδ( ew)Tδη)(s, θ) = η(s, θ) and we are therefore left to examine
intervals of the type [svj + ℓj − 3R/2, suj −R/2] and [suj +R/2, svj−1 − ℓj−1 +
3R/2]. We treat only the first case since the second one is entirely similar.
Upon applying the operator DGδ( ew) to ζ we obtain five types of terms as
following.
• DGδ( ew)x̃j(+∞, θ). Since x̃j(+∞, θ) does not depend on s we can view
DGδ( ew) as a family of operators on S
1. Then we have
‖DGδ( ew)x̃j(+∞, θ)‖δ = ‖(DGδ( ew) −Dvj(+∞,θ))x̃j(+∞, θ)‖δ
≤ ‖DGδ( ew) −Dvj(+∞,θ)‖δ‖x̃j(+∞, θ)‖
≤ C(δ)‖η‖δ.
Symplectic homology for autonomous Hamiltonians 60
Here Dvj(+∞,θ) denotes the linearized operator at the constant cylinder
vj(+∞, θ), the norm ‖x̃j(+∞, θ)‖ is induced from the (1-dimensional)
space V ′vj , and
C(δ) → 0, δ → 0.
This last statement and the last inequality follow from
‖DGδ( ew)−Dvj(+∞,θ)‖δ
≤ C(‖v̂j−vj(+∞, θ)‖L1,p,d([(Tj+ǫj)/2δ−R/2,(Tj+ǫj)/2δ]×S1)
+ ‖ûj−uj(−∞, θ)‖L1,p,d([−R,−R/2]×S1))
and the fact that the intervals of integration migrate to ±∞. The above
inequality makes crucial use of the fact that the weight gδ on the necks is
given by the exponential weight of the ambient spaces Bδ, B′δ. Moreover,
we have ‖x̃j(+∞, θ)‖ ≤ ‖x̃‖ ≤ C‖η‖δ because Q ew, P and S are uniformly
bounded with respect to δ.
• −β′
(s− svj − ℓj +R)
x̃j(s− svj , θ)− x̃j(+∞, θ)
, as well as β′
suj − R))
ỹj(s − suj , θ) − ỹj(−∞, θ)
. The δ-norm of each of these two
terms is bounded by C(δ)‖η‖δ, with C(δ) → 0 as δ → 0. To see this we
first use that |β′
| ≤ 4/R → 0, δ → 0. Secondly we use that ‖x̃j(s −
svj , θ) − x̃j(+∞, θ)‖ ≤ ‖x̃‖ ≤ C‖η‖δ and ‖ỹj(s − suj , θ) − ỹj(−∞, θ)‖ ≤
‖x̃‖ ≤ C‖η‖δ.
• (1 − βR/2(s− svj − ℓj + R))DGδ( ew)
x̃j(s− svj , θ)− x̃j(+∞, θ)
and (1−
βR/2(−s+suj−R))DGδ( ew)
ỹj(s−suj , θ)−ỹj(−∞, θ)
. The parts involving
x̃j(+∞, θ) = ỹj(−∞, θ) are bounded by C(δ)‖η‖δ as above. On the other
hand we write
DGδ( ew)x̃j = (DGδ( ew) −D ewδ)x̃j +D ewδ x̃j
and similarly for DGδ( ew)ỹj. The first term of such a sum is bounded by
C(δ)‖η‖δ as above, with C(δ) → 0, δ → 0. We are left with
(1− βR/2)D ewδ x̃j(s− svj , θ) + (1− βR/2)D ewδ ỹj(s− suj , θ)
(P ◦ S)vjη
(s− svj , θ) +
(P ◦ S)ujη
(s− suj , θ) = η.
Here we denote by (P ◦ S)vj , (P ◦ S)uj the components of P ◦ S in
Lp,d(vδj
TŴ ) and Lp,d(uδj
TŴ) respectively. The first equality uses the
fact that 1− βR/2 ≡ 1 on the support of (P ◦S)vjη and on the support of
(P ◦ S)ujη, as well as D ewδ ◦Q ewδ = 1l.
As a conclusion we have
‖DGδ( ew)Tδη − η‖δ ≤ C(δ)‖η‖δ, C(δ) → 0, δ → 0,
and the estimate (72) holds for δ small enough.
Symplectic homology for autonomous Hamiltonians 61
We shall use the following quantitative form of the implicit function theorem
from McDuff and Salamon [21, A.3.4].
Theorem 4.19. Let X and Y be Banach spaces, U ⊂ X be an open set, and
f : U → Y be a continuously differentiable map. Let x0 ∈ U be such that
D := df(x0) : X → Y is surjective and has a bounded right inverse Q : Y → X.
Choose positive constants ε and c such that ‖Q‖ ≤ c, Bε(x0) ⊂ U , and
‖x− x0‖ < ε =⇒ ‖df(x)−D‖ ≤ 1/2c. (74)
Then, for any x1 ∈ X satisfying
‖f(x1)‖ < ε/4c, ‖x1 − x0‖ < ε/8, (75)
there exists a unique x ∈ X such that
f(x) = 0, x− x1 ∈ imQ, ‖x− x0‖ ≤ ε. (76)
Moreover, ‖x− x1‖ ≤ 2c‖f(x1)‖.
The above theorem will be used within the following setup. Consider an
element [u] ∈ MA(p, q;H, {fγ}, J) and denote u0 := Gδ,ǫ(u). Given ε > 0 we
denote by Bε(0) the ball of radius ε centered at 0 inW
1,p(R×S1, u∗0TŴ ; ‖·‖1,δ),
where ‖ · ‖1,δ is defined by (61). For ζ ∈W 1,p(R× S1, u∗0TŴ ; ‖ · ‖1,δ) we write
ζ = ζ1 +
κjβ(−s+ suj )β(s− suj + 2R)XH
κjβ(s− suj )β(−s+ suj + 2R)XH
κiβ(s− svi + ℓi − 2R)β(−s+ svi + ℓi − 2R)ζi,δ(· − svi , ·)
with ℓi = R+ (Ti + ǫi)/2δ and ζi,δ the generator of ker Dvi whose value at 0 is
the vector field XH along γi. Then
‖ζ‖1,δ = ‖ζ1‖W 1,p(gδ,ǫ) +
(|κj |+ |κj |) +
|κi|.
We denote
ζ̃ := ζ1+
κjβ(−s+suj )β(s−suj+2R)XH+κjβ(s−suj )β(−s+suj+2R)XH
so that ζ̃(svi , ·) is L2-orthogonal to ζi,δ(0, ·). For each i = 1, . . . ,m − 1 we
consider the smooth cutoff function
ρi,δ,ǫ(s) := β(s− svi + ℓi − 2R)β(−s+ svi + ℓi − 2R),
Symplectic homology for autonomous Hamiltonians 62
so that ρi,δ,ǫ vanishes outside [svi − Ti+ǫi2δ , svi +
Ti+ǫi
] and ρi,δ,ǫ ≡ 1 on the
interval [svi − Ti+ǫi2δ + 1, svi +
Ti+ǫi
− 1].
We define ϕζ(u0) : R× S1 → Ŵ by
ϕζ(u0)(s, θ) :=
u0(s, θ), suj −R ≤ s ≤ suj +R,
ρi,δ,ǫ(s)κi
(u0(s, ·))(θ), svi − Ti+ǫi2δ ≤ s ≤ svi +
Ti+ǫi
Note that the last formula can also be written in the chart (ϑ, z) around Sγi as
ϑ ◦ϕζ(u0)(s, θ) = ϑ ◦ϕ
ρi,δ,ǫ(s)κi
(u0(s, 0))+ θ. Given a vector field ξ along u0 we
define the vector field ϕζ∗ξ along ϕζ(u0) by
ϕζ∗ξ(s, θ) :=
ξ(s, θ), suj −R ≤ s ≤ suj +R,
ρi,δ,ǫ(s)κi∗
(u0)ξ(s, θ), svi − Ti+ǫi2δ ≤ s ≤ svi +
Ti+ǫi
We define a map
Φ : Bε(0) → Bδ = B1,p,dδ (γp, γq, A;H, {fγ}) (77)
Φ(ζ) := expϕζ(u0)(ϕζ∗ζ̃).
Since ρi,δ,ǫ is precisely the coefficient of ζi,δ in our splitting for ζ, it follows that
dΦ(0) = Id. Hence, for ε > 0 small enough the map Φ is a diffeomorphism onto
its image, i.e. a chart.
We denote X := W 1,p(R × S1, u∗0TŴ ; ‖ · ‖1,δ), U := Bε(0) ⊂ X , Y :=
Lp(R × S1, u∗0TŴ ; gδ,ǫdsdθ), x0 = 0. For ε > 0 small enough the Banach
bundle E → Bδ can be trivialized over the image of Φ as Bε(0) × Y , and we
denote by f : Bε(0) → Y the section ∂̄Hδ,J ◦Φ read in this trivialization. Then
df(0) = Du0 is surjective and has a right inverse Qδ whose δ-norm is uniformly
bounded with respect to δ → 0 by Proposition 4.18. In order for the hypotheses
of Theorem 4.19 to be satisfied we need to check that (74) holds.
Lemma 4.20. There exists a constant C > 0 independent of δ such that, for
all x ∈ Bε(0), we have
‖df(x)− df(0)‖ ≤ C‖x‖1,δ.
Remark 4.21. The motivation for introducing the chart Φ is that we must
use the “compensated” norm ‖ · ‖1,δ. The lemma would fail if one used the
usual exponential chart ζ 7→ expu0(ζ) instead of Φ, because the estimate for the
expression (82) in the proof below would not hold.
Proof. We need to prove the existence of a uniform constant C > 0 such that
‖D(∂̄Hδ,J ◦ Φ)(x) · ζ −D(∂̄Hδ ,J ◦ Φ)(0) · ζ‖δ ≤ C‖x‖1,δ‖ζ‖1,δ (78)
Symplectic homology for autonomous Hamiltonians 63
for all ζ ∈ X . We recall the decomposition ζ = ζ̃ +
i=1 κiρi,δ,ǫζi,δ, which
satisfies ‖ζ‖1,δ = ‖ζ̃‖1,δ +
i=1 |κi|. It is therefore enough to prove (78) sep-
arately for ζ = ζ̃ and for ζ = ρi,δ,ǫζi,δ, i = 1, . . . ,m− 1. We abbreviate in the
following computations ∂̄ = ∂̄Hδ,J .
We first assume ζ = ρi,δ,ǫζi,δ. Given x = x̃+
j=1 κjρj,δ,ǫζj,δ we have
D(∂̄ ◦ Φ)(x)ζ −D(∂̄ ◦ Φ)(0)ζ
= D(∂̄ ◦ Φ)(x)ζ −D(∂̄ ◦ Φ)(
κjρj,δ,ǫζj,δ)ζ (79)
+D(∂̄ ◦ Φ)(
κjρj,δ,ǫζj,δ)ζ −D(∂̄ ◦ Φ)(0)ζ. (80)
The term (79) is further equal to
∂̄(expϕx+tζ(u0)(ϕx+tζ∗x̃))−
∂̄(expϕx+tζ(u0)(0))
= Dexpϕx(u0)(ϕx∗ex)
·D2 expϕx(u0)(ϕx∗x̃) · ∇tϕx+tζ∗x̃
+ Dexpϕx(u0)(ϕx∗ex)
·D1 expϕx(u0)(ϕx∗x̃) · ρi,δ,ǫ∇fγi(ϕx(u0))
− Dϕx(u0) ·D1 expϕx(u0)(0) · ρi,δ,ǫ∇fγi(ϕx(u0))
= Dexpϕx(u0)(ϕx∗ex)
·D2 expϕx(u0)(ϕx∗x̃) · ∇tϕx+tζ∗x̃ (81)
+ Dexpϕx(u0)(ϕx∗ex)
· (D1 expϕx(u0)(ϕx∗x̃)− T ·D1 expϕx(u0)(0))
·ρi,δ,ǫ∇fγi(ϕx(u0))
+ (Dexpϕx(u0)(ϕx∗ex)
· T −Dϕx(u0)) ·D1 expϕx(u0)(0) · ρi,δ,ǫ∇fγi(ϕx(u0)).
Here T is the parallel transport in Ŵ along the geodesic τ 7→ expϕx(u0)(τϕx∗x̃),
τ ∈ [0, 1], and we have ρi,δ,ǫ∇fγi(ϕx(u0)) = ρi,δ,ǫ(ϕ
ρi,δ,ǫκi)∗ζi,δ.
We study the first term in (81). We have pointwise bounds
|∇tϕx+tζ∗x̃| ≤ C(1 + |κi|)|x̃|,
|∇∇tϕx+tζ∗x̃| ≤ C(1 + |κi|)(|x̃|+ |∇x̃|)
for some universal constant C > 0. In particular
‖∇tϕx+tζ∗x̃‖W 1,p(gδ,ǫ) ≤ C‖x̃‖1,δ
if |κi| ≤ ‖x‖1,δ ≤ ε, with C > 0 a universal constant. On the other hand the
operators D2 expϕx(u0)(ϕx∗x̃) : W
1,p(gδ,ǫ) → W 1,p(gδ,ǫ) and Dexpϕx(u0)(ϕx∗ex) :
W 1,p(gδ,ǫ) → Lp(gδ,ǫ) are uniformly bounded if ‖x‖∞ ≤ C‖x‖1,δ ≤ Cε (we use
here the Sobolev inequality). This implies that the δ-norm of the first term
in (81) is bounded by a constant multiple of ‖x̃‖1,δ.
