Table Of ContentQUANTIZATION OF THE SERRE SPECTRAL SEQUENCE
7
0 JEAN-FRANCOISBARRAUDAND OCTAV CORNEA
0
2
Abstract. The present paper is a continuation of [1] and [2]. It ex-
n
plores how the spectral sequence introduced in [1] interacts with the
a
presence of bubbling. As consequences are obtained some relations be-
J
tween binary Gromov-Witten invariants and relative Ganea-Hopf in-
2
variants,acriterionfordetectingthemonodromyofbubblingaswellas
2
algebraic criteria for the detection of periodic orbits.
]
G
S
.
h
t
a Contents
m
1. Introduction 1
[
1.1. Acknowledgment. 3
1
2. Notation and statement of results 3
v
5 2.1. Setting and recalls 3
2 2.2. Main statement. 6
6
3. Proof of the main theorem. 7
1
3.1. Construction of the truncated spectral sequence. 7
0
7 3.2. Invariance of the truncated spectral sequence. 11
0 3.3. Detection of monodromy. 13
/
h 3.4. Quantum perturbation of the Serre spectral sequence. 15
at 4. Examples, applications and further comments 19
m 4.1. Extensions 19
: 4.2. Examples 20
v
4.3. Fibrations over S2 22
i
X
4.4. Non trivial periodic orbits for Morse functions 24
r References 27
a
1. Introduction
In [1] has been introduced an algebraic way to encode the properties of
high dimensional moduli spaces of trajectories in Morse-Floer type theories.
The basic idea is that, by making use of a “representation” theory of the
relevant moduli spaces
lx,y
M(x,y) −→ G
1
2 J.-F.BARRAUDANDO.CORNEA
into some sufficiently large topological monoid G, one can define a “rich”
Morse type chain complex whose differential is of the usual form
dx = a y
x,y
X
y
but a , the coefficient “measuring” the moduli space M(x,y), belongs to
x,y
a graded ring (for example, the ring of cubical chains of G) and is in general
not zero when dim(M(x,y)) >0. By representation theory it is meant here
notonly that themapsl are continuous butalso that they are compatible
x,y
in the obvious way with compactification and with the crucial boundary
formula:
(1) ∂M(x,y) = M(x,z)×M(z,y) .
[
z
The complex constructed this way comes with a natural filtration induced
by the grading of the generators x,y,.... The pages of order greater than 1
of the associated spectral sequence are invariant with respect to the various
choices made in the construction and their differentials encode algebraically
the properties of the M(x,y)’s.
This construction is described in the absence of bubbling in [1] and, in
[2], it is shown to be easily extendable to cases when pseudo-holomorphic
spheres and disks exist as long as we work under the threshold of bubbling.
The present paper explores what happens when bubbling does occur.
It is obvious that to study this case it is natural to start with the Hamil-
tonian version of Floer homology and this is indeed the setting of this pa-
per. In particular, the modulispaces M(x,y) consist of Floer tubes and the
monoid G is the space of pointed Moore loops on M, ΩM, with (M2n,ω)
our underlying symplectic manifold. We will also restrict to the monotone
case even if the machinery described here appears to extend to the general
case. The reason for this is that the main phenomena we have identified
are already present in this case and, at the same time, in this way we avoid
to deal with the well-know transversality issues which are present in full
generality.
Here is a short summary of our findings. Firstly, it is not surprising that
when bubbling is possible, only some of the pages of the spectral sequence
mentioned before exist. It is also expectable that the number of pages that
are defined should roughly be the minimal Chern class, c , and that,
min
moreover, some of these pages should again be independent of the choices
made in the construction.
What is remarkable is that, in general, these pages do not coincide with
those associated to a Morse function: a quantum deformation is generally
present. Given that in the Morse case the resulting spectral sequence is,
as shown in [1], the Serre spectral sequence of the path-loop fibration over
QUANTIZATION OF THE SERRE SPECTRAL SEQUENCE 3
M, we see that this construction provides a new symplectic invariant which
consists of the first c pages (together with their differential) of a spectral
min
sequence which is a quantum deformation of the Serre spectral sequence.
One additional important point is that, on the last defined page, the pre-
sumptive differential, dr, is still defined and invariant but might not verify
(dr)2 = 0.
