Homotopy equivalence of nearby Lagrangians and the Serre spectral sequence

Let $L\subset T^{*}N$ be an exact Lagrangian embedding with $L$ and $N$ closed (compact without boundary). We will always assume that $N$ is connected, but for generality we will not assume that $L$ is connected. In [8], Fukaya, Seidel, and Smith constructed a spectral sequence converging to the Lagrangian intersection Floer homology of $L$ with itself, and used this to prove that exact relative spin Lagrangians in simply connected cotangent bundles with vanishing Maslov class are homology equivalent to the base (a similar result was simultaneously obtained by Nadler in [13]). This was extended by Abouzaid in [1] to prove that vanishing Maslov class implies homotopy equivalence (combined with the result in [11] this actually proves homotopy equivalence for all exact Lagrangians). These approaches are rather technical and the goal of this paper is to prove a slightly weaker version in a much simpler way. To be precise we reprove the following theorem.

We will prove the theorem by constructing a spectral sequence similar to the one used by Fukaya, Seidel, and Smith. We will construct this for any exact Lagrangian $L$ with any local coefficient system of vector spaces over some field $\mathbb{F}$ (and with a relative pin structure when needed).

The construction of this spectral sequence goes as follows. We start with a Morse function (with some restrictions that we will not write out here) $g:N\to\mathbb{R}$ and consider two large scale perturbations of $L$ given by

	$\displaystyle K_{t}$	$\displaystyle=tL$
	$\displaystyle L_{t}$	$\displaystyle=tL+dg$

for very small $t>0$. So $K_{t}$ is a scaling of $L$ by a very small constant making it very close to the zero section, and $L_{t}$ is the same but pushed off the zero-section using the Morse function $g$, so that it is close to the graph of $dg$ instead. This is illustrated in Figure 1 close to a critical point $q$ of $g$.

As the figure illustrates all the intersection points of the two Lagrangians will “bunch” around the critical points of $g$. Each of the intersections points in the bunch close to $q$ will have action close to the critical value $g(q)$ (up to an overall shift that we thus fix). For small $t$ we now consider an action filtration such that we have a non-trivial filtration level for each critical value of $g$ and it contains all the intersection points in all bunches with action value close to this critical value. So, each filtration level contains an unspecified number of these bunches.

The main technical part of the construction is carried out in Section 3. There we basically prove that each of the bunches on the same filtration level do not interact (with respect to the differential), and that restricting the differential to any bunch is well-defined and that this always produces the same homology groups - up to a shift by the Morse index of the associated critical point of $g$. In fact, we will use this “bunching” construction to create a local system on $N$. This we will use in Section 4 to prove that, for $g$ self-indexing, page two of the associated spectral sequence looks a lot like a Serre spectral sequence. In fact we will identify page 1 as the Morse homology complex of $g$ with coefficients in the local system defined in Section 3.

The original spectral sequence by Fukaya, Seidel, and Smith did not look as much as a Serre spectral as the one in this paper - so we now explain the difference. Consider the following two filtrations defined for a fibration $\pi:E\to X$ where $X$ is a finite cell complex by:

• •

  $\pi^{-1}(X_{0})\subset\cdots\subset\pi^{-1}(X_{j})\subset\pi^{-1}(X_{j+1})\subset\cdots\subset E$, where $X_{j}$ denotes the $j$-skeleton. This defines the Serre spectral sequence (see Hatcher [10]).

• •

  Similar, except $X_{j}$ is not the $j$-skeleton, but $X_{j+1}$ is $X_{j}$ with a single new cell. This leads to a spectral sequence analogous to the one by Fukaya, Seidel, and Smith with a filtration level per cell - not necessarily ordered by dimension.

In a Morse theoretic construction this corresponds to the two cases:

• •

  Self-indexing Morse function - where the critical value equals the Morse index.

• •

  Any Morse function with distinct critical value for each critical point.

The relation between these two viewpoints (and its analogy to the bunching of critical points) can be described as follows. Assume $E\to X$ is a fiber-bundle of closed manifolds. Let $f$ be a Morse function on $X$. This makes $f^{\prime}=(f\circ\pi):E\to\mathbb{R}$ a Morse-Bott function. Perturbing $f^{\prime}$ slightly to make it Morse we get bunches of critical points each close to one of the original critical fibers, and we may define a filtration on the Morse complex of $f^{\prime}$ by using a sequence of values intertwining the original critical values of $f$ in such a way that all the bunches associated to the same original critical value are in the same filtration level. By standard perturbation arguments the individual bunches close to fibers over critical points with the same critical value do not interact in the differential, and the local system produced by a construction similar to that in Section 3 will simply be the homologies of the fibers.