We now study the second term in (81). Let ‖| · ‖| be the operator norm for
continuous linear maps
W 1,p(ϕx(u0)
∗TŴ ; ‖ · ‖1,δ) →W 1,p(expϕx(u0)(ϕx∗x̃)
∗TŴ ; gδ,ǫdsdθ).
Symplectic homology for autonomous Hamiltonians 64
We claim that ‖|D1 expϕx(u0)(ϕx∗x̃)− T ·D1 expϕx(u0)(0)‖| ≤ C‖x̃‖1,δ for some
uniform constant C > 0, provided ‖x‖1,δ ≤ ε. Indeed, since the metric on Ŵ
varies smoothly, for any ξ = ξ̃ +
ℓ=1 κ
ℓρℓ,δ,ǫ∇fγℓ(ϕx(u0)) we have pointwise
bounds
∣∣(D1 expϕx(u0)(ϕx∗x̃)− T ·D1 expϕx(u0)(0))ξ̃
∣∣ ≤ C|x̃||ξ̃|,
∣∣∇(D1 expϕx(u0)(ϕx∗x̃)− T ·D1 expϕx(u0)(0))ξ̃
∣∣ ≤ C(|∇x̃||ξ̃|+ |x̃||∇ξ̃|),
∣∣(D1 expϕx(u0)(ϕx∗x̃)− T ·D1 expϕx(u0)(0))ρℓ,δ,ǫ∇fγℓ(ϕx(u0))
∣∣ ≤ C|x̃|,
∣∣∇(D1 expϕx(u0)(ϕx∗x̃)−T ·D1 expϕx(u0)(0))ρℓ,δ,ǫ∇fγℓ(ϕx(u0))
∣∣ ≤ C(|x̃|+|∇x̃|).
The claim then follows by integration with respect to the weight gδ,ǫ and by
using the Sobolev inequalities ‖x̃‖L∞ ≤ C‖x̃‖1,δ and ‖ξ̃‖L∞ ≤ C‖ξ̃‖1,δ. On the
other hand, as already seen above, the operator Dexpϕx(u0)(ϕx∗ex)
acting from
the space W 1,p(gδ,ǫ) to L
p(gδ,ǫ) is uniformly bounded for ‖x‖1,δ ≤ ε, since its
coefficients are bounded. We infer that the δ-norm of the second term in (81)
is bounded by a constant multiple of ‖x̃‖1,δ.
We finally study the third term in (81). We claim that ‖Dexpϕx(u0)(ϕx∗ex) ·
T −Dϕx(u0)‖ ≤ C‖x̃‖1,δ for some uniform constant C > 0, provided ‖x‖1,δ ≤ ε.
This follows from the pointwise bounds
∣∣(Dexpϕx(u0)(ϕx∗ex) · T −Dϕx(u0))ξ̃
∣∣ ≤ C|x̃|(|ξ̃|+ |∇ξ̃|),
∣∣(Dexpϕx(u0)(ϕx∗ex) · T −Dϕx(u0))ρℓ,δ,ǫ∇fγℓ(ϕx(u0))
∣∣ ≤ C|x̃|
by integrating with respect to the weight gδ,ǫ and by using the previous Sobolev
inequalities. Since D1 expϕx(u0)(0) = Id, we infer that the δ-norm of the third
term in (81) is bounded by a constant multiple of ‖x̃‖1,δ.
As a conclusion, the δ-norm of the expression in (79) is bounded by a con-
stant multiple of ‖x̃‖1,δ.
We now consider the expression in (80), which can be written as
D(∂̄ ◦ Φ)(κiρi,δ,ǫζi,δ)ζ −D(∂̄ ◦ Φ)(0)ζ (82)
∂̄(ϕκiζ+tζ(u0))−
∂̄(ϕtζ(u0))
∂̄(u0(·+ (κi + t)ρi,δ,ǫ, ·))−
∂̄(u0(·+ tρi,δ,ǫ, ·)).
Each term in the above difference is supported in the intervals [svi − Ti+ǫi2δ , svi −
Ti+ǫi
+1] and [svi+
Ti+ǫi
−1, svi+ Ti+ǫi2δ ]. Moreover, their difference is pointwise
bounded by C|κi| for some uniform constant C > 0. Since the weight gδ,ǫ is
uniformly bounded on the above intervals of length 1, we infer that the δ-norm
of the expression in (80) is bounded by C|κi|, hence by C‖x‖1,δ for some uniform
constant C > 0.
Symplectic homology for autonomous Hamiltonians 65
We now assume ζ = ζ̃ and we again decompose D(∂̄ ◦Φ)(x)ζ −D(∂̄ ◦Φ)(0)ζ
as the sum of the expressions in (79) and (80).
The expression in (79) can be written
∂̄(expϕx(u0)(ϕx∗(x̃ + tζ̃)))−
∂̄(expϕx(u0)(ϕx∗tζ̃))
= Dexpϕx(u0)(ϕx∗ex)
·D2 expϕx(u0)(ϕx∗x̃) · ϕx∗ζ̃
− Dϕx(u0) ·D2 expϕx(u0)(0) · ϕx∗ζ̃
= Dexpϕx(u0)(ϕx∗ex)
· (D2 expϕx(u0)(ϕx∗x̃)− T ·D2 expϕx(u0)(0)) · ϕx∗ζ̃
+ (Dexpϕx(u0)(ϕx∗ex)
· T −Dϕx(u0)) ·D2 expϕx(u0)(0) · ϕx∗ζ̃. (83)
Here T denotes the same parallel transport map as above.
We claim that the δ-norm of the first term in the expression (83) is bounded
by C‖x̃‖1,δ‖ζ̃‖1,δ when ‖x‖1,δ ≤ ε, for some uniform constant C > 0. We have
the pointwise estimates
|ϕx∗ζ̃| ≤ C(1 +
|κj |)|ζ̃|,
|∇ϕx∗ζ̃| ≤ C(1 +
|κj |)(|ζ̃|+ |∇ζ̃|),
which imply ‖ϕx∗ζ̃‖W 1,p(gδ,ǫ) ≤ C‖ζ̃‖1,δ for some uniform constant C > 0,
provided ‖x‖1,δ ≤ ε. On the other hand, the pointwise estimates
∣∣(D2 expϕx(u0)(ϕx∗x̃)− T ·D2 expϕx(u0)(0))ξ
∣∣ ≤ C|x̃||ξ|,
∣∣∇(D2 expϕx(u0)(ϕx∗x̃)− T ·D2 expϕx(u0)(0))ξ
∣∣ ≤ C(|∇x̃||ξ|+ |x̃||∇ξ|)
show that the norm of the operator D2 expϕx(u0)(ϕx∗x̃) − T · D2 expϕx(u0)(0)
acting from W 1,p(gδ,ǫ) to itself is bounded by C‖x̃‖1,δ. Finally, we have already
seen that the operator Dexpϕx(u0)(ϕx∗ex)
acting between W 1,p(gδ,ǫ) and L
p(gδ,ǫ)
is uniformly bounded, and the claim follows.
We now claim that the δ-norm of the second term in the expression (83)
is also bounded by C‖x̃‖1,δ‖ζ̃‖1,δ when ‖x‖1,δ ≤ ε, for some uniform constant
C > 0. We have the pointwise estimate
∣∣(Dexpϕx(u0)(ϕx∗ex) · T −Dϕx(u0))ξ
∣∣ ≤ C|x̃|(|ξ|+ |∇ξ|),
which implies that the norm of the operator Dexpϕx(u0)(ϕx∗ex)
·T −Dϕx(u0) acting
from W 1,p(gδ,ǫ) to L
p(gδ,ǫ) is bounded by C‖x̃‖1,δ for some uniform constant
C > 0. Since ‖ϕx∗ζ̃‖W 1,p(gδ,ǫ) ≤ C‖ζ̃‖1,δ and D2 expϕx(u0)(0) = Id, the claim
follows.
Symplectic homology for autonomous Hamiltonians 66
We finally study the term (80) in the decomposition of D(∂̄ ◦Φ)(x)ζ−D(∂̄ ◦
Φ)(0)ζ, which can be written
∂̄(expϕx(u0) ϕx∗tζ̃)−
∂̄(expu0 tζ̃)
= Dϕx(u0) ·D2 expϕx(u0)(0) · ϕx∗ζ̃ −Du0 ·D2 expu0(0) · ζ̃
= Dϕx(u0) · ϕx∗ζ̃ −Du0 · ζ̃.
This last expression is pointwise bounded by C(
j=1 |κj |)(|ζ̃| + |∇ζ̃|), which
implies that its δ-norm is bounded by C‖x‖1,δ‖ζ̃‖1,δ for some uniform constant
C > 0.
This proves the lemma.
Proposition 4.22. Let [u] ∈ MA(p, q;H, {fγ}, J). There exists δ1 > 0 and a
one-parameter family [uδ] ∈ MA(γp, γq;Hδ, J), 0 < δ < δ1 such that
[uδ] → [u], δ → 0.
Here convergence is understood in the sense of Definition 4.2. Moreover, if
dim MA(p, q;H, {fγ}, J) = 0 then the intermediate gradient fragments in [u]
are nonconstant and the above one-parameter family is unique.
Remark 4.23. The fact that the intermediate gradient fragments in [u] are
nonconstant is the reason why we had to prove the gluing theorem only in the
case where the intermediate lengths of gradient trajectories are strictly positive:
Ti > 0, i = 1, . . . ,m− 1, where m is the number of sublevels in [u].
Proof. We choose a representative u = (cm, um, . . . , u1, c0) of [u] and we apply
Theorem 4.19 in a chart of Bδ as above. By Proposition 4.18 the operator D
admits a right inverse Qδ which is uniformly bounded with respect to δ by
some constant c. By Lemma 4.20 there exists ε > 0 independent of δ such that
condition (74) is satisfied. We set x0 := Gδ(u). By Proposition 4.15 we have
‖f(x0)‖ = 0
and therefore condition (75) is satisfied on some open neighbourhood of x0 if δ
is small enough. Taking x1 := x0 in the statement of Theorem 4.19 provides us
with an element x ∈ X satisfying (76). We set
uδ := x.
Then [uδ] ∈ MA(γp, γq;Hδ, J). Because ‖x − x0‖ ≤ 2c‖f(x0)‖ → 0 and x0 =
Gδ(u) → u by construction, we infer by Proposition 4.16 that [uδ] → [u], δ → 0.
We now assume that the dimension of MA(p, q;H, {fγ}, J) is zero. We have
dimMA(γp, γq;Hδ, J) =
= µ(γp)− µ(γq) + 2〈c1(TW ), A〉 − 1
= µ(γ)− µ(γ) + 2〈c1(TW ), A〉 − 1 + ind(p)− ind(q),
Symplectic homology for autonomous Hamiltonians 67
hence µ(γ)−µ(γ)+2〈c1(TW ), A〉 = 1− ind(p)+ ind(q) ≤ 2. On the other hand
µ(γ) − µ(γ) + 2〈c1(TW ), A〉 =
i=1 µ(γi) − µ(γi−1) + 2〈c1(TW ), Ai〉, where
m ≥ 0 is the number of sublevels of u. Each of the summands is nonnegative
by transversality, and the only possibilities occuring are the following:
(i) each summand is zero, which means that all the Floer trajectories involved
in u are rigid;
(ii) one of the summands is 1 and the others vanish. Since [u] is rigid, the
only nonrigid summand must be u0 or um, while c0, respectively cm have
to be constant.;
(iii) two of the summands are 1, and the others vanish. As above, the nonrigid
summands must be u0 and um, while c0 and cm are constant;
(iv) one of the summands is 2 and the others vanish. Since [u] is rigid we must
have m = 1 and c0, c1 have to be constant.
In each of the cases (i-iii) the intermediate gradient trajectories have to be
nonconstant by transversality of the evaluation maps (34).
Let now [ṽδ] → [u], δ → 0. Since the only possible intermediate gradient
trajectory in [u] is nonconstant, we can apply Proposition 4.16. We obtain
representatives vδ ∈ [ṽδ], v ∈ [u] and functions ǫ = ǫ(δ) = (ǫ1(δ), . . . , ǫm−1(δ))
such that vδ, Gδ,ǫ(v) belong to some ‖ · ‖1,δ,ǫ -chart in Bδ and
‖vδ −Gδ,ǫ(v)‖1,δ,ǫ → 0, δ → 0. (84)
We have to prove that uδ and vδ differ by a shift for δ > 0 sufficiently small.