Ofcourse,thenextstageistounderstand-atleastinpart-thisquantum
deformation in terms of classical Gromov-Witten invariants. In this respect
we obtain that the first interesting differential dr, can be expressed in terms
of binary Gromov-Witten invariants (these are those associated to spheres
with two marked points) and Ganea-Hopf invariants (these control the clas-
sical part of the differential). Moreover, in this case, the relation (dr)2 = 0
becomes a relation between these two types of invariants which takes place
in the Pontryaguin ring H (ΩM). Undoubtedly, this is just a first step
∗
towards understanding the deeper relationships between the combinatorics
of Gromov-Witten invariants and classical algebraic topology invariants en-
coded in the ring structure of H (ΩM).
∗
The next interesting point is to understand what happens for the first r
when (dr)2 6= 0. Clearly, the culprit is bubbling but interestingly enough
what this non-vanishing relation detects is monodromy - the fact that in
the appropriate moduli space the attachment point of the bubbles turns
non-trivially around Floer cylinders - which turns out to interfere with the
representation maps l . The fact that dr is invariant but, simultaneously,
x,y
(dr)2 might not vanish is quite remarkable and, indeed, this morphism dr
has interest in itself and is seen to be, in fact, a generalization of the Seidel
homomorphism [10]. Finally, we also discuss an application of this struc-
ture to the detection of periodic orbits. This provides a sort of algebraic
counterpart to the result of Hofer-Viterbo [5].
The paper is structured as follows. In the second section we introduce
the main notation and give the precise statements of our results. The third
section contains the proofs. Inthe last section we firstshortly mention some
possible extensions of the construction we then provide some examples and,
finally, we discuss the aplication to periodic orbits.
1.1. Acknowledgment. It is our great pleasure to dedicate this paper to
Dusa McDuff on the occasion of her 60st birthday. This is even more appro-
priateas,earlyinthisproject,webelieved,aposteriori withoutjustification,
thatthemonodromyof bubblingismuch lessrelevant anditisoneofDusa’s
questions which made us reconsider the issue and appreciate the full impor-
tance of this phenomenon.
2. Notation and statement of results
2.1. Setting and recalls. Fix the symplectic manifold (M2n,ω) and we
suppose for now that M is closed. We assume that M is monotone in
4 J.-F.BARRAUDANDO.CORNEA
the sense that the two morphisms ω : π (M) → R and c : π (M) → Z
2 1 2
are proportional with a positive constant of proportionality ρ. We denote
by c the minimal Chern class and by ω the corresponding minimal
min min
symplectic area (so that we have ω = ρc ).
min min
2.1.1. Binary Gromov-Witten invariants. Fix on M a generic almost com-
plex structure J which tames ω. The binary Gromov Witten invariants
we are interested in can be described as follows: pick a generic Morse
function f and metric on M. Denote by i(x) = ind (x) for each x ∈
f
Crit(f). For two critical points x and y and a class α ∈ π (M) such that
2
i(x) − i(y) + 2c (α) − 2 = 0, we define GW (x,y) as the number of ele-
1 α
ments in the moduli space M(J,α;x,y) which consists of J-holomorphic
spheres in the homology class α with two marked points, one lying on the
unstable manifold of x and the other on the stable manifold of y, mod-
ulo reparametrization. As such GW (x,y) is not an invariant (because x,y
α
might not be Morse cycles). However, if for two classes [x] = [ λ x ]
i i
and [y] = [ µ y ], we define GW ([x],[y]) = λ µ GW (x ,y ) tPhen we
i i α i j α i j
obtain an iPnvariant. For α ∈ π , let [α] be iPts image by the morphism
2
π (M) → H (ΩM).
2 1
2.1.2. The Novikov ring. Let L(M) bethe space of contractible loops in M.
Let Γ be the image of the Hurevicz morphism π (M) → H (M,Z/2).
2 2
The two forms ω and c define morphisms Γ :−ω−,c→1 R,Z which under our
1
monotonicity assumption are proportional. Let Γ = Γ/ker(ω). We let Λ
0
be the associated Novikov ring which is defined as follows
Λ = λ eα .
α
X
α∈Γ0
where the coefficients λ belongto Z/2 suchthat
α
∀c> 0, ♯{α,λ 6= 0,ω(α) ≤ c} < +∞ .
α
The grading of the elements in Λ is given by |eλ| = −2c (λ).