The final piece to proving Theorem 1 is essentially to establish a version of Poincaré duality fiber-wise. A heuristic description of why this might be a useful property is as follows; the fibers represent the relative difference $L\to N$. However if the fibers also behave as a manifold - then this is homologically supposed to look like a fiber-bundle, and since $L$ and $N$ have the same dimension the fiber basically (homologically) has to be a $0$ dimensional manifold. A similar argument was used in the simply connected case by Fukaya, Seidel, and Smith.

The general layout of the paper is as follows. In section 2 we briefly describe Lagrangian intersection Floer homology of two exact Lagrangians $K$ and $L$ and how to apply local coefficients. In Section 3 we define the fiber-wise (over $N$) intersection Floer homology of any Lagrangian $L$ with itself using the idea of the bunches described above. We also prove that this fiber-wise Floer homology defines a graded local system on $N$, and satisfies other natural properties that we will need - most importantly the Poincaré duality mentioned above. In Section 4 we use action filtrations to construct the spectral sequence as described above, converging to the full Lagrangian intersection Floer homology; and we also identify page 1 of this spectral sequence for special cases of $g$. In Section 5 we extend the type of local coefficient systems we allow to include local systems on the universal cover of $N$. The reader only interested in the case where $N$ is simply connected can skip this section. Then in Section 6 we prove Theorem 1 starting with the simply connected case not requiring Section 5.

It should be noted that the ideas used in this construction are similar to the original ideas behind the spectral sequence constructed by Fukaya, Seidel, and Smith.

In [5], Floer introduced the Lagrangian intersection Floer homology $HF_{*}(K,L)$; and proved that it is a Hamiltonian isotopy invariant. In this section we recall this construction for two exact Lagrangians $K,L\subset T^{*}N$. This also serves to fix some conventions regarding signs, gradings and orientations. We will consider some ground field $\mathbb{F}$. However, we will consider any local coefficient systems of vector spaces over $\mathbb{F}$ defined on $K$ or $L$, and describe (Corollary 2.3) the generalization of Floer’s result that

$\displaystyle HF_{*}(L,L;\mathbb{F})\cong H_{*}(L;\mathbb{F}).$

(1)

to such local coefficient system. Formally we consider the local systems on $K$ and $L$ to be non-graded or, equivalently, graded in degree 0.

The reader only interested in the case of both $L$ and $N$ simply connected can ignore the local coefficients in this section. However, we note that we still need to specifically identify a certain differential in the spectral sequence in Proposition 4.1, which means we need to understand the fiber-wise Floer homology defined in the next section as a graded local system on the base $N$. So, one cannot avoid local coefficients in this argument, and hence it does not simplify matters much to ignore them here.

Let $N$ be any closed (compact without boundary) manifold. The canonical 1-form (or Liouville form) $\lambda\in\Omega^{1}(T^{*}N)$ on the cotangent bundle is defined by

$\displaystyle\lambda_{q,p}(v)=p(\pi_{*}(v)),\qquad q\in N,\,p\in T^{*}_{q}N,\,\pi:T^{*}N\to N.$

The canonical symplectic form is then given by $\omega=-d\lambda$. Pick a Riemannian structure on $N$, then we get an induced almost complex structure $J$ on $T^{*}N$ (which is compatible with the symplectic structure). This canonically identifies $T_{(q,p)}T^{*}N\cong\mathbb{C}\otimes T_{q}N$, where the real part is horizontal and the imaginary part is vertical.

For such a $J$ there is a canonical map from the space of linear Lagrangians subspaces $V\subset T_{q,p}(T^{*}N)$ to $S^{1}$ given by the square determinant. Indeed, pick any orthonormal basis for $T_{q}N$ then this represents a basis of the horizontal Lagrangian at $T_{q,p}(T^{*}N)$. Now also pick an orthonormal basis for $V$. The complex unitary linear map changing from the first basis to the second describes a unique element in $U(n)$ of which we can take the square determinant. This is independent on the choice of both bases since it is invariant under both actions by $O(n)$.