Let us choose a continuous path vt ∈ M̂A(p, q;H, {fγ}, J), t ∈ [0, 1] with
v0 = u, v1 = v. We denote yt = yt(δ) := Gδ,tǫ(vt) and ‖ · ‖t := ‖ · ‖1,δ,tǫ. Note
that, for each δ > 0, there exists a continuous function Cδ : [0, 1]× [0, 1] → R+
such that
Cδ(t, t′)
‖ · ‖t′ ≤ ‖ · ‖t ≤ Cδ(t, t′)‖ · ‖t′ , t, t′ ∈ [0, 1]
satisfying Cδ(t, t) = 1, t ∈ [0, 1].
As yt(δ) and Dyt(δ) vary continuously with t and δ, we can choose ε > 0 and
c > 0 so that the hypotheses of the implicit function theorem 4.19 are satisfied
for each yt(δ), t ∈ [0, 1], 0 < δ < δ1 and some suitable constant δ1 > 0. After
further shrinking δ1 we can also assume that ‖f(yt)‖t < ε/4c for all t ∈ [0, 1]
and 0 < δ < δ1. Finally, in view of (84), for some smaller δ1 we can achieve
‖vδ − y1(δ)‖1 ≤ ε.
We define
Iδ := {t ∈ [0, 1] : ∃x ∈ [vδ], ‖x− yt(δ)‖t ≤ ε}, 0 < δ < δ1.
We prove that Iδ = [0, 1] by showing that it is a nonempty open and closed
subset of [0, 1]. Note that Iδ is nonempty since 1 ∈ Iδ. We prove now that Iδ
Symplectic homology for autonomous Hamiltonians 68
is closed. Assume that tn ∈ Iδ is such that tn → t. Let xn ∈ [vδ] be such that
‖xn−ytn(δ)‖tn ≤ ε. By the triangular inequality we see that ‖xn−yt(δ)‖t stays
bounded, hence the sequence of shifts defining xn is also bounded and, up to a
subsequence, we may assume that xn → x ∈ [vδ]. Then
‖xn − yt(δ)‖t ≤ Cδ(t, tn)‖xn − yt(δ)‖tn
≤ Cδ(t, tn)(‖xn − ytn(δ)‖tn + ‖ytn(δ)− yt(δ)‖tn)
≤ Cδ(t, tn)(ε+ ‖ytn(δ)− yt(δ)‖tn).
We pass to the limit n→ ∞ and obtain ‖x−yt(δ)‖t ≤ ε, hence t ∈ Iδ. We prove
now that Iδ is open. Let t ∈ Iδ and choose an open interval J containing t such
that Cδ(t
′, t) < 2 and ‖yt′(δ)−yt(δ)‖t < ε/8 for all t′ ∈ J . Theorem 4.19 applied
to x0 := yt(δ) and x1 := yt′(δ) yields x such that f(x) = 0 and ‖x−yt(δ)‖t ≤ ε.
The uniqueness statement in the implicit function theorem ensures that the
intersection of the space of solutions with the ‖ · ‖t-ball of radius ε centered
at x0 is a graph over ker D. Since dim ker D = 1 and since translation in
the s-variable already provides a 1-parameter family of solutions, we infer that
x ∈ [vδ]. Moreover, the last statement in Theorem 4.19 gives ‖x−yt′(δ)‖t ≤ ε/2.
Then ‖x− yt′(δ)‖t′ ≤ Cδ(t′, t)ε/2 < ε, so that t′ ∈ Iδ and Iδ is open.
The upshot is that there exists x ∈ [vδ] such that ‖x−y0(δ)‖0 ≤ ε, 0 < δ < δ1.
But y0(δ) = Gδ(u) and, again by the uniqueness statement in the implicit
function theorem, we get that x and uδ differ by a shift. Hence [uδ] = [vδ].
Proof of Theorem 3.7. We first prove (i) and show the existence of δ1. As-
sume by contradiction that there exists a sequence δn → 0 and Floer trajec-
tories vn ∈ M̂A(γp, γq;Hδn , J) such that J is not regular for vn. By Propo-
sition 4.7 we may assume, up to shifting and passing to a subsequence, that
vn → u ∈ M̂A(p, q;H, {fγ}, J). As seen in the proof of Proposition 4.22, the
limit u has nonconstant intermediate gradient trajectories since J is regular for
u. We can therefore apply Proposition 4.16 and get parameters ǫn and vector
fields ζn such that vn = expGδn,ǫn (u)
(ζn) and ‖ζn‖1,δn,ǫn → 0. By Proposi-
tion 4.18 the operator DGδn,ǫn (u) is surjective and admits a right inverse which
is uniformly bounded with respect to δ. We infer that the operator Dvn is also
surjective for n large enough, a contradiction.
Let us prove (ii). Let (δ, vδ) ∈ M̂A]0,δ1[(γp, γq;H, {fγ}, J) and let I(δ) ⊂]0, δ1[
be a small relatively compact open interval containing δ. Since the norms ‖·‖1,δ′
are equivalent for δ′ ∈ I(δ), the space BI(δ) :=
δ′∈I(δ){δ′} × Bδ′ is a Banach
manifold. Similarly, there is a Banach vector bundle EI(δ) → BI(δ) endowed with
an obvious section ∂̄HI(δ),J whose restriction to Bδ′ is ∂̄Hδ′ ,J . The restriction
of its linearization D(δ,vδ) at (δ, vδ) to TvδBδ is the surjective operator Dvδ
of index 1, hence D(δ,vδ) is surjective and has index 2. Therefore ker D(δ,vδ)
projects surjectively onto TδI(δ) = R and the projection in (ii) is a submersion.
Symplectic homology for autonomous Hamiltonians 69
We now prove (iii). Let us note that, by Proposition 4.22, we have a map
MA(p, q;H, {fγ}, J) → π0(MA]0,δ1[(γp, γp;H, {fγ}, J)),
[u] 7→ C[u] :=
δ∈]0,δ1[
{(δ, [uδ])},
where [uδ] is the uniquely defined one-parameter family of Proposition 4.22 such
that [uδ] → [u]. This map is injective because the limit of such a family [uδ]
as δ → 0 is unique. In order to prove surjectivity, let C = {(δ, [vδ])} be a
connected component of MA
]0,δ1[
(γp, γp;H, {fγ}, J). By Proposition 4.7 there
exists a sequence δn → 0 and [u] ∈ MA(p, q;H, {fγ}, J) such that [vδn ] → [u].
By the uniqueness statement in Proposition 4.22 we get that C = C[u].
4.4 Coherent orientations
The structure of this section is as follows. We first present the construction
of coherent orientations in the usual Floer setting for (Hδ, J) by adopting the
point of view of [5]. We construct coherent orientations on the moduli spaces of
Morse-Bott trajectories, out of which we get orientations on the space of Morse-
Bott trajectories with gradient fragments. Finally, we prove Proposition 3.9.
We denote S1 := R/Z and, for a path of symmetric matrices S : S1 →
M2n(R), we denote by ΨS the unique solution of the Cauchy problem
Ψ̇(θ) = J0S(θ)Ψ(θ), Ψ(0) = 1l, θ ∈ [0, 1], (85)
where J0 is the standard complex structure on R
2n. Then ΨS is a path of
symplectic matrices and we denote
S := {S : S1 →M2n(R) : tS = S and det(1l−ΨS(1)) 6= 0}.
Let us denote by E a symplectic vector bundle of rank 2n over CP 1, or
R×S1, or C, with fixed trivializations in neighbourhoods of infinity in the case
of R× S1 and C. We denote by
O(CP 1, E)
the space of linear operators D : W 1,p(CP 1, E) → Lp(CP 1,Λ0,1E) of the form
(∂x + J0∂y + S(z))dz̄ in a local trivialization of E, where z = x + iy is a local
coordinate on CP 1. Given S, S ∈ S we denote by
O(R× S1, E;S, S)
the space of linear operators D : W 1,p(R × S1, E) → Lp(R × S1,Λ0,1E) of the
form (∂s + J0∂θ + S(s, θ))(ds − idθ) in a local trivialization of E, such that
lims→−∞ S(s, ·) = S and lims→∞ S(s, ·) = S in the given trivializations of E.
Given S0 ∈ S we denote by
O±(C, E;S0)
Symplectic homology for autonomous Hamiltonians 70
O+ O+
O(R× S1)O−O+O(CP1)
Figure 7: The four possibilities of gluing (O = O(R × S1)).
the space of linear operators D :W 1,p(C, E) → Lp(C,Λ0,1E) of the form (∂x +
J0∂y + S(z))dz̄ in a local trivialization of E and such that, when expressed in
holomorphic cylindrical coordinates (s, θ) with e±2π(s+iθ) = z as (∂s + J0∂θ +
S(s, θ))(ds − idθ), we have lims→±∞ S(s, θ) = S0(θ) in the given trivialization
of E. Intuitively, the space O+ corresponds to the sphere with one positive
puncture, while O− corresponds to the sphere with one negative puncture.
It is a standard fact in the literature that each of the above spaces O is
contractible and consists of Fredholm operators. Moreover, they each come
equipped with a canonical real line bundle Det(O) whose fiber atD is Det(D) :=
(Λmax ker D)⊗ (ΛmaxcokerD)∗. Each of the bundles Det(O) is trivial since the
base is contractible.
We now define gluing operations between elements of the above spaces (see
Figure 7). Let K ∈ O+(C, E;S0) or K ∈ O(R × S1, E;S, S0), and L ∈
O−(C, F ;S0) or L ∈ O(R × S1, F ;S0, S). Let us choose a cutoff function
β : R → [0, 1] such that β(s) = 0 if s ≤ 0 and β(s) = 1 if s ≥ 1. Given
R > 0 large we define operators KR and LR by replacing S in the asymptotic
expressions ofK and L by S0+β(R−s)(S−S0) and S0+β(R+s)(S−S0) respec-
tively. We cut out semi-infinite cylinders {s > R} from the base of E, {s < −R}
from the base of F , then identify their boundaries using the coordinate θ. We
glue the vector bundles E and F using their given trivializations near infinity
and denote the resulting vector bundle by E#F . We define K#RL by concate-
nating KR and LR, so that K#RL belongs to one of the spaces O(CP 1, E#F ),
O+(C, E#F ;S), O−(C, E#F ;S), or O(R× S1, E#F ;S, S).
Following [5, Corollary 7], for R large enough there is a natural isomor-
phism Det(K)⊗Det(L) ∼→ Det(K#RL) defined up to homotopy. In particular,
given orientations oK of Det(K) and oL of Det(L), we induce a canonical ori-
entation oK#oL of Det(K#RL). Moreover, this operation on orientations is
associative [13, Theorem 10].
We describe now, following [5], a procedure for constructing orientations on
Symplectic homology for autonomous Hamiltonians 71
the spaces O(R × S1, E;S, S) which are coherent with respect to the gluing
operation, in the sense of [13, Definition 11]. We denote by θn a trivial sym-
plectic vector bundle of rank 2n. We first note that each determinant bundle
Det(O(CP 1, E)) is naturally oriented since O(CP 1, E) contains the connected
space of complex linear operators and the latter have kernels and cokernels
which are canonically oriented as complex vector spaces. We now choose arbi-
trary orientations of the determinant bundles Det(O+(C, E;S0)) such that the
trivialization of E at infinity extends to C.
Remark 4.24. Note that, if S0 commutes with J0, the set of C-linear operators
in O+(C, E;S0) forms a nonempty convex set, hence Det(O+(C, E;S0)) has a
canonical orientation.
We induce orientations on the determinant bundles Det(O−(C, E;S0)) such that
the trivialization of E at infinity extends to C by requiring that the orientation
induced by gluing on Det(O(CP 1, θn)) is the canonical one. Finally, we induce
orientations on Det(O(R×S1, E;S, S)) by requiring that the orientation induced
on Det(O(CP 1, θn#E#θn)) by the gluing operation
O+(C, θn;S)×O(R× S1, E;S, S)×O−(C, θn;S) → O(CP 1, θn#E#θn)
is the canonical one. It is proved in [5] that this defines a system of coherent
orientations.
The general procedure for inducing orientations of the spaces of Floer trajec-
tories M̂A(γp, γq;Hδ, J) out of a system of coherent orientations goes as follows.
Let Ψp, Ψq denote the linearizations of the Hamiltonian flow of Hδ along γp, γq
in their fixed respective trivializations and let Sp, Sq ∈ S be the corresponding
paths of symmetric matrices as in (85). Let E be a symplectic vector bundle
over R × S1 with fixed trivializations at infinity and relative first Chern class
equal to 〈c1(TŴ ), A〉. For each u ∈ M̂A(γp, γq;Hδ, J) there is an isomorphism
of symplectic vector bundles Φu : u
∼→ E, chosen to depend continuously
on u. There is a map
M̂A(γp, γq;Hδ, J) → O(R× S
1, E;Sp, Sq), u 7→ Φu ◦ D̃u ◦Φ−1u ,
where D̃u has the same analytical expression as the linearized operator Du :
W 1,p(R×S1, u∗TŴ ; ed|s|ds dθ)⊕V u⊕V u → Lp(R×S1, u∗TŴ ; ed|s|ds dθ) con-
sidered in Section 4.3. Under the assumption that J is regular and because of
elliptic regularity, the operators D̃u and Du have the same kernel, consisting of
smooth elements. Hence their determinant lines are naturally isomorphic. It fol-
lows that the pull-back of Det(O) under the above map is naturally isomorphic
to ΛmaxTM̂ = Λmax ker Du, and we get an orientation on M̂.