1
We also denote by L˜(M) the covering of L(M) associated to Γ : it is the
0
quotient of the space of couples (γ,∆), where γ ∈ L(M) and ∆ is a disk
bounded by γ, under the equivalence relation (γ,∆) ∼ (γ′,∆′) if γ = γ′ and
ω([∆−∆′]) = c ([∆−∆′]) = 0.
1
Remark 1. Here and later in the paper we could also use, alternatively,
rational coefficients as all the moduli spaces involved are orientable and the
orientations are compatible with our formulae.
2.1.3. Moduli spaces of Floer tubes. Let H :M ×S1 → R be a Hamiltonian
function. The Hamiltonian flow associated to H is the flow of the (time
dependent) vector field X defined by :
H
ω(X ,·) = −dH .
Ht t
QUANTIZATION OF THE SERRE SPECTRAL SEQUENCE 5
All along this paper, the periodic orbits of X will be supposed to be non
H
degenerate. We denote by P ⊂ L(M) the set of all contractible periodic
H
orbitsof thehamiltonianflowassociated toH andweletP˜ bethecovering
H
of P which is induced from L˜(M).
H
For each periodic orbit x ∈ P we fix a lift (x,∆ ) ∈ P˜ . For a generic
H x H
pair (H,J) and x,y ∈ P , λ ∈Γ we now consider the moduli spaces:
H 0
M′(x,y;λ) = {u :R×S1 :u verifies (2)}
so that the pasted sphere ∆ ∪u∪(−∆ ) is of class λ and
x y
(2)
∂ u+J(u)∂ u−J(u)X (u) = 0 , lim u(s,t) = x(t), lim u(s,t) =y(t) .
s t H
s→−∞ s→+∞
Of course, these moduli spaces are quite well-known in the subject and we
refer to [9] for their properties. In particular, they have natural orientations
and, when (x,∆ ) 6= (y,∆ ) they admit a free R action. We denote the
x y
quotient by this action by M(x,y;λ) and we have
dimM(x,y;λ) = µ((x,∆ ))−µ((y,∆ ))+2c (λ)−1
x y 1
where µ((x,∆ ) is the Conley-Zehnder index of the orbit x computed with
x
respect to the capping disk ∆ .
x
2.1.4. Monodromy of bubbling. Among the standard properties of the mod-
uli spaces above we recall that they admit a natural topology as well as
natural compactifications, M(x,y;λ), so that the following formula is valid:
(3) ∂M(x,y;λ) = M(x,z;λ′)×M(z,y;λ′′) ∪ Σ
x,y,λ
z,λ′+[λ′′=λ
Here Σ is a set of codimension 2 which consists of Floer tubes with
x,y,λ
at least one attached bubble.
We will say that (H,J) has bubbling monodromy if there exist x,y ∈ P
H
and λ ∈ Γ so that:
0
H1(Σ ;Z) 6= 0
x,y,λ
Thismeans,inparticular,thatπ (Σ ) 6= 0sothattherearenon-contractible
1 x,y,λ
loops in the space of Floer tubes with bubbles.
2.1.5. Truncated differentials and spectral sequences. The following alge-
braic notions will be useful in the formulation of our results.
We say that the sequence of graded vector spaces (Er,dr), 0 ≤ r ≤ k is
a truncated spectral sequence of order k if (Er,dr) is a chain complex for
each r ≤ k − 1 which verifies H (Er,dr) = Er+1 and dk is a linear map
∗
of degree −1. A truncated spectral sequence of ∞-order is a usual spectral
sequence. A morphismof order k truncated spectral sequences is a sequence
of chain maps φ : (Er,dr) → (Fr,dr), 0 ≤ r ≤ k, so that H (φ ) = φ for
r ∗ r r+1
0≤ r ≤ k−1. We say that two truncated spectral sequences are isomorphic
6 J.-F.BARRAUDANDO.CORNEA
starting from page s is they admit a morphism which is an isomorphism on
page s (and, hence, on each later page).
The typical example of a truncated spectral sequence appears as follows.
AssumethatC isagradedrationalvector spaceandthatFiC isanincreas-
∗
ing filtration of C . We say that a linear map d : C → C is a truncated
∗ ∗ ∗−1
differential of order k compatible with the given filtration if d(FiC)⊂ FiC
∀i and
(d◦d)(FrC)⊂ Fr−2kC
for all r ∈ Z. It is easy to see that a truncated differential of order k
induces a truncated spectral sequence of the same order. Indeed, by using
the standard descriptions of the r-cycles
Zr = {v ∈ FpC : dv ∈ Fp−rC}+Fp−1C
p
and r-boundaries
Br = {dFp+r−1C ∩FpC}+Fp−1C
p
it is immediate to see that Br ֒→ Zr for 0 ≤ r ≤ k which allows us to define
p p
the pages of the truncated spectral sequence by Er = Zr/Br. Obviously, d
p p p
induces differentials dr on Er when r < k as well as a degree −1 linear map
dk on Ek.