This is smooth in $V$ and $(q,p)$, and thus it induces a smooth map from any Lagrangian submanifold $L\subset T^{*}N$ to $S^{1}$, by sending $z\in L$ to the number defined by $T_{z}L$. The induced map on $\pi_{1}$ or $H_{1}$ is known as the Maslov class. For a Lagrangian submanifold $L\subset T^{*}N$ with vanishing Maslov class a grading $\psi$ (defined in [14]) is a lift $\psi:L\to\mathbb{R}$ of the map $L\to S^{1}\cong\mathbb{R}/\mathbb{Z}$ defined above. From now on we assume that $K$ and $L$ are two exact Lagrangians with vanishing Maslov classes and gradings $\psi_{K}$ and $\psi_{L}$. Notice that when one has an isotopy of Lagrangians $L_{t},t\in I$ then a grading on $L_{0}$ “parallel transports” to a unique grading on each $L_{1}$.

Now let $z\in K\cap L$ be a transverse intersection point of $K$ and $L$. Since the space of linear Lagrangians in $T_{z}T^{*}N$ which are transverse to $T_{z}K$ is contractible there is a path unique up to homotopy in this space from $T_{z}L$ to $J(T_{z}K)$. This path lifts using the determinant construction above to a path from $\psi_{L}(z)$ to some other real number $a$. We now define the grading of $z$ (dependent on the order $K$ before $L$) by the formula

$\displaystyle\deg_{(K,L)}(z)=\psi_{K}(z)-a+\tfrac{n}{2}.$

This is an integer since $\det(JA)^{2}=(-1)^{n}\det(A)^{2}$, and each $(-1)$ represents a half turn around $S^{1}$. Notice that with this convention it is an easy exercise to see that pushing the zero-section $N$ off itself using a Morse function $g$ makes the intersection points between $N$ and $dg$ (in that order) have grading equal to the Morse index of $g$ (here the grading on $dg$ is induced from $N$ by the obvious isotopy). This grading also satisfies:

$\displaystyle\deg_{(K,L)}(z)=n-\deg_{(L,K)}(z).$

(2)

However, when the order is implied from context we will simply write $\deg(z)$.

Now assume that we have two transverse intersection points $z_{-1},z_{1}\in K\cap L$. Let $\mathbb{H}$ be the upper half plane in $\mathbb{C}$ and consider the space $\chi(z_{-1},z_{1})$ of maps $u:D^{2}\to T^{*}N$ such that $u$

• •

  maps $\pm 1$ to $z_{\pm 1}$,

• •

  maps the lower edge to $K$ – i.e. $u(S^{1}\cap\overline{\mathbb{H}})\subset K$, and

• •

  maps the upper edge to $L$ – i.e. $u(S^{1}\cap\mathbb{H})\subset L$.

and let $\mathcal{M}(z_{-1},z_{1})\subset\chi(z_{-1},z_{1})$ be the subspace of pseudo-holomorphic maps. For generic $J$ (usually achieved by a small perturbation) this subspace is a manifold of dimension $\deg(z_{1})-\deg(z_{-1})$. Indeed, this is the Fredholm index of the linearization of the $\overline{\partial}$ operator.

The space $\mathcal{M}(z_{-1},z_{1})$ has an $\mathbb{R}$ action (symmetries of holomorphic disc with two marked points on the boundary), which when the map is non-constant is free. So in the case where $\deg(z_{1})-\deg(z_{-1})=1$ the quotient $\mathcal{M}(z_{-1},z_{1})/\mathbb{R}$ is for generic $J$ a manifold of dimension $0$ - we refer to these points as rigid discs. For $K$ transverse to $L$ and generic $J$ (which we assume for the rest of this section) Floer defined the chain complex:

$\displaystyle CF_{*}(K,L;\mathbb{F}_{2})=(\mathbb{F}_{2}[K\cap L],\partial)$

Here

• •

  $\mathbb{F}_{2}=\mathbb{Z}/2$, but we will describe more general coefficients later,

• •

  the grading is given by $\deg_{(K,L)}$, and

• •

  $\partial$ counts the number of rigid discs between the intersection points going down in degree - I.e. $\deg(z_{1})=\deg(z_{-1})+1$.

By the assumptions the space $\mathcal{M}(z_{-1},z_{1})/\mathbb{R}$ when $\deg(z_{1})-\deg(z_{-1})=2$ is a 1-manifold. A version of Gromov compactness and gluing of discs shows that it can be compactified to a manifold with boundary by adding the boundary:

$\displaystyle\smashoperator[]{\bigsqcup_{z\in K\cap L,\deg(z)=\deg(z_{1})-1}^{}}\mathcal{M}(z_{1},z)\times\mathcal{M}(z,z_{-1}),$

which is a complete analogue of the Morse homology complex situation when gluing gradient trajectories. Indeed, the boundary structure is given by gluing discs together in a similar fashion. This is thus used to prove that $\partial^{2}=0$ - as in Morse homology. Floer also proved that this homology is invariant under Hamiltonian isotopy of either $K$ or $L$.