If the dimension is one, the space M̂ has a canonical orientation given at each
point u by the vector field ∂su. Comparing this with the orientation constructed
above associates to each connected component [u] of M̂ a sign ǫ(u).
Symplectic homology for autonomous Hamiltonians 72
Lemma 4.25. Let S1 ∈ S and define Sm(θ) := S1(mθ). Assume Sm ∈ S and
define an automorphism φm of O+(C, E;Sm) by conjugation with the map z 7→
e2iπ/mz. Then φm is orientation reversing for Det(O+(C, E;Sm)) if and only if
m is even and the difference of Conley-Zehnder indices µCZ(ΨSm)− µCZ(ΨS1)
is odd.
Proof. We start by explaining how φm acts on the orientations of the deter-
minant bundle. The operators K ∈ O+(C, E;Sm) which are invariant under
conjugation by φm, i.e. K(ζ ◦φm) ◦φ−1m = K(ζ) for all ζ, form a convex and, in
particular, connected set. Since φm acts on ker K and cokerK, it also acts on
Det(K) and the induced action on orientations extends to Det(O+(C, E;Sm)).
There is a bijective correspondence between operators K1 ∈ O+(C, E;S1)
and operators K ∈ O+(C, E;Sm) which are invariant under conjugation by φm,
in which case the pull-back of ker K1 under z 7→ zm is the 1-eigenspace of φm
acting on ker K. Since ker K splits as a direct sum of eigenspaces corresponding
to them-th roots of unity and since imaginary roots give rise to even-dimensional
eigenspaces, we infer that the dimension of the −1-eigenspace has the parity of
dimker K − dimker K1. This fact is relevant in our situation since φm reverses
the orientation of ker K if and only if this dimension is odd. Similarly, φm
reverses the orientation of cokerK if and only if dim cokerK − dim cokerK1
is odd. As a conclusion, φm reverses the orientation of Det(K) if and only if
ind(K)− ind(K1) = µCZ(ΨSm)− µCZ(ΨS1) is odd. This can happen of course
only if −1 is an m-th root of unity, i.e. m is even.
Remark 4.26. The proof of Lemma 4.25 shows that, if m is odd, the difference
of Conley-Zehnder indices is automatically even.
Lemma 4.27. Let S1 ∈ S, define Sm(θ) := S1(mθ) and assume Sm ∈ S. Let
T ∈ O(R× S1, θn;Sm, Sm) be an element of the form
T := ∂s + J0∂θ + Sm(θ − β(s)/m),
with β : R → [0, 1] a smooth function satisfying β(s) = 0 near −∞, β(s) = 1
near +∞ and with derivative uniformly bounded by some small constant c. We
denote by O one of the spaces O+(C, E;Sm) or O(R × S1, E;S, Sm), S ∈ S.
The family ψ = {ψR}, R > 0 of automorphisms of O defined by
ψR(D) := D#RT
induces an action on the orientations of Det(O) which is reversing if and only if
m is even and the difference of Conley-Zehnder indices µCZ(ΨSm)− µCZ(ΨS1)
is odd.
Proof. Note that T is an isomorphism if c is small enough, by the same argument
as the one for D′′u in the proof of Proposition 4.9.
We now explain what is the action of ψ on the orientations of Det(O).
Let D ∈ O and let V ⊂ Lp be a finite dimensional vector space spanned by
Symplectic homology for autonomous Hamiltonians 73
smooth sections with compact support, such that V + imD = Lp. We define
the stabilization of D by V as
DV : V ⊕W 1,p → Lp, (v, ζ) 7→ v +Dζ.
ThenDV is a surjective Fredholm operator and there is a canonical isomorphism
Det(D) ≃ Λmax ker DV ⊗ ΛmaxV ∗. For R large enough the glued operator
DR = D
V #RT : V ⊕W 1,p → Lp is surjective with a uniformly bounded right
inverse QR, and moreover the projection onto ker DR given by 1l − QRDR is
an isomorphism when restricted to ker DV (see [5, Corollary 6], as well as [13,
Proposition 9] for a slightly different setup). Since DV #RT = (D#RT )
V , this
induces a natural isomorphism between Det(D) and Det(ψR(D)).
The gluing of orientations is associative, hence it is enough to prove the
statement for O = O+(C, θn;Sm). We claim that the action induced by ψ is
the same as the one induced by φm in Lemma 4.25. Let us choose D ∈ O which
is s-independent for s large enough, and let DV be a surjective stabilization.
We construct a continuous path in O from ψR(DV ) := ψR(D)V to φm(DV ) :=
φm(D)
V as follows. Let DVt be the conjugation of D
V by rt : C → C, z 7→
e−2iπt/mz, and let Tt be the operator ∂s + J0∂θ + Sm(θ − (t+ (1− t)β(s))/m).
Then DVt #RTt interpolates between ψR(D
V ) and φm(D
V ) as t varies from 0 to
1. This is a path of surjective operators admitting a continuous family of right
inverses Qt. Given a basis (ζ1, . . . , ζk) of ker D
V , a basis of ker DVt is given by
(ζ1, . . . , ζk) ◦ rt. By projecting along imQt we obtain a basis of ker DVt #RTt.
For t = 1, since DV1 = D
1 #RT1, the elements ζi ◦ r1 are preserved by the
projection and form a basis of φ−1m (D
V ) which is exactly the one giving the
action of φ−1m (or φm) on orientations, as explained in Lemma 4.25.
Lemma 4.28. Let γ ∈ P≤αλ and γp, γq be the orbits corresponding to the
minimum p and maximum q of fγ respectively. For δ > 0 small enough, the
moduli space MA(γp, γq;Hδ, J) is empty if A 6= 0, while for A = 0 it consists
of exactly two elements u1, u2 corresponding to the two gradient trajectories of
fγ running from p to q. Moreover, they satisfy
ǫ(u1) + ǫ(u2) =
0, if γ is a good orbit,
±2, if γ is a bad orbit.
Proof. Let c1, c2 be the gradient trajectories of fγ running from p to q. By
Theorem 3.7, for δ > 0 small enough each element [uδ] ∈ MA(γp, γq;Hδ, J)
corresponds to a unique Floer trajectory with gradient fragments [u] whose
endpoints are p and q. For energy reasons there can be no nonconstant Floer
trajectory involved in [u] and therefore [u] is either c1 or c2. Since the cylinders
u1 and u2 of the form uδ,γ,−∞,+∞ associated to c1 and c2 are already Floer
trajectories forHδ, we infer that [uδ] equals either [u1] or [u2], and the homology
class A is necessarily zero. Let us introduce the notation ǫ(γ) := 1 if γ is a good
orbit and ǫ(γ) := −1 if γ is a bad orbit. The conclusion of the Lemma is
equivalent to the relation
ǫ(u1) = −ǫ(γ)ǫ(u2). (86)
Symplectic homology for autonomous Hamiltonians 74
Let us choose a symplectic trivialization Φγ : TŴ |Sγ → Sγ×(R×R2n−1) such
that Φγ(XH) = (1, 0). We assume without loss of generality that ċ1 is a positive
multiple of XH , so that Φγ(∂su1) = (f1, 0) with f1 > 0 and Φγ(∂su2) = (f2, 0)
with f2 < 0. We denote by Du1 , Du2 the elements of O(R× S1, θn;Sp, Sq) ob-
tained by conjugation of D̃u1 , D̃u2 with Φγ . The main point is to consider the
operator ψ(Du1) = Du1#RT , with T as in Lemma 4.27. A basis of Det(Dui)
corresponding to the coherent orientation is by definition ǫ(ui)(fi, 0), i = 1, 2.
The image of this basis under the action of ψ is given by ǫ(u1)(f
1 , 0), for some
1 ∈ W 1,p(R× S1,R) with ‖f
1 − f1‖1,p arbitrarily small with R → ∞, hence
1 > 0 for R large enough. By Lemma 4.27, a basis of Det(Du1#RT ) corre-
sponding to the coherent orientation is ǫ(γ)ǫ(u1)(f
1 , 0). Finally, the operators
Du1#RT and Du2 can be connected by a continuous path of operators Dt,
t ∈ [0, 1] satisfying properties (ii)-(iv) in the proof of Proposition 4.9, as well as
the following weaker form of property (i) therein.
(i’) there exists a smooth path c : R → Sγ with c(±∞) being fixed critical
points of fγ , such that ‖S(s, θ)− S(θ+ ϑ ◦ c(s)− ϑ ◦ c(−∞))‖ is bounded
by a constant multiple of δ.
The connected components of the set of operators satisfying (i’) and (ii)-(iv)
are indexed by homotopy classes of paths c as above. Gluing Du1 to T has
precisely the effect of concatenating c1 with (γq|[0,1/m])−1, which is homotopic
to c2. The proof of Proposition 4.9 works the same with the weaker assumption
(i’) and shows that the operators Dt are surjective and that ker Dt is generated
by an element of the form (ft, 0), where ft ∈ W 1,p(R×S1,R) has constant sign
for t ∈ [0, 1]. We conclude that ǫ(u1)ǫ(γ)f#1 and ǫ(u2)f2 have the same sign,
hence (86) is proved.
We now generalize the construction of coherent orientations to the moduli
spaces of Morse-Bott trajectories with gradient fragments. We define S̃ to be
the space of loops of symmetric matrices S : S1 → M2n(R) such that the
symplectic matrix ΨS(1) defined by (85) has exactly one eigenvalue equal to 1,
corresponding to the eigenspace R⊕0 ⊂ R⊕R2n−1 = R2n. Let β : R → [0, 1] be
a smooth function equal to 0 near −∞ and equal to 1 near +∞. We define V ,
V to be the one-dimensional real vector spaces generated by the vector-valued
functions (1−β(s))(1, 0) and β(s)(1, 0) respectively. In the following we denote
by W 1,p,d = W 1,p(ed|s|ds dθ), Lp,d = Lp(ed|s|ds dθ). Given S, S ∈ S̃ we denote
Õ(R× S1, E;S, S)
the space of linear operators D : W 1,p,d(R × S1, E) ⊕ V ⊕ V → Lp,d(R ×
S1,Λ0,1E) of the form (∂s + J0∂θ + S(s, θ))(ds− idθ) in a local trivialization of
E, for which there exist θ, θ ∈ R/Z such that lims→−∞ S(s, θ) = S(θ + θ) and
lims→∞ S(s, θ) = S(θ + θ) in the given trivializations at infinity of E. Given
S0 ∈ S, S ∈ S̃ we denote by
Õu(R× S1, E;S0, S)
Symplectic homology for autonomous Hamiltonians 75
the space of linear operators
D :W 1,p(R× S1, E; g+(s)ds dθ) ⊕ V → Lp(R× S1,Λ0,1E; g+(s)ds dθ)
with g+(s) := max(1, e
ds), which are of the form (∂s + J0∂θ + S(s, θ))(ds −
idθ) in a local trivialization of E, and for which there exists θ ∈ R/Z such
that lims→−∞ S(s, θ) = S0(θ) and lims→∞ S(s, θ) = S(θ + θ) in the given
trivializations at infinity of E. Given S ∈ S̃, S0 ∈ S we denote by
Õs(R× S1, E;S, S0)
the space of linear operators
D :W 1,p(R× S1, E; g−(s)ds dθ) ⊕ V → Lp(R× S1,Λ0,1E; g−(s)ds dθ)
with g−(s) := max(1, e
−ds), which are of the form (∂s + J0∂θ + S(s, θ))(ds −
idθ) in a local trivialization of E, and for which there exists θ ∈ R/Z such
that lims→−∞ S(s, θ) = S(θ + θ) and lims→∞ S(s, θ) = S0(θ) in the given
trivializations at infinity of E. Given S̃ ∈ S̃ we denote by
Õ±(C, E; S̃)
the space of linear operators D : W 1,p,d(C, E) ⊕ V± → Lp,d(C,Λ0,1E) of the
form (∂x + J0∂y + S(z))dz̄ in a local trivialization of E and such that, when
expressed in holomorphic cylindrical coordinates (s, θ) with e±2π(s+iθ) = z as
(∂s+J0∂θ+S(s, θ))(ds− idθ), there exists θ± ∈ R/Z so that lims→±∞ S(s, θ) =
S̃(θ+θ±) in the given trivialization of E near infinity. Here we use the notation
V+ := V and V− := V .
Due to the exponential weights, each of the above spaces Õ consists of Fred-
holm operators and comes equipped with a canonical real line bundle Det(Õ)
whose fiber at D is Det(D). Unlike in the nondegenerate case, the spaces Õ
are not generally contractible, hence we have to investigate the orientability of
Det(Õ).
Given S ∈ S̃ we define m = m(S) to be the maximal positive integer such
that S(θ + 1/m) = S(θ), θ ∈ R/Z. The number m is infinite if and only if the
loop S is constant, in which case the spaces Õ±(C, E;S), Õu(R×S1, E;S0, S),
Õs(R×S1, E;S, S0) are contractible. In the following we shall restrict ourselves
to nonconstant loops S ∈ S̃, in which case the above spaces have the homotopy
type of S1, while Õ(R× S1, E;S, S) has the homotopy type of S1 × S1 (this is
because they fiber over S1, respectively S1 × S1 with contractible fibers). We
denote by S1 ∈ S̃ the unique loop such that S(θ) = S1(mθ).