2.2. Main statement. We will formulate our main statement in a simple
case and we will discuss various extensions at the end of the paper. There-
fore, we assume here that (M,ω) is closed, simply-connected and monotone
with c ≥ 2.
min
Theorem 2.1. There exists a truncated spectral sequence of order c ,
min
(Er(M),dr), whose isomorphism type starting from page 2 is a symplectic
invariant of (M,ω) and which has the following additional properties:
i. As a bi-graded vector space we have an isomorphism:
E2 ∼= H (M)⊗H (ΩM)⊗Λ .
∗ ∗
ii. The differential d2 has the decomposition
d2 = d2+d2
0 Q
where d2 is the differential appearing in the classical Serre spectral
0
sequence of the path loop fibration ΩM → PM → M and
d2x= GW (x,y)y[α]eα .
Q α
X
y,α
iii. If (dcmin)2 6= 0, then any regular pair (H,J) has bubbling mon-
odromy.
Remark 2. Clearly, if d2◦d2 = 0 - for example if c ≥ 3 - the vanishing of
min
the square of d2◦d2 translates into some relations between binary Gromov-
Witten invariants and the classical Serre spectral sequence differential d2.
0
In turn, this differential is quite well known and rather easy to compute and
QUANTIZATION OF THE SERRE SPECTRAL SEQUENCE 7
itcan beexpressedin many cases in terms of relative Ganea-Hopf invariants
(see [4]). The interesting part about these relations is that they take place
in the Pontryaguin algebra H (ΩM). Indeed, in the formula at ii. α ∈
∗
H (ΩM), eα ∈ Λ and x,y ∈ H (M) so that in the square of the differential
1 ∗
appears the Pontryaguin product H (ΩM)⊗H (ΩM) → H (ΩM).
1 1 2
The relation with the Seidel homomorphism is seen by considering the
spectral sequence in the case of a symplectic fibration over CP1.
We also formulate here a very simple version of our application to the
detection of periodic orbits. We specialize to the case when the manifold M
admits a perfect Morse function (that is a Morse function whose associated
Morse complex has trivial differential). We also need the following notion.
Letx,y ∈ H (M)andλ ∈ Λ. Wewillsay thatxandyeλ (which existonthe
∗
E2 page of thespectralsequencein Theorem2.1) are dr-related if xsurvives
to the r-th level of the spectral sequence and there is some γ ∈ C (ΩM)
∗
so that the product γ ⊗yeλ also survives to the r-th page of the spectral
sequence and we have dr([x]) = [γ ⊗yeλ]+...
Corollary 2.2. Assume that there are homology classes x,z ∈ H (M),
∗
|x| < |z|, so that x is dr-related to zeλ and H (M) ⊗ Λ = 0 for |x| >
k q
k + q > |zeλ|). Then any self-indexed perfect Morse function on M has
some non-trivial closed characteristic.
By a self-indexed Morse function f we mean here that the critical points
ofthesameindexhavethesamecriticalvalueandind (x) > ind (y)implies
f f
f(x)> f(y).
There are many ways in which this statement can be extended and some
will be discussed at the end of the paper.
3. Proof of the main theorem.
3.1. Construction of the truncated spectral sequence. In this sec-
tion we fix the 1-periodic hamiltonian H and almost complex structure J
compatible with ω so that the pair (H,J) is generic (of course, both are in
general time-dependent). For simplicity, we will also assume to start that
the manifold M is simply-connected but we will see later on that this con-
dition can be dropped with the price that the construction becomes more
complicated.
As in [1] the truncated spectral sequence we intend to discuss is induced
by a natural filtration of an enriched Floer type pseudo-complex. We use
the term pseudo-complex here to mean that we will not have here a true
differential but rather a truncated one. The construction of this pseudo-
complex is a refinement of the classical Floer construction in which the
coefficient ring is replaced with the ring of cubical chains over the Moore
loops on M. Here is this construction in more detail.