Now consider any local coefficient system $C$ of $\mathbb{F}_{2}$ vector spaces on $K$ (or $L$), and define the chain complex

$\displaystyle CF_{*}(K,L;C)=(\smashoperator[r]{\bigoplus_{z\in K\cap L}^{}}C_{z},\partial).$

Here the differential is again defined by counting the rigid discs, but using the parallel transport in the local system along the boundary path of the disc in $K$ (or $L$). The proof that $\partial^{2}=0$ easily extends to this case since the boundary path in $K$ of any glued disc is up to homotopy given by the concatenations. It should be noted that this works even when the local system is infinite dimensional, which we will make use of in Section 5 and Section 6. This was first observed by Damian in [3] and Abouzaid in [1].

When $K$ and $L$ are not transverse one defines this by perturbing one of them by a Hamiltonian flow. Floer proved that if $K=L$ with the same grading then

$\displaystyle HF_{*}(L,L;\mathbb{F}_{2})\simeq H_{*}(L;\mathbb{F}_{2}).$

(3)

In fact, by using a $C^{2}$ small Morse function $g:L\to\mathbb{R}$ to push $L$ off itself (and changing $J$) Floer proved that

$\displaystyle CF_{*}(L,L;\mathbb{F}_{2})\cong CM_{*}(L;\mathbb{F}_{2}).$

Here $CM_{*}$ denotes Morse complex of $g$. Floer proved this by proving that the gradient trajectories of $g$ are in bijective correspondence with the (now very narrow) pseudo-holomorphic discs with both boundaries equal to the gradient trajectory, and as we saw above the degree matches the Morse index. Since this also explicitly describes the boundaries of the discs involved in the differential we conclude that for a local system this proof extends to proving that

$\displaystyle CF_{*}(L,L;C)\cong CM_{*}(L;C),$

when $C$ is a local system on either of the two copies of $L$, implying that

$\displaystyle HF_{*}(L,L;C)\cong H_{*}(L;C).$

To define the intersection Floer homology with coefficients not $2$ torsion (local or not) one needs to count the rigid discs with signs, and to do this one needs that the Fredholm index bundle (which leads to the above discussed Fredholm index) has a trivialization of its determinant line bundle (as discussed in [4]) which is compatible with gluing. To achieve this we need to choose relative pin structures on $K$ and $L$ (see e.g. [15] or [7]). We will use the conventions from [15] and define

$\displaystyle CF_{*}(L,L;C)=(\smashoperator[r]{\bigoplus_{z\in K\cap L}^{}}\lvert o_{z}\rvert\otimes C_{z},\partial),$

where $\lvert o_{z}\rvert$ denotes the infinite cyclic group generated by the orientations (representing generators with opposite signs) of a certain line $o_{z}$ as defined in [15] (sections 12b and 12f). We note that in the case of $N$ and $dg$ we can canonically identify the two possible generators of $\lvert o_{z}\rvert$ with orientations on the negative eigen-space of the Hessian of $g$. This is a key ingredient in defining the signs in Morse homology away from characteristic 2. The pin structures allow us to associate a canonical isomorphism

$\displaystyle o_{z_{-1}}\cong o_{z_{1}}\qquad\Rightarrow\qquad\lvert o_{z_{-1}}\rvert\cong\lvert o_{z_{1}}\rvert$

to each rigid disc as above. To define the differential $\partial$ we now sum the latter maps tensored with the induced maps on the local system $C$ from before. The relative pin structures also allow us to associate a compatible orientation on the 1-manifolds in the proof of $\partial^{2}=0$, which means that that proof extends to this case. Even the Hamiltonian invariance generalizes.

Floer’s proof extends to signs (given the same relative pin structure on both copies of $L$) in the sense that the signs equal the signs in the Morse complex. So, his proof immediately generalizes to show the following corollary.