Lemma 4.29. Let S ∈ S̃ be nonconstant. Then Det(Õ±(C, E;S)) is nonori-
entable if and only if m is even and µRS(S)− µRS(S1) is odd.
Proof. We prove the statement only for Õ+ := Õ±(C, E;S), the proof of the
other case being similar. The following two remarks will allow us to apply
Symplectic homology for autonomous Hamiltonians 76
Lemma 4.25. First, Det(D) is naturally isomorphic to Det(D|W 1,p,d) ⊗ V+ and
V+ is a trivial bundle over Õ+. Second, the operator D|W 1,p,d is conjugated to
an operator D̃ ∈ O+(C, E;S− dp1l). Hence it is enough to study the orientability
of the bundle D̃et(Õ+) over Õ+ with fiber Det(D̃).
The bundle D̃et(Õ+) is orientable if and only if its restriction to a loop
generating π1(Õ) is orientable. After choosing D ∈ Õ+ which is invariant
under conjugation with z 7→ e−2iπ/m, the conjugation of D by rt : C → C,
z 7→ e−2iπt/mz provides such a loop Dt, t ∈ [0, 1] with D0 = D1 = D. The
orientation on Det(D̃1) obtained by continuation along the path Dt from an
orientation on Det(D̃0) is the same as the one induced by the action of φ
m (or
φm) in Lemma 4.25. Since µCZ(S − dp1l) = µRS(S)− 1/2 and µCZ(S1 −
1l) =
µRS(S1)− 1/2, the statement follows from Lemma 4.25.
The same kind of argument gives the following result.
Lemma 4.30. Let S, S, S ∈ S̃ be nonconstant and S0, S0 ∈ S. The line bundles
Det(Õ(R× S1, E;S, S)), Det(Õu(R× S1, E;S0, S)), Det(Õs(R× S1, E;S, S0))
are nonorientable if and only if the condition in Lemma 4.29 holds for S and
for one of S, S. �
The previous results motivate the following definition. We denote
S̃good := {S ∈ S̃ : S constant or µRS(S)− µRS(S1) is even},
S̃bad := {S ∈ S̃ : S nonconstant and µRS(S)− µRS(S1) is odd},
so that S̃bad = S̃ \ S̃good. Although the determinant lines over the various
spaces Õ are nonorientable if one of the asymptotes is in S̃bad, we can construct
covers
O of Õ over which the determinant lines become orientable. Let S0 ∈ S
and S ∈ S̃bad with S(θ) = S1(mθ), θ ∈ R/Z. We define
Os(R× S1, E;S, S0) to
consist of pairs (D, θ) such that θ ∈ R/ 2
Z, D = (∂s+J0∂θ+S(s, θ))(ds−idθ) ∈
Õs(R× S1, E;S, S0) with lims→−∞ S(s, θ) = S(θ + θ). The obvious projection
Os → Õs is a double cover and the lift of the determinant bundle to
Os is
orientable. We define in a completely analogous manner double covers
O±(S) →
Õ±(S),
Ou(S0, S) → Õu(S0, S), S ∈ S̃bad and a cover
O(S, S) → Õ(S, S) which
is double if exactly one of S, S is in S̃bad, and quadruple if both S, S are in S̃bad.
We define now gluing operations between elements of the various spaces Õ.
Let K in Õ+(C, E;S), Õ(R × S1, E;S, S), or Õu(R × S1, E;S0, S), and L in
Õ−(C, F ;S), Õ(R × S1, F ;S, S), or Õs(R × S1, F ;S, S0). We denote by SK ,
respectively SL the matrix valued functions involved in K and L near infinity.
We assume that
SK(s, ·) = lim
SL(s, ·) = S(·+ θ0) =: Sθ0
Symplectic homology for autonomous Hamiltonians 77
for some θ0 ∈ R/Z. We choose a cutoff function β : R → [0, 1] such that
β(s) = 0 if s ≤ 0 and β(s) = 1 if s ≥ 1. Given R > 0 large we define
operators KR and LR by replacing SK and SL by Sθ0 +β(R− s)(SK −Sθ0) and
Sθ0+β(R+s)(SL−Sθ0) respectively. We cut out semi-infinite cylinders {s > R}
from the base of E, {s < −R} from the base of F , then identify their boundaries
using the coordinate θ. We glue the vector bundles E and F using their given
trivializations near infinity and denote the resulting vector bundle by E#F . We
define K#RL by concatenating KR and LR, so that K#RL belongs to one of
the spaces O(CP 1), Õ+(C;S), O+(C;S0), or Õ−(R × S1;S), Õ(R × S1;S, S),
Õs(R × S1;S, S0), or O−(C;S0), Õu(R × S1;S0, S), O(R × S1;S0, S0), where
we have omitted the symbol E#F from the notation.
The above gluing operations admit a straightforward extension to the spaces
O. For example, two elements (K, θ) ∈
Ou(S0, S), (L, θ) ∈
Os(S, S0) can be
glued if θ = θ, in which case they give rise to an element K#RL ∈ O(S0, S0).
Recall that the domain of an operator D in some Õ contains a canoni-
cally oriented 1-dimensional summand for each asymptote in S̃, together with
a canonical isomorphism with R. We denote by VK , VL the summands corre-
sponding to the asymptote S of K and L respectively, and we let V := VK⊕RVL
be their (canonically oriented) fibered sum. By [5, Corollary 6], for R > 0 large
enough there is a natural isomorphism Det(K ⊕R L) ≃ Det(K#RL) defined up
to homotopy, where K ⊕R L is the restriction of K ⊕ L to the fibered sum of
their domains. Since V is canonically oriented, it follows that Det(K ⊕R L) is
canonically isomorphic to Det(K ⊕ L) ≃ Det(K) ⊗ Det(L). Hence we obtain
a canonical isomorphism Det(K) ⊗ Det(L) ∼→ Det(K#RL) defined up to ho-
motopy, and inducing an associative gluing operation for orientations. Similar
considerations apply to the elements of the spaces
Remark 4.31. We can construct a system of coherent orientations on the
determinant line bundles Det(Õ±(C, E;S)) and Det(Õ(R × S1, E;S, S)) with
S, S, S ∈ S̃good by the same procedure as for the spaces O. We can moreover
extend this to a system of coherent orientations involving all spaces O, Õ and
O. Nevertheless, if we want that certain orientations have a geometric meaning,
we have to impose compatibility conditions which seem ad-hoc in such a general
setup. This is why we restrict ourselves in the sequel to the spaces O, Õ and
which are relevant for our geometric situation.
We use now the notations of Section 3. Given γ ∈ P(H) we denote by
Ψγ the linearization of the Hamiltonian flow along γ given by (21) and let
Sγ : R/Z → M2n(R) be the corresponding loop of symmetric matrices defined
by Ψ̇γ = J0SγΨγ . Then Sγ ∈ S̃good if and only if γ is a good orbit. We similarly
define Sγq for each γq ∈ P(Hδ), with q ∈ Crit(fγ). For γ ∈ P(H), γq ∈ P(Hδ)
we denote Õs(R× S1, E; γ, γ
) := Õs(R× S1, E;Sγ , Sγ
) etc.
Convention. In what follows the spaces Õ will be understood to be indexed
Symplectic homology for autonomous Hamiltonians 78
only by good orbits, whereas if one of the asymptotic orbits is bad we use the
corresponding double or quadruple cover
We construct orientations on the determinant bundles over all spaces O, Õ,
O indexed by the elements of P(H) and P(Hδ) as follows. We start by choosing
arbitrary orientations of Det(Õ+(C, E; γ)), respectively Det(
O+(C, E; γ)), γ ∈
P(H) such that the trivialization of E at infinity extends to C. We then choose
orientations of Det(Õs(R×S1, E; γ, γq)), respectively Det(
Os(R×S1, E; γ, γq)),
γ ∈ P(H), q ∈ Crit(fγ) such that the trivializations of E at infinity extend
to R × S1, as follows. If γ is good, the space Õs(R × S1, θn; γ, γq) contains
a distinguished family of operators of the form Φγ ◦ Du ◦ Φ−1γ , where u =
uδ,γ,−1,∞ is the cylinder corresponding to a semi-infinite gradient trajectory
ending at q and Φγ : TŴ |Sγ → Sγ × R2n is a fixed trivialization satisfying
Φγ(XH) = (1, 0) ∈ R⊕R2n−1. This family is naturally parametrized by W s(q),
hence it is connected. As seen in Proposition 4.9 the above Fredholm operators
are surjective and have index 1 − ind(q). If the index is zero we choose the
orientation sign to be +1. If the index is one the kernel is generated by a
nonvanishing section of the form (f, 0), hence is canonically isomorphic to R⊕0
and therefore admits a canonical orientation. If γ is bad, we choose in an
arbitrary way a lift of the operator Φγ ◦Du ◦ Φ−1γ , where u = uδ,γ,−1,∞ is the
cylinder corresponding to a constant semi-infinite gradient trajectory at q. This
determines a lift of the whole path of operators described above, and hence an
orientation of Det(
Os(R× S1, E; γ, γq)) by the previous rule.
We induce orientations on Det(O+(θn)) by gluing orientations on the line
bundles Det(Õ+(θn)) and Det(Õs(θn)). The orientations on Det(Õ+(θn)) and
Det(O+(θn)) determine orientations on Det(Õ±(E)) and Det(O±(E)) by re-
quiring that the glued orientation on Det(O(CP 1, E)) is the canonical one. We
get orientations of Det(Õ(R×S1, E)) by requiring that the orientation induced
on Det(O(CP 1, θn#E#θn)) by the gluing operation
Õ+(C, θn; γ) ev ×ev Õ(R× S1, E; γ, γ)ev ×ev Õ−(C, θn; γ)→ O(CP 1, θn#E#θn)
is the canonical one. Here we have denoted by ev, ev the evaluation maps to
S1 at −∞ and +∞ respectively. Similarly, we get orientations on Det(O(R ×
S1, E)), Det(Õu(R×S1, E)) and Det(Õs(R×S1, E)), as well as orientations on
O) for the various spaces
Lemma 4.32. The above recipe defines a system of coherent orientations.
Proof. We have to prove that, given operators K, L that can be glued lying
in one of the spaces O, Õ or
O, the coherent orientations oK , oL of Det(K)
and Det(L) induce an orientation oK#oL that coincides with the coherent ori-
entation of Det(K#L). In the case when K, L belong to some O(R × S1),
Õ(R × S1) or
O(R × S1) this means that, for a suitable choice of operators
Symplectic homology for autonomous Hamiltonians 79
O+(C, θn), A ∈ Õ+(C, θn) or A ∈ O+(C, θn), and B ∈
O−(C, θn),
B ∈ Õ−(C, θn) or B ∈ O−(C, θn), with oA, oB the coherent orientations on
the respective determinant line bundles, oA#(oK#oL)#oB is the canonical ori-
entation on Det(O(CP 1)).
Let E and F be the symplectic vector bundles corresponding to K and L
respectively. If E = θn or F = θn the conclusion is a direct consequence of the
definitions and of the associativity of gluing. In the general case E 6= θn and F 6=
θn we give the proof whenK ∈ Õu(R×S1, E; γp, γ) and L ∈ Õs(R×S1, F ; γ, γq),
the other cases being similar. Let us introduce an auxiliary loop of symmetric
matrices S0 ∈ S such that [S0, J0] = 0, and we define the orientations on
Det(O±(C, E′;S0)) to be the canonical ones (see Remark 4.24). This determines
in turn orientations on Det(Õu(R×S1, E′;S0, Sγ)), γ ∈ P(H) by requiring that
gluing induces the coherent orientation on Det(Õ+(C, θn#E′; γ)).
Let A1 ∈ O+(C, E1;S0), K1 ∈ Õu(R× S1, θn;S0, Sγ) with E1#θn = θn#E.
By the above definition, we have oA1#oK1 = oA#oK . We obtain
oA#(oK#oL)#oB = (oA#oK)#oL#oB = (oA1#oK1)#oL#oB
= oA1#oK1#L#oB = oA1#oK1#L#B.
The operators A1 and K1#L#B are homotopic to C-linear operators with
asymptotic condition S0. The main observation now is that the gluing of two
C-linear operators is again C-linear, hence the gluing of the above orientations
is the canonical one on Det(O(CP 1)).
Let γ, γ be good orbits. In this case the procedure for orienting the Morse-
Bott spaces of Floer trajectories M̂A(Sγ , Sγ ;H, J) is entirely similar to the
corresponding procedure in the nondegenerate case (it is actually simpler since
we do not need the intermediate transition from Lp,d to Lp spaces). Namely,
we pull back the orientation on Det(Õ) using the natural map M̂ → Õ. This
in turn induces orientations on the quotient spaces MA(Sγ , Sγ ;H, J). Recall
that, given oriented vector spaces V ⊂W , we define an orientation on W/V by
requiring that the isomorphism V ⊕ (W/V ) ≃W is orientation preserving.
Since the stable and unstable manifolds of the functions fγ are canoni-
cally oriented, one gets orientations (i.e. signs) on all zero-dimensional moduli
spaces of Floer trajectories with gradient fragments MA(p, q;H, {fγ}, J) which
involve only good orbits. This is done by the following fibered sum rule.
Let fi : Wi → W , i = 1, 2 be linear maps of oriented vector spaces such that
f : W1 ⊕W2 → W , (w1, w2) 7→ f1(w1) − f2(w2) is surjective. The orientation
on the fibered sum W1f1⊕f2W2 := ker f is defined such that the isomorphism
of vector spaces (W1 ⊕W2)/ ker f
∼→ W induced by f changes orientations by
the sign (−1)dim W2·dim W . Note that this rule is such that the fibered sum
operation is associative for oriented vector spaces, and moreover, if f2 is an ori-
entation preserving isomorphism, the natural isomorphismW1f1⊕f2W2 ≃W1 is
orientation preserving. Similarly, if f1 is an orientation preserving isomorphism,
the natural isomorphism W1f1⊕f2W2 ≃W2 is orientation preserving.
Symplectic homology for autonomous Hamiltonians 80
The important remark now is that, although the spaces M̂A(Sγ , Sγ ;H, J)
with γ or γ being a bad orbit may not be orientable, we can nevertheless define
orientations (i.e. signs) on all zero-dimensional moduli spaces of Floer trajec-
tories with gradient fragments MA(p, q;H, {fγ}, J). The sign of an (isolated)
point [u] = (cm, [um], . . . , c1, [u1], c0) in this moduli space is determined as fol-
lows. For each operator Dui , i = 1, . . . ,m with at least one bad asymptote we
choose a lift in the corresponding space
O(R×S1). For each ci, i = 0, . . . ,m ly-
ing on a bad orbit γi the corresponding operatorDuδ,γi,−Ti/2,Ti/2 admits a unique
lift to the space
O(Sγi , Sγi) such that it can be glued with both Dui+1 and Dui .
Since all these operators are surjective, the orientations of the determinant line
bundles over the spaces Õ and
O induce orientations on TuiM̂Ai(Sγi , Sγi−1),
respectively TWu(p), TW s(q) and T(ci(−Ti/2),Ti)(Sγi × R+), i = 1, . . . ,m − 1.
By the fibered sum rule we get an orientation on TuM̂A(p, q;H, {fγ}, J) which
we call “the coherent orientation”. On the other hand this vector space carries
the “geometric orientation” of the basis (∂sum, . . . , ∂su1). We define the sign
ǫ(u) = ǫ([u]) (87)
to be +1 if these two orientations coincide, and −1 if they are different.
We now want to compare the signs ǫ(u) with the signs ǫ(uδ) of the glued
trajectories uδ corresponding to u. The situation is expressed by the follow-
ing diagram, in which we dropped the decorations A, (H, {fγ}, J) and (Hδ, J)
and in which we have indicated on the morphism arrows the way in which the
corresponding isomorphisms of vector spaces act on orientations.
Coherent
orientation
// TM̂(p, q)
Id ǫ(u)
TM̂(γp, γq)⊕ R
Id ǫ(uδ)
Coherent
orientation
Geometric
orientation
〈∂sum,...,∂su1〉
// TM̂(p, q)
? // TM̂(γp, γq)⊕ R
Geometric
orientation
〈∂suδ〉⊕ R
The map φ is defined from gluing as follows. The tangent space TM̂(p, q)
is the kernel of the operator D ew, w̃ = (vm, um, . . . , v1, u1, v0) considered in
Lemma 4.17. Moreover, since the cokernel ofD ew is naturally oriented, the coher-
ent orientation of Det(D ew) induces a “coherent” orientation on ker D ew. Recall
that the analytical expression ofD ew isDvm⊕Dum⊕D′vm−1⊕. . .⊕D
⊕Du1⊕Dv0 ,
and note that D ew admits a natural stabilization D
ew obtained by replacing
D′vi , i = 1, . . . ,m− 1 with D{∂̄T }(vi, Tvi) (see Remark 4.11 for the definitions).
By [5, Corollary 6] there is a natural isomorphism φ̃ : ker DR
∼→ ker DRm−1
Gδ( ew)
which preserves the coherent orientations. We denote by φ : ker DR
ker DR
the composition of φ̃ with the projection Π on ker DR
along the
image of the right inverse Qδ of DGδ( ew) given by Proposition 4.18. Since DGδ( ew)
Symplectic homology for autonomous Hamiltonians 81
and Duδ are close in the relevant δ-norm, we get that φ is an isomorphism pre-
serving coherent orientations.
The vertical maps change orientations by ǫ(u), respectively ǫ(uδ) by defi-
nition, and the whole work now goes into determining the action of φ on the
geometric orientations.
Remark 4.33. If γ is a good orbit and p ∈ Crit(fγ), the geometric orientations
on Wu(p) and W s(p) coincide with the coherent ones. Indeed, the unstable
manifold Wu(p) is naturally identified with the zero set of the section ∂̄−∞,1
defined on B1,p,dδ (p, Sγ ; fγ) by (54), whereas the stable manifold W s(p) is natu-
rally identified with the zero set of the section ∂̄−1,∞ defined on B1,p,dδ (Sγ , q; fγ).
The assertion for W s(p) is then a direct consequence of the definition of the ori-
entation on Det(Õs(R×S1, θn; γ, γq)). As for Wu(p), let us consider the gluing
operation
Õu(R× S1, θn; γp, γ)ev ×ev Õs(R× S1, θn; γ, γp) → O(R× S1, θn; γp, γp).
We choose the surjective operators D1 := Duδ,γ,−∞,1 , D2 := Duδ,γ,−1,∞ corre-
sponding to the constant gradient trajectory at p. With these choicesD1#D2 =
Duδ,γ,−∞,∞ =: D also corresponds to the constant gradient trajectory at p. The
operator D is an isomorphism and, by the coherent choice of the orientations,
the determinant line Det(D) ≃ R is positively oriented. If p is the maximum of
fγ then ker D2 ≃ TpSγ as oriented vector spaces (by definition), the kernel of
D1 is trivial and its determinant line must be positively oriented. If p is the min-
imum of fγ then ker D2 is trivial and its determinant line is positively oriented
by definition, therefore ker D1 ≃ TpSγ must have the geometric orientation.
Lemma 4.34. Assume dim MA(p, q;H, {fγ}, J) = 0 and fix an element [u] ∈
MA(p, q;H, {fγ}, J) with m ≤ 2 sublevels. Then ǫ(u) = ǫ(uδ) if m = 0, 1 and
ǫ(u) = −ǫ(uδ) if m = 2.
Proof. If m = 0 the statement is obvious since u consists of a single gradient
trajectory and u = uδ (see Lemma 4.28). We now have to show that the map φ in
the previous diagram preserves the geometric orientation if m = 1, respectively
reverses it if m = 2. Since a shift σ on u1 produces a glued trajectory uδ shifted
by the same amount σ, we infer that φ(∂su1) = ∂suδ and, for m = 1, the
statement follows from the commutativity of the diagram.
Let us now examine the case m = 2. We recall that φ = Π ◦ φ̃, where the
isomorphism φ̃ is the composition of the gluing map G in the proof of Propo-
sition 4.18 with the projection to ker DR
Gδ( ew)
along the image of Qδ (see [5]).
We first show that φ(∂su1 + ∂su2) is close in ‖ · ‖1,δ-norm to ∂suδ. We denote
w̃σ,σ := (v2, u2(·+ σ), v1, u1(·+ σ), v0), σ ∈ R. Then G(0⊕ ∂su2 ⊕ 0⊕ ∂su1 ⊕ 0)
is ‖ · ‖1,δ-close to ddσ
Gδ(w̃
σ,σ), which is ‖ · ‖1,δ-close to ddσ
Gδ(w̃)(·+ σ),
which is in turn close to d
vδ(· + σ) = ∂svδ. Then φ(∂su1 + ∂su2) =
Π(G(0 ⊕ ∂su2 ⊕ 0⊕ ∂su1 ⊕ 0)) is ‖ · ‖1,δ-close to Π(∂suδ) = ∂suδ.
We now show that φ(∂su1−∂su2) ∈ ker Duδ ⊕R is a vector having a negative
component in the R direction and whose component on ker Duδ is ‖ · ‖1,δ-close
to −∂su1. Then the conclusion follows.
Symplectic homology for autonomous Hamiltonians 82
Let w̃−σ,σ := (v2, u2(· − σ), v1, u1(·+ σ), v0), σ ∈ R. Then G(0⊕ (−∂su2)⊕
0 ⊕ ∂su1 ⊕ 0) is ‖ · ‖1,δ-close to ddσ
Gδ(w̃
−σ,σ). We define ǫ(σ) := 2δσ and
the section
Gδ,ǫ(σ)(w̃
−σ,σ) =
Gδ(w̃
−σ,σ) + 2δ
Gδ,ǫ(w̃) (88)
is by construction ‖ · ‖1,δ-close to ddσ
Gδ(w̃
−σ,−σ), hence ‖ · ‖1,δ-close to
−∂suδ. By adapting the arguments in the proof of Proposition 4.15 one sees
that the section
∂̄Tv1+ǫGδ,ǫ(w̃) = D∂̄Tv1
Gδ,ǫ(w̃) +D{∂̄T }(Gδ(w̃), Tv1) · (0, 1)
is ‖ · ‖δ-small. Here the sections ∂̄T are of the form ∂̄HT ,J , where HT is the s-
dependent Hamiltonian given respectively by (54) on the intervals of definition
of v2, v1, v0, and equal toH on the intervals of definition of u1, u2. The previous
equation shows that ( d
Gδ,ǫ(w̃), 1) ∈ dom(DRGδ( ew)) is ‖·‖1,δ-close to ker D
On the other hand, equation (88) shows that G(0 ⊕ (−∂su2)⊕ 0⊕ ∂su1 ⊕ 0) is
‖ · ‖1,δ-close to −∂suδ − 2δ ddǫ
Gδ,ǫ(w̃). Hence, after projecting to ker D
ker Duδ ⊕ R, we get a vector having a negative component in the R direction
and whose component on ker Duδ is ‖ · ‖1,δ-close to −∂suδ.
Proof of Proposition 3.9. The special statement concerning the case m = 0
was proved in Lemmas 4.28 and 4.34, whereas the equality ǫ(u) = (−1)m−1ǫ(uδ)
in case m = 1, 2 was the content of Lemma 4.34. The proof in the case m ≥ 3
is just a more elaborate version of the proof of Lemma 4.34. We consider the
basis of TuM̂(p, q) given by
e0 := ∂sum + ∂sum−1 + . . .+ ∂su2 + ∂su1,
e1 := −∂sum + ∂sum−1 + . . .+ ∂su2 + ∂su1,
em−2 := −∂sum − ∂sum−1 − . . .+ ∂su2 + ∂su1,
em−1 := −∂sum − ∂sum−1 − . . .− ∂su2 + ∂su1.
It is easy to see that the orientation determined by (e0, . . . , em−1) is the same
as the geometric orientation determined by (∂sum, . . . , ∂su1). We have to show
that the orientation of the basis (φ(e0), . . . , φ(em−1)) differs from the canonical
orientation of 〈∂suδ〉 ⊕ Rm−1 by (−1)m−1.
As in Lemma 4.34 we see that φ(e0) is ‖ · ‖1,δ-close to ∂suδ. We now show
that φ(ek) ∈ kerDuδ ⊕Rm−1, k = 1, . . . ,m− 1 has a negative component which
is bounded away from zero along the corresponding factor R ⊂ Rm−1, that
the other components in Rm−1 are close to zero, whereas the component along
kerDuδ is close to −∂suδ in ‖ · ‖1,δ-norm. Then the conclusion will follow since
the orientation defined by (φ(e0), . . . , φ(em−1)) is the same as the orientation
defined by
(∂suδ, 0, . . . , 0), (−∂suδ,−1, 0, . . . , 0), . . . , (−∂suδ, 0, . . . , 0,−1)).
Symplectic homology for autonomous Hamiltonians 83
Let us fix k = 1, . . . ,m−1. We shall freely use the notation ek for the vector
0⊕ (−∂sum)⊕ 0⊕ . . .⊕ (−∂sum−k+1)⊕ 0⊕∂sum−k⊕ . . . ∂su1⊕ 0 in the domain
of the gluing map G defined in the proof of Proposition 4.18. For σ > 0 we
denote
k :=(vm, um(·−σ), . . . , um−k+1(·−σ), vm−k, um−k(·+σ), . . . , u1(·+σ), v0),
w̃−σ,−σ :=(vm, um(·−σ), . . . , um−k+1(·−σ), vm−k, um−k(·−σ), . . . , u1(·−σ), v0).
Then G(ek) is ‖ · ‖1,δ-close to ddσ
Gδ(w̃
k ). We denote
ǫk(ǫ) := (0, . . . , ǫ, . . . , 0),
where the parameter ǫ > 0 appears on position m− k. The section
Gδ,ǫk(2δσ)(w̃
k ) =
Gδ(w̃
k ) + 2δ
Gδ,ǫk(w̃) (89)
is by construction ‖·‖1,δ-close to ddσ
Gδ(w̃
−σ,−σ), hence ‖·‖1,δ-close to −∂suδ.
As in Lemma 4.34, by adapting the arguments in the proof of Proposition 4.15
one sees that the section
∂̄Tvm−k+ǫGδ,ǫk(ǫ)(w̃) = D∂̄Tvm−k
Gδ,ǫk(ǫ)(w̃)
+ D{∂̄T }(Gδ(w̃), Tvm−k) · (0, 1)
is ‖ · ‖δ-small. As before, the sections ∂̄T are of the form ∂̄HT ,J , where HT
is the s-dependent Hamiltonian given respectively by (54) on the intervals of
definition of vm, vm−1, . . . , v0, and equal to H on the intervals of definition of
um, . . . , u1. The previous equation shows that
Gδ,ǫk(ǫ)(w̃), 0, . . . , 1, . . . , 0) ∈ dom(D
Gδ( ew)
is ‖·‖1,δ-close to ker DR
. On the other hand, equation (89) shows that G(ek)
is ‖·‖1,δ-close to −∂suδ−2δ ddǫ
Gδ,ǫk(ǫ)(w̃). After projecting to ker D
get a vector whose k-th component in Rm−1 is negative, whose other components
in Rm−1 are small, and whose component on ker Duδ is ‖ · ‖1,δ-close to −∂suδ.
Remark 4.35. We chose to define the signs ǫ(u) by comparing the orientation
induced on TuM̂A(p, q;H, {fγ}, J) by the fiber sum rule from the coherent ori-
entations on TM̂Ai(Sγi , Sγi−1 ;H, J), i = 1, . . . ,m with the orientation of the
basis (∂sum, . . . , ∂su1). Another possible recipe would have been the following:
induce orientations on TMAi(Sγi , Sγi−1 ;H, J) out of the coherent orientations
by quotienting out 〈∂s〉, then apply the fiber sum rule in order to get a sign on
the zero-dimensional spaces T[u]MA(p, q;H, {fγ}, J). The sign obtained in this
way would have differed from the previously defined ǫ(u) by a factor ±1 which
Symplectic homology for autonomous Hamiltonians 84
can be explicitly computed and which depends on the combinatorics of the lev-
els of u. The curious reader can test this procedure in the case m = 1: it gives
a sign equal to ǫ(u) if p, q are both minima, respectively equal to −ǫ(u) if p, q
are both maxima. The following two properties of the fibered sum constitute
a useful tool for making the verification (here W1 and W2 are oriented vector
spaces).
• the natural isomorphism W1 0
∼→ W1 ⊕ ker f2 changes the orien-
tation by (−1)dim W1·(dim W2+1);
• the natural isomorphism W1 f1
∼→ ker f1 ⊕W2 preserves the orien-
tation.
A Appendix: Asymptotic estimates
For all γ ∈ P(H), we choose coordinates (ϑ, z) ∈ S1 × R2n−1 parametrizing a
tubular neighbourhood of γ, such that ϑ ◦ γ(θ) = θ and z ◦ γ(θ) = 0. Given a
smooth function fγ : Sγ → R, we denote by ϕ
s the gradient flow of fγ with
respect to the natural metric on S1.
In a neighbourhood of γ ∈ P(H) the Floer equation ∂su+ J∂θu− JXH = 0
becomes ∂sZ+J∂θZ+J
−JXH = 0, where Z(s, θ) := (ϑ◦u(s, θ)−θ, z◦u(s, θ)).
Since XH =
on {z = 0} this can be rewritten as
∂sZ + J∂θZ + Sz = 0
for some matrix-valued function S = S(ϑ, z). The matrix S∞(θ) := S(θ, 0) is
symmetric. Let A∞ : H
k(S1,R2n) → Hk−1(S1,R2n) be the operator defined by
A∞Z := J
Z + S∞(θ)z.
The kernel of A∞ has dimension one and is spanned by the constant vector
e1 := (1, 0, . . . , 0). We denote by Q∞ the orthogonal projection onto (ker A∞)
and we set P∞ := 1l − Q∞. Then A∞ is invertible when restricted to imQ∞
and Q∞A∞ = A∞.
Proposition A.1. Let H ∈ H′ be fixed. There exists r > 0 such that for all
J ∈ J ℓ and for all u ∈ MA(Sγ , Sγ ;H, J), γ, γ ∈ P(H) we have
ϑ ◦ u(s, θ)− θ − θ0 ∈ W 1,p(]−∞,−s0]× S1,R; er|s|ds dθ),
z ◦ u(s, θ) ∈ W 1,p(]−∞,−s0]× S1,R2n−1; er|s|ds dθ),
ϑ ◦ u(s, θ)− θ − θ0 ∈ W 1,p([s0,∞[×S1,R; er|s|ds dθ),
z ◦ u(s, θ) ∈ W 1,p([s0,∞[×S1,R2n−1; er|s|ds dθ),
for some θ0, θ0 ∈ S1 and some s0 > 0 sufficiently large.
Symplectic homology for autonomous Hamiltonians 85
Proof. We make the proof only at +∞ since the case of −∞ is entirely similar.
For s large enough we set S(s, θ) := S(ϑ ◦ u(s, θ), z ◦ u(s, θ)), so that S∞(θ) =
lims→∞ S(s, θ) and lims→∞ |∂sS(s, θ)| = 0.
Let A(s) : Hk(S1,R2n) → Hk−1(S1,R2n) be the operator defined by
A(s)Z := J
Z + S(s, θ)z,
so that A∞ = lims→∞A(s). We have A(s) = A(s)Q∞, ∂sQ∞ = Q∞∂s. Since
A∞ is invertible when restricted to imQ∞ and Q∞A∞ = A∞, the operators
A(s) andQ∞A(s) are also invertible when restricted to imQ∞ for s large enough
and there exists c > 0 such that
‖A(s)Q∞Z‖2 ≥ ‖Q∞A(s)Q∞Z‖2 ≥ c‖Q∞Z‖2
for all Z ∈ Hk(S1,R2n). For s large enough we define
f(s) :=
‖Q∞Z(s)‖2.
We have
f ′′(s) = ‖∂sQ∞Z‖2 + 〈Q∞Z, ∂2sQ∞Z〉
= ‖∂sQ∞Z‖2 − 〈Q∞Z, ∂sQ∞A(s)Q∞Z〉
= ‖Q∞A(s)Q∞Z‖2 − 〈Q∞Z,Q∞(∂sA(s))Q∞Z −Q∞A(s)2Q∞Z〉
≥ (c− ε)‖Q∞Z‖2+〈(A(s)∗ −A(s))Q∞Z,A(s)Q∞Z〉+‖A(s)Q∞Z‖2
≥ (2c− 2ε)‖Q∞Z‖2 ≥ 4ρ2f(s).
Here A(s)∗ is the adjoint of A(s) and we used the fact that ‖∂sA(s)‖ → 0,
A(s)∗ −A(s) → 0 for s→ ∞ and ‖A(s)‖ is uniformly bounded.
Let now s0 be large enough and define g(s) := f(s0)e
−2ρ(s−s0). Then g′′ =
4ρ2g, (f − g)′′ ≥ 4ρ2(f − g), (f − g)(s0) = 0 and lims→∞ f(s) − g(s) = 0.
Then f − g ≤ 0 on [s0,∞[ because it cannot have a strictly positive maximum.
Therefore
‖Q∞Z(s)‖ ≤ ‖Q∞Z(s0)‖e−ρ(s−s0).
It is important to note that this estimate holds for any Sobolev normHk. By
the Sobolev embedding theorem this implies the following pointwise estimate
|Q∞Z(s, θ)| ≤ Ce−ρs, |∂θQ∞Z(s, θ)| = |∂θZ(s, θ)| ≤ Ce−ρs, s ≥ s0.
Because ∂sZ +A(s)Z = ∂sZ +A(s)Q∞Z = 0 we obtain
|∂sZ(s, θ)| ≤ Ce−ρs, s ≥ s0
and, by integration on [s,∞[ and taking into account that Z(s, θ) converges to
(θ0, 0, . . . , 0) for s→ ∞, we obtain the pointwise estimate
|(ϑ− θ − θ0, z)| ≤ Ce−ρs.
This implies the conclusion for any r < ρ.
Symplectic homology for autonomous Hamiltonians 86
Proposition A.2. Let H ∈ H′ and {fγ : Sγ → R} be a collection of perfect
Morse functions indexed by γ ∈ Pλ. There exist r > 0 and δ0 > 0 such that
for all J ∈ J , γ, γ ∈ P(H), p ∈ Crit(fγ), q ∈ Crit(fγ) and for all (δ, u) ∈
]0,δ0]
(γp, γq;H, {fγ}, J), we have
ϑ ◦ u(s, θ)− θ − ϕδfγs (θ0) ∈ W 1,p(]−∞,−s0]× S1,R; er|s|ds dθ),
z ◦ u(s, θ) ∈ W 1,p(]−∞,−s0]× S1,R2n−1; er|s|ds dθ),
ϑ ◦ u(s, θ)− θ − ϕδfγs (θ0) ∈ W 1,p([s0,∞[×S1,R; er|s|ds dθ),
z ◦ u(s, θ) ∈ W 1,p([s0,∞[×S1,R2n−1; er|s|ds dθ),
for some θ0, θ0 ∈ S1 and some s0 > 0 sufficiently large.
Proof. The proof is similar to the one of Proposition A.1. With the same nota-
tions as before the Floer equation satisfied by u can be written in local coordi-
nates Z = (ϑ− θ, z) as
∂sZ + J∂θZ + Sz − δ∇fγ(Z1) = 0, (90)
where Z1 := ϑ−θ. We again show that f(s) = 12‖Q∞Z‖
2 satisfies an inequality
of the form f ′′(s) ≥ 4ρ2f(s). There are two additional terms to estimate in the
expression of f ′′(s), namely
〈Q∞Z, δQ∞A(s)∇fγ(Z1)〉 (91)
〈Q∞Z, δQ∞∂s(∇fγ(Z1))〉. (92)
Let P∞ := 1l−Q∞ be the orthogonal projection on ker A∞. The main observa-
tion is that Q∞∇fγ(P∞Z1) = 0. As a consequence there exists a matrix-valued
function L = L(s, θ) such that
Q∞∇fγ(Z1) = LQ∞(Z1).
The term (91) is then estimated by
〈Q∞Z, δQ∞A(s)∇fγ(Z1)〉 = 〈Q∞Z, δQ∞A(s)Q∞∇fγ(Z1)〉
≤ Cδ‖Q∞Z‖2
for s ≥ s0, where s0 depends on u, but C depends only on γ and fγ . Similarly,
the term (92) is estimated by
〈Q∞Z, δQ∞∂s(∇fγ(Z1))〉 = 〈Q∞Z, δ∂sQ∞∇fγ(Z1)〉
≤ Cδ‖Q∞Z‖‖∂sQ∞(Z1)‖
‖Q∞Z‖2 + ‖∂sQ∞Z‖2
The norm of ∂sQ∞Z = Q∞∂sZ satisfies
‖∂sQ∞Z‖ = ‖Q∞A(s)Z − δQ∞∇fγ(Z1)‖ ≤ C‖Q∞Z‖.
Symplectic homology for autonomous Hamiltonians 87
As a consequence, there exists δ0 > 0 and ρ > 0 such that f
′′(s) ≥ 4ρ2f(s) for
s ≥ s0 and 0 < δ ≤ δ0. As before, we infer the pointwise bounds
|Q∞Z(s, θ)| ≤ Ce−ρs, |∂θQ∞Z(s, θ)| = |∂θZ(s, θ)| ≤ Ce−ρs, s ≥ s0. (93)
It remains to estimate P∞Z. For that we write ∇fγ(Z1) = ∇fγ(P∞(Z1)) +
KQ∞(Z1) for some matrix-valued function K = K(s, θ). Again, for s ≥ s0, the
norm ‖K‖ is uniformly bounded by a constant depending only on γ and fγ . By
applying P∞ to the equation (90) and using the fact that P∞∇fγ(P∞(Z1)) =
∇fγ(P∞(Z1)) and P∞(Z1) = P∞(Z) we obtain
|∂s(P∞Z)− δ∇fγ(P∞Z)| ≤ Ce−ρs. (94)
We claim that this implies
|P∞Z(s)− ϕ
s (θ0)| ≤ Ce−ρs, s ≥ s0 (95)
for a suitable θ0. We choose a Morse coordinate x on Sγ around the critical
point q of fγ in which the gradient ∇fγ(x) = ±Mx,M > 0. Then equation (94)
becomes
∂s(P∞Z)(s)∓ δMP∞Z(s) = G(s)
with |G(s)| ≤ Ce−ρs. Then P∞Z(s) = c(s)e±δMs with e±δMs∂sc(s) = G(s).