3.1.1. Coefficient rings. Let C denote the “cubical” chain complex, let ΩX
∗
betheMooreloopspaceoverX (thespaceofloopsparametrizedbyintervals
8 J.-F.BARRAUDANDO.CORNEA
of arbitrary length). Consider the space M′ obtained from M by collapsing
to a point a simple path γ going through the starting point of each periodic
orbit. NoticethatC (ΩM′)isadifferentialringwheretheproductisinduced
∗
by the concatenation of loops. Finally, our coefficient ring is:
R = C (ΩM′)⊗Λ .
∗ ∗
This is a (non abelian) differential ring, and its differential will be denoted
by ∂.
The (pseudo)- complex we are interested in is a (left) differential module
generated by the contractible periodic orbits of H over this ring:
C(H,J) = ⊕ R x˜/ ∼
x˜∈P˜H ∗
with the identification x˜eλ ∼ x˜♯λ, where x˜♯λ stands for the capping of x
obtained by gluing a sphere in the class λ to x˜. The grading of an element
in x˜ ∈ P˜ is given by the respective Conley-Zehnder index. There is a
H
natural filtration of this complex which is given by
FrC(H,J) = R < x˜ ∈P˜ : µ(x˜) ≤ r > .
∗ H
We will call this the canonical filtration of C(H,J).
3.1.2. Truncated boundary operator. The next step is to introduce a trun-
cated differential on C(H,J). We recall from §2.1.3 the definition of the
moduli spaces M(x,y;λ) of Floer tubes. We recall also that this definition
requires a choice of lift x˜ ∈ P˜ for each x ∈ P . With these conventions
H H
and-as assumedbefore-foragenericchoice ofJ andH -themodulispaces
are smooth manifolds of dimension |x˜|−|y˜|−1 when |x˜| =6 |y˜|, and they
have a natural compactification involving “breaks” of the tubes on interme-
diate orbits, or bubbling off of holomorphic spheres. We will write M(x˜,y˜)
for the moduli space of Floer tubes which lift to paths inside L˜(M) joining
x˜ ∈ P˜ to y˜ ∈ P˜ and we let M(x˜,y˜) be the respective compactification.
H H
In our monotone situation these compactifications are pseudo-cycles with
boundary.
To define the truncated boundary operator we proceed as in the usual
Floer complex, but we intend to take into consideration the moduli spaces
of arbitrary dimensions instead of restricting to the 0 dimensional ones. To
associate to the (compactification of the) moduli spaces coefficients in our
ring R, we first need to represent them into the loop space Ω(M′), and
then pick chains representing them (i.e. defining their fundamental classes
relative to their boundary).
Let us start with “interior” trajectories, i.e. elements v ∈ M(x˜,y˜). Let
u : R×S1 → M be a parametrization of v. Since the value of the action
functional
a : L˜(M) → R, a ((γ,∆)) = − ∆∗ω+ H(t,γ(t))dt
H H
Z Z
D2 S1
QUANTIZATION OF THE SERRE SPECTRAL SEQUENCE 9
is strictly decreasing along the R direction, it can be used to reparametrize
u by the domain [−a(x˜),−a(y˜)]×S1, and the restriction of u to the interval
[−a(x˜),−a(y˜)]×{0} defines a Moore loop in M′. This defines a map
(4) σ : M(x˜,y˜) → Ω(M′)
x˜,y˜
which is continuous. We will call it the “spine” map.
This map should then be extended to the compactification M(x˜,y˜) of
M(x˜,y˜).
It is well-known that the objects in M(x˜,y˜) are constituted by Floer
trajectories possibly broken on some intermediate periodic orbits to which
might be attached some J-holomorphic spheres that have bubbled off.
It is easy to see that the map σ extends continuously over the part of
x˜,y˜
this set where no spheres are attached to some tube in a point belonging
to the line R× {0}. Indeed, as in [1], except for these types of elements,
the spine map is compatible with the breaking of Floer tubes in the sense
that the loop associated to a broken trajectory is the product of the loops
associated to each “tube” component.
Let α ∈Γ be the class so that c (α ) = c (by our monotonicity
min 0 1 min min
assumption there is a single such class). By using again the monotonicity
assumption we see that bubbling off of a sphere in class α ∈ Γ can occur
0
in a moduli space M(x˜,y˜) with y˜6= x˜♯α only if
|x˜|−|y˜| ≥ 2c (α)+1 .