Let $q\in N$ be any point, and let $V^{m}\subset T_{q}N$ be an $m$-dimensional linear subspace. In this section we define the fiber-wise self-intersection Floer homology

$\displaystyle HF_{*}(L,q,V^{m};C)$

of a graded exact Lagrangian $L$ (with relative pin structure when necessary). Here $C$ is a field $\mathbb{F}$ or more generally a local coefficient system of $\mathbb{F}$-vector spaces on $L$ (potentially infinite dimensional). Initially this fiber-wise Floer homology will depend on a lot of other choices, of which the most important is a function $g$ with $q$ as a Morse critical point with unstable manifold tangent to $V^{m}$. We then prove that these fiber-wise intersection Floer homology groups are independent of the auxiliary choices and satisfy the following properties, which we will need in the proof of Theorem 1.

• •

  Invariance: $HF_{*}(L,\bullet,-;C)$ canonically defines a graded local system on the Grassmann bundle of choices $(q,V^{m})$.

• •

  Morse shifting: $HF_{*}(L,q,V^{m};C)\cong HF_{*+m}(L,q,0;C)$ (sign dependent on a choice of an orientation of $V^{m}$).

• •

  Poincare duality: $HF_{*}(L,q,V^{m};C^{\dagger})\cong HF_{n-*}(L,q,(V^{m})^{\perp};C)^{\dagger}$.

Here the latter $(-)^{\dagger}$ is vector space dual. However, $C^{\dagger}$ denotes the dual local system, which is defined by taking the fiber-wise dual and tensoring with the rank 1 local system associated to local orientations of $L$. This latter local system is trivial if $L$ is orientable with respect to $\mathbb{F}$.

In Section 2 we fixed a Riemannian structure on $N$ inducing an almost complex structure $J$ on $T^{*}N$. Let

$\displaystyle g:N\to\mathbb{R}$

(5)

be a smooth function which has $q$ as a non-degenerate critical point, and whose Hessian has negative eigenspace equal to $V^{m}$. This is easily constructed using a normal neighborhood of $q$, and it is a contractible choice.

Define

	$\displaystyle K_{t}$	$\displaystyle=tL$
	$\displaystyle L_{t}$	$\displaystyle=tL+dg,$

which we will consider for very small $t>0$. Here $dg:N\to T^{*}N$ is a Lagrangian, but $+dg$ means that we shift a point $v\in T^{*}N$ to $v+dg(\pi(v))$. This is the same as the Hamiltonian time 1 flow using the Hamiltonian $(g\circ\pi):T^{*}N\to\mathbb{R}$. So, $K_{t}$ and $L_{t}$ are both Hamiltonian isotopic to $L$. Fix a primitive $f^{L}:L\to\mathbb{R}$ for the restrictions of $\lambda$, then

	$\displaystyle f^{K_{t}}(z)$	$\displaystyle=tf^{L}(t^{-1}z)$
	$\displaystyle f^{L_{t}}(z)$	$\displaystyle=tf^{L}(t^{-1}(z-dg_{\pi(z)}))+g(\pi(z))$

will be used as primitives for $\lambda$ on $K_{t}$ and $L_{t}$ respectively.

Using the canonical identification we can transport $C$ and the gradings to corresponding structures on $K_{t}$ and $L_{t}$. The intersection Floer homology with these structures can be defined as in Section 2. However, for small $t$ we get that the intersections of $K_{t}$ and $L_{t}$ are close to critical points of $g$. Indeed, $K_{t}$ is close to the zero-section and $L_{t}$ is close to $dg$ so only when $dg$ is close to $0$ do they intersect (see Figure 1). For small $t>0$ we will call the intersection points close to $q$ the bunch of intersection points associated to $q$.

The action of an intersection point $z\in K_{t}\cap L_{t}$ is given by the difference of the primitives:

$\displaystyle f^{K_{t}}(z)-f^{L_{t}}(z)=t(f^{L}(z_{+})-f^{L}(z_{-}))+g(\pi(z)),$

(6)

where $z_{-}=t^{-1}z\in K$ and $z_{+}\in L$ is the solution to $tz_{+}+dg=z$. Since $f^{L}$ is bounded this means that the critical action values will for small $t$ cluster around the critical values of $g$. More importantly, the action values of the bunch associated to $q$ cluster around $g(q)$. This means that the action interval of the bunch is very narrow, and the following lemma will be used to argue that restricting the Floer chain complex to only include the intersection points in this bunch gives a well-defined complex for small $t$. However, we formulate it using any function $f$ with any isolated singularity at $q$ since we will need this later.