As a consequence, for δ < ρ/M the function c admits a limit c∞ as s→ ∞ and
c(s) = c∞ −
G(σ)e∓δMσ dσ. Let θ0 be such that ϕ
s (θ0) = c∞e
±δMs (note
that c∞ = 0 if q is a maximum). Then
|P∞Z(s)− ϕ
s (θ0)| = |e±δMs
G(σ)e∓δMσ dσ|
≤ Ce−ρs.
The estimates (93) and (95) imply the conclusion.
Proposition A.3. Let δ ∈]0, δ0] and let uδ ∈ M̂A(γp, γq;Hδ, J). Let Iδ =
[s0(δ), s1(δ)] ⊂ R be an interval such that uδ(Iδ×S1) is contained in the domain
of a coordinate chart Z = (ϑ, z) around Sγ for some γ ∈ P(H).
There exist ρ > 0, θ0 ∈ S1, C > 0 and M > 0 such that z ◦ u(s, θ) and its
(first order) derivatives are bounded by
Cmax(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖)
cosh(ρ(s− s0+s1
cosh(ρ(s1 − s0)/2)
for s ∈ Iδ, θ ∈ S1. If P∞Z(s), s ∈ Iδ stays away from all but one of the critical
points of fγ, then ϑ ◦ u(s, θ)− θ − ϕ
s (θ0) and its (first order) derivatives are
bounded by
Cmax(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖)eδM(s1−s0)
cosh(ρ(s− s0+s1
cosh(ρ(s1 − s0)/2)
Moreover, if P∞Z(s), s ∈ Iδ stays away from all critical points of fγ , the above
bound is improved to (96).
Symplectic homology for autonomous Hamiltonians 88
Proof. With the notations of Proposition A.2, the Floer equation satisfied by u
can be written in local coordinates Z = (ϑ− θ, z) as
∂sZ + J∂θZ + Sz − δ∇fγ(Z1) = 0, (97)
where Z1 := ϑ − θ. Let A∞ = J ddθ + S∞(θ) the asymptotic operator at γ, let
Q∞ be the orthogonal projection onto (ker A∞)
⊥ and P∞ := 1l − Q∞. Then,
as in Proposition A.2, the quantity f(s) = 1
‖Q∞Z‖2 satisfies an inequality of
the form f ′′(s) ≥ 4ρ2f(s). Define
g(s) := max(f(s0), f(s1))
cosh(2ρ(s− s0+s1
cosh(ρ(s1 − s0))
Then (f − g)′′ ≥ 4ρ2(f − g) and f − g cannot have a strictly positive maximum.
Since f−g ≤ 0 at s0 and s1, we infer that f−g ≤ 0 on Iδ. As in Proposition A.1,
we infer the pointwise bounds for s ≥ s0
|Q∞Z(s, θ)| ≤ Cg1(s), (98)
|∂θQ∞Z(s, θ)| = |∂θZ(s, θ)| ≤ Cg1(s),
|∂s(P∞Z)(s)− δ∇fγ(P∞Z)(s)| ≤ C1g1(s),
where
g1(s) := max(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖)
cosh(2ρ(s− s0+s1
cosh(ρ(s1 − s0))
If P∞Z(s) stays away from Crit(fγ), we can assume that ∇fγ(P∞Z(s)) = M
in some suitable coordinate on S1. Then the last equation becomes
∂s(P∞Z)(s)− δM = G(s),
where |G(s)| ≤ C1g1(s). By direct integration we obtain
|(P∞Z)(s)− δMs− c0| =
s0+s1
G(σ) dσ
s0+s1
cosh(2ρ(s− s0 + s1
)) dσ
∣∣ sinh(ρ(s− s0 + s1
cosh(ρ(s−
s0 + s1
Here C2 = C1 max(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖)/
cosh(ρ(s1 − s0)) and we have
used the inequality
coshx ≤
2 cosh(x/2). Therefore, there exists a uniquely
Symplectic homology for autonomous Hamiltonians 89
determined θ0 ∈ S1 such that
|(P∞Z)(s)− ϕδfγs (θ0)|
max(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖)
cosh(ρ(s− s0+s1
cosh(ρ(s1 − s0))
max(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖)
cosh(ρ(s− s0+s1
cosh(ρ(s1 − s0)/2)
The last inequality follows from cosh(x/2) ≤
coshx. A similar manipulation
on (98) gives
|Q∞Z(s, θ)| ≤ C
2max(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖)
cosh(ρ(s− s0+s1
cosh(ρ(s1 − s0)/2)
The last two inequalities imply the conclusion of the Proposition in the case
when P∞Z(s), s ∈ Iδ stays away from Crit(fγ).
If P∞Z(s) is allowed to approach one of the critical points of fγ , the estimate
on |Q∞Z(s)| stays the same, but the estimate involving P∞Z(s) has to be
modified as follows. In a suitable Morse coordinate chart around the critical
point we can assume that ∇fγ(x) = ±Mx, M > 0 and we have to study the
equation
∂s(P∞Z)(s)∓ δMP∞Z(s) = G(s),
with |G(s)| ≤ C1g1(s). As in Proposition A.2 we have P∞Z(s) = c(s)e±δMs
with e±δMs∂sc(s) = G(s). Then c(s) = c0 +
s0+s1
e∓δMσG(σ) dσ and there
exists a θ0 ∈ S1 such that ϕ
s (θ0) = c0e
±δMs. We obtain
|(P∞Z)(s)− ϕδfγs (θ0)| ≤
s0+s1
e±δM(s−σ)G(σ) dσ
≤ eδM(s1−s0)
s0+s1
G(σ) dσ
The last integral is bounded by
max(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖)
cosh(ρ(s− s0+s1
cosh(ρ(s1 − s0)/2)
as in the previous case and the conclusion follows.
References
[1] F. Bourgeois, A Morse-Bott approach to contact homology, Ph.D. dissertation, Stanford Uni-
versity, 2002.
[2] ——, Contact homology and homotopy groups of the space of contact structures, Math. Res.
Lett. 13 (2006), 71–85.
Symplectic homology for autonomous Hamiltonians 90
[3] F. Bourgeois, A. Oancea, An exact sequence for contact and symplectic homology. Preprint
arXiv: 0704.2169.
[4] ——, The Gysin exact sequence for S1-equivariant symplectic homology. Work in progress.
[5] F. Bourgeois, K. Mohnke, Coherent orientations in symplectic field theory, Math. Z. 248
(2004), 123–146.
[6] K. Cieliebak, Handle attaching in symplectic homology and the chord conjecture, J. Eur.
Math. Soc. (JEMS) 4 (2002), 115–142.
[7] K. Cieliebak, A. Floer, and H. Hofer, Symplectic homology. II. A general construction, Math.
Z. 218 (1995), 103–122.
[8] K. Cieliebak, A. Floer, H. Hofer, and K. Wysocki, Applications of symplectic homology II:
Stability of the action spectrum, Math. Z. 223 (1996), 27–45.
[9] K. Cieliebak, K. Mohnke, Compactness for punctured holomorphic curves, J. Symplectic
Geom. 3 (2005), 589–654.
[10] D. Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in
symplectizations, Comm. Pure Appl. Math. 57 (2004), 726–763.
[11] Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom.
Funct. Anal., Special Volume (2000), Part II, 560–673.
[12] A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989),
575–611.
[13] A. Floer, H. Hofer, Coherent orientations for periodic orbit problems in symplectic geometry,
Math. Z. 212 (1993), 13–38.
[14] ——, Symplectic homology. I. Open sets in Cn, Math. Z. 215 (1994), 37–88.
[15] A. Floer, H. Hofer, and D. Salamon, Transversality in elliptic Morse theory for the symplectic
action, Duke Math. J. 80 (1995), 251–292.
[16] U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology, Int. Math.
Res. Not. 2004, no. 42, 2179–2269.
[17] H. Hofer, D. Salamon, Floer homology and Novikov rings, in The Floer memorial volume,
Eds. H. Hofer et al. Progress in Math., vol. 133, Birkhäuser, 1995, pp. 483–524.
[18] H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplecti-
zations. I. Asymptotics. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 337–379.
[19] H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectiza-
tions. IV. Asymptotics with degeneracies. in Contact and symplectic geometry (Cambridge,
1994), Ed. C.B. Thomas. Publ. Newton Inst., vol. 8, Cambridge Univ. Press, Cambridge,
1996, 78–117.
[20] H. Hofer, K. Wysocki, and E. Zehnder, Polyfolds and Fredholm theory. Part I, Preliminary
preprint, 182 pp., 2005.
[21] D. McDuff, D. Salamon, J-holomorphic curves and symplectic topology, AMS Colloquium
Publications, vol. 52, AMS, 2004.
[22] A. Oancea, A survey of Floer homology for manifolds with contact type boundary or sym-
plectic homology, Ensaios Mat. 7 (2004), 51–91.
[23] J. Robbin, D. Salamon, The Maslov index for paths, Topology 32 (1993), 827–844.
[24] D. Salamon, Lectures on Floer Homology, in Symplectic Geometry and Topology, Eds.
Y. Eliashberg and L. Traynor. IAS/Park City Math. Series, vol. 7, AMS, 1999, pp. 143–229.
[25] D. Salamon, E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the
Maslov index, Comm. Pure Appl. Math. 45 (1992), 1303–1360.
Symplectic homology for autonomous Hamiltonians 91
[26] M. Schwarz, Cohomology operations from S1-cobordisms in Floer homology, PhD thesis, ETH
Zurich, 1995.
[27] I. Ustilovsky, Contact homology and contact structures on S4m+1, Ph.D. dissertation, Stan-
ford University, 1999.
[28] C. Viterbo, Functors and computations in Floer homology with applications. I. Geom. Funct.
Anal. 9 (1999), 985–1033.
Index
A∞, asymptotic operator, 27, 84
BCa∗ (H), BC
∗(H), Morse-Bott chain groups, 20
Du, linearized operator, 9
D ew, fibered sum linearized operator, 54
G, gluing map, 58
Gδ,ǫ( ew), pre-glued curve, 46
H′a,b,ǫ, 34
Hδ , perturbed Hamiltonian, 14
P , mixing map, 57
P∞, asymptotic operator, 27, 84
Qδ , right inverse for gluing theorem, 56
Q∞, asymptotic operator, 27, 84
R = R(δ), gluing profile, 46
Rλ, Reeb vector field, 6
S, splitting map, 58
SCa∗ (H), symplectic chain group, 8
SHa∗ (W,ω), symplectic homology, 11
Sγ , circle of orbits, 14
S∞, asymptotic matrix, 27, 84
X, Liouville vector field, 6
XH , Hamiltonian vector field, 6
Λω , Novikov ring, 8
ǭ(u), 20
β, cutoff function, 23, 31, 34, 46, 48, 70
βL, cutoff function, 57
AH , action functional, 7
B′δ, 33
BA = B1,p,d(Sγ , Sγ , A;H), 22
BAδ = B
1,p,d
(γp, γq
, A;H, {fγ}), 30
E(u), energy, 9
Freg(H, J), regular families {fγ}, 18
H, admissible Hamiltonians, 7
H′, autonomous admissible Hamiltonians, 13
J , admissible a.c. structures, 7
J ′, time-indep. admissible a.c. structures, 17
J ′reg(H), 17
Jreg(H), 9, 17
M0(γ, γ;H, J), cM0(γ, γ;H, J), 9
MA(Sγ , Sγ ;H, J), cM
A(Sγ , Sγ ;H, J), 16–17
MA(Sγ , eq;H, J), cM
A(Sγ , eq;H, J), 16–17
MA(γ, γ;H, J), cMA(γ, γ;H, J), 8–9
MA]0,δ0]
(γp, γq
;H, {fγ}, J), 16
MAm(p, q;H, {fγ}, J), M
A(p, q;H, {fγ}, J), 18–
O, space of CR operators, 69–70
P(H),Pa(H), 7
i−1(a)
, 11–12
S, loops of symmetric matrices without degen-
erate directions, 69
∂̄H , 22
∂̄Hδ,J , 31
∂̄a,b,ǫ := ∂̄H′
a,b,ǫ
,J , 34
ǫ(u), 21, 80
ǫ(u), ǫ(uδ), 10, 20
γmin , γMax , surviving orbits, 15
ind(p), index of critical point of fγ , 18
µ(γ), index of Reeb orbit, 8, 15
ev, ev, evaluation maps, 17
∂, Floer differential, 10
∂, Morse-Bott differential, 20
eO, space of CR operators, 74–75
eS, loops of symmetric matrices with one degen-
erate direction, 74
Spec(M,λ), 6
O, cover of space of CR operators, 76
cW , symplectic completion, 6
bω, symplectic form on the completion cW , 6
eBδ , 44
ξ, contact distribution, 6
{∂̄T }, 43
fγ , Morse function on Sγ , 14, 18
gδ,ǫ(s), weight function for gluing, 47
ha,b,ǫ, 32
ka,b,ǫ, 32
uδ,γ,a,b,ǫ, gradient cylinder, 32
	Introduction
	Symplectic homology
	The Morse-Bott chain complex
	Morse-Bott moduli spaces
	Transversality
	Compactness for Morse-Bott trajectories
	Gluing for Morse-Bott moduli spaces
	Coherent orientations
	Appendix: Asymptotic estimates