1
It is also important to note that bubblingof a spherein the class α is also
possible inside the space M(x˜,x˜♯α). In all cases, bubbling of an α sphere is
never possible if |x˜|−|y˜| ≤ 2c (α)−1.
1
We summarize this discussion:
Lemma 3.1. The spine map σ extends continuously to M(x˜,y˜) if
x˜,y˜
|x˜|−|y˜| ≤ 2c −1 .
min
In case |x˜|−|y˜|= 2c and if σ does not have such a continuous extension
min
to M(x˜,y˜), then y˜= x˜♯α .
min
Thespinemapobtainedinthiswaysatisfiesalsoacompatibility condition
which we now make explicit. If M(x˜,z˜) × M(z˜,y˜) ⊂ M(x˜,y˜), then the
restrictionofσ onthesetontheleftoftheinclusionequalsm◦(σ ×σ )
x˜,y˜ x˜,z˜ z˜,y˜
where
m : ΩM′×ΩM′ → ΩM′
is loop concatenation.
For pairs (x˜,y˜) with |x˜|−|y˜| ≤ 2c −1, we use the map σ to rep-
min x˜,y˜
resent the moduli spaces M(x˜,y˜) inside the loop space Ω(M′). We then
choose “chain representatives” m(x˜,y˜) ∈ C (ΩM′), i.e. chains generating
∗
the fundamental class of σ(M(x˜,y˜)) relative to its boundary, in such a way
10 J.-F.BARRAUDANDO.CORNEA
that:
(5) ∂m(x˜,y˜) = m(x˜,z˜)∗m(z˜,y˜)
X
|y|<|z|<|x|
where∗is theoperation induced on C (ΩM′) by the concatenation of loops.
∗
The key point regarding this formula is that, under our assumption |x˜|−
|y˜| ≤ 2c −1, the compactified moduli space M(x˜,y˜) is a manifold with
min
boundary. Moreover, its boundary verifies the usual formula valid in the
absence of bubblingso that the construction of the m(−,−)’s is the same as
that in the non-bubbling setting. We refer to [1] for a complete discussion
of this construction.
We now define the boundary operator d by:
(6) dx˜ = m(x˜,y˜) y˜
X
1≤|x˜|−|y˜|≤2cmin−1
and extend it to the full complex using the Leibnitz rule.
It is easy to check that d has degree −1 with respect to the total grading
and that it is compatible with the canonical filtration. Notice first that if
γ ⊗x˜ ∈ C (ΩM)⊗P˜ we have d◦d(γ ⊗x˜) = (γ ⊗(d◦d)(x˜)). We now
∗ H
compute:
d◦d(x˜)= d(m(x˜,y˜) y˜)
X
|x˜|−|y˜|≤2cmin−1
= ∂m(x˜,y˜) y˜+m(x˜,y˜) dy˜
X
1≤|x˜|−|y˜|≤2cmin−1
= m(x˜,z˜)m(z˜,y˜) y˜+ m(x˜,y˜)m(y˜,z˜) z˜
X X
1≤|x˜|−|y˜|≤2cmin−1 1≤|x˜|−|y˜|≤2cmin−1
|y˜|+1≤|z˜|≤|x˜|−1 1≤|y˜|−|z˜|≤2cmin−1
= m(x˜,y˜)m(y˜,z˜) z˜
X
1≤|x˜|−|y˜|≤2cmin−1
|y˜|−2cmin+1≤|z˜|≤|x˜|−2cmin
and we see that d2 drops the filtration index by at least 2c . In the
min
algebraic terms of §2.1.5 we obtain:
Lemma 3.2. With the definition above, d is a truncated differential of order
c with respect to the canonical filtration on C(H,J) and thus it induces
min
a truncated spectral sequence Er(H,J) of the same order so that
E2(H,J) ∼= H (M)⊗H (ΩM)⊗Λ
∗ ∗
TheisomorphisminthelemmaisobviousbecauseE1(H,J) ∼= CF (H,J)⊗
∗
H (ΩM) and as d1 only involves the 0 dimensional moduli spaces of Floer
∗
tubes we obtain that d1 is just: d ⊗id where (CF (H,J),d ) is the usual
F ∗ F
Floer complex (with coefficients in the Novikov ring Λ). Thus we have con-
structed our truncated spectral sequence and have proved property i. in
Theorem 2.1