Now let $a>0$ and $\delta>0$ be as in Lemma 3.2 for $g$ with some $R>0$ isolating $q$ from other critical points. We now use these to define the fiber-wise intersection Floer homology. Firstly, pick $t$ so small that any disc $u$ with upper boundary on $L_{t}$ and lower boundary on $K_{t}$ satisfies the $\delta$ distance bound in the lemma, but also such that the critical actions of the bunch associated to $q$ lies in the interval $[g(q)-a/4,g(q)+a/4]$. Then define the fiber-wise intersection Floer homology:

$\displaystyle CF_{*}(L,q,V^{m};C)=$

$\displaystyle CF_{*}(L,q,V^{m},g,t,H,J^{\prime};C)=(\smashoperator[]{\bigoplus_{z\in K_{t}\cap L_{t}^{\prime}\cap T^{*}B_{R/2}(q)}^{}}\lvert o_{z}\rvert\otimes C_{z},\partial_{\mid}).$

Here $L_{t}^{\prime}$ is a small Hamiltonian perturbation of $L_{t}$ using a $C^{2}$ small Hamiltonian $H$, and $J^{\prime}$ is a generic small perturbation of $J$. The differential $\partial_{\mid}$ is the restriction of the differential discussed in Section 2.

Let $E^{m}\to N$ be the Grassmann bundle with fibers $E^{m}_{q}$ the $m$ dimensional linear sub-spaces of $T_{q}N$. Hence $E^{0}=E^{n}=N$.

In the above lemma there are essentially two different isomorphisms for fixed $V^{m-1}\subset V^{m}$ - one for $v$ and one for $-v$. However, when dealing with a birth-death bifurcation, which of these is involved is uniquely determined by how the two critical points cancel.

By picking orientations of the unstable manifolds (corresponds to orientations of $V^{m}$ and $V^{m-1}$) the sign of the differential in the usual Morse chain complex for $f$ in the above lemma is determined by the direction of $v$ given by the cancellation. Indeed, the sign is given by whether $V^{m-1}\oplus\mathbb{R}[v]=V^{m}$ is orientation preserving or not.

For the proofs in Section 6 the most important consequence of this section is the following “0-dimensional” Poincare duality for the fiber-wise Floer homology.

In this section we construct the spectral sequence described in the introduction. However, as mentioned we will not do this for an arbitrary Morse function $g:N\to\mathbb{R}$. So, we start by describing the Morse function we are going to use in more detail.

By taking product with a large dimensional sphere we may assume that $N$ has dimension at least 6. Indeed, if the dimension is less than 6 then all the results follow from the same results for $L\times S^{9}\subset T^{*}(N\times S^{9})$. So, we may pick a Morse function $g:N\to\mathbb{R}$ and a pseudo-gradient $X:N\to TN$ such that:

• •

  The pair is Morse-Smale,

• •

  the function $g$ is self-indexing (i.e. Morse index = critical value), and

• •

  if $x$ and $y$ are critical points of $g$ with adjacent Morse indices then there are either no pseudo-gradient trajectories connecting them or precisely 1.

Note that this last requirement can always be accomplished - by introducing a birth of two critical points along any unwanted gradient trajectory. This replaces a single gradient trajectory with 3, but also introduces two new critical points of which one can control the rigid trajectories down to lower dimensional strata (see e.g. [12]).

Let $q_{i}$ denote the critical points of $g$. As in Section 3 let $L$ be an exact Lagrangians and define $K_{t}=tL$ and $L_{t}=tL+dg$. However, in this section we consider the global situation for this specific $g$ and do not focus on a specific critical point $q$. We therefore (for small $t$) introduce the filtration on the entire complex:

$\displaystyle F^{p}(CF_{*}(K_{t},L_{t};C))$

given by restricting to all the intersection point with action less than $p+\tfrac{1}{2}$. We are now suppressing all small perturbations needed to properly define these. The continuation maps for perturbations of these will preserve the filtration as long as the action of an intersection point never crosses $p+\tfrac{1}{2}$ for any $p$. Since the action values of the intersection points bunch around the critical values of $g$ (all integers) for small $t$ this is true for small $t$.

This filtration defines a spectral sequence converging to the intersection Floer homology of $L$ with $L$ with coefficients in $C$.

Here $CM_{*}(g;A)$ denotes the Morse homology complex of $g$ using the pseudo-gradient $X$ with coefficients in the graded local system $A$. Notice, that unlike the fiber bundle example above this may be non-trivial in negative $*_{2}$-gradings.