Homotopy equivalence of nearby Lagrangians and the Serre spectral sequence
We construct using relatively basic techniques a spectral sequence for exact Lagrangians in cotangent bundles similar to the one constructed by Fukaya, Seidel, and Smith. That spectral sequence was used 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 also obtained by Nadler). The ideas in that paper were extended by Abouzaid who proved that vanishing Maslov class alone implies homotopy equivalence.
In this paper we present a short proof of the fact that any exact Lagrangian with vanishing Maslov class is homology equivalent to the base and that the induced map on fundamental groups is an isomorphism. When the fundamental group of the base is pro-finite this implies homotopy equivalence.
Let be an exact Lagrangian embedding with and closed (compact without boundary). We will always assume that is connected, but for generality we will not assume that is connected. In [FSS], Fukaya, Seidel, and Smith constructed a spectral sequence converging to the Lagrangian intersection Floer homology of 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 [Nadler]). This was extended by Abouzaid in [Abou1] to prove that vanishing Maslov class implies homotopy equivalence (combined with the result in [MySympfib] 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.
If is a closed exact Lagrangian submanifold with vanishing Maslov class, then the map is a homology equivalence and induces an isomorphism of fundamental groups.
Note that, the theorem implies (by applying it to finite covers) that if the fundamental group of is pro-finite then is a homotopy equivalence.
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 with any local coefficient system of vector spaces over some field (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) and consider two large scale perturbations of given by
for very small . So is a scaling of by a very small constant making it very close to the zero section, and is the same but pushed off the zero-section using the Morse function , so that it is close to the graph of instead. This is illustrated in Figure 1 close to a critical point of .
As the figure illustrates all the intersection points of the two Lagrangians will “bunch” around the critical points of . Each of the intersections points in the bunch close to will have action close to the critical value (up to an overall shift that we thus fix). For small we now consider an action filtration such that we have a non-trivial filtration level for each critical value of 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 . In fact, we will use this “bunching” construction to create a local system on . This we will use in Section 4 to prove that, for 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 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 where is a finite cell complex by:
, where denotes the -skeleton. This defines the Serre spectral sequence (see Hatcher [HatchSpec]).
Similar, except is not the -skeleton, but is 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 is a fiber-bundle of closed manifolds. Let be a Morse function on . This makes a Morse-Bott function. Perturbing 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 by using a sequence of values intertwining the original critical values of 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 . However if the fibers also behave as a manifold - then this is homologically supposed to look like a fiber-bundle, and since and have the same dimension the fiber basically (homologically) has to be a 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 and and how to apply local coefficients. In Section 3 we define the fiber-wise (over ) intersection Floer homology of any Lagrangian 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 , 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 . In Section 5 we extend the type of local coefficient systems we allow to include local systems on the universal cover of . The reader only interested in the case where is simply connected can skip this section. Then in Section LABEL:sec:consequences 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 the paper [sequel] with Abouzaid we use the same large scale perturbations of above together with some additional structure to prove the new result that any exact Lagrangian is in fact simple homotopy equivalent to the base.
I would like to thank Tobias Ekholm for many insightful discussions on the topic. I would also like to thank the anonymous referee and Maksim Maydanskiy for suggestions which led to a much better exposition of the material.
2. Lagrangian Intersection Floer homology and local coefficients
In [MR965228], Floer introduced the Lagrangian intersection Floer homology ; and proved that it is a Hamiltonian isotopy invariant. In this section we recall this construction for two exact Lagrangians . This also serves to fix some conventions regarding signs, gradings and orientations. We will consider some ground field . However, we will consider any local coefficient systems of vector spaces over defined on or , and describe (Corollary 2.3) the generalization of Floer’s result that
to such local coefficient system. Formally we consider the local systems on and to be non-graded or, equivalently, graded in degree 0.
The reader only interested in the case of both and 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 . So, one cannot avoid local coefficients in this argument, and hence it does not simplify matters much to ignore them here.
Let be any closed (compact without boundary) manifold. The canonical 1-form (or Liouville form) on the cotangent bundle is defined by
The canonical symplectic form is then given by . Pick a Riemannian structure on , then we get an induced almost complex structure on (which is compatible with the symplectic structure). This canonically identifies , where the real part is horizontal and the imaginary part is vertical.
For such a there is a canonical map from the space of linear Lagrangians subspaces to given by the square determinant. Indeed, pick any orthonormal basis for then this represents a basis of the horizontal Lagrangian at . Now also pick an orthonormal basis for . The complex unitary linear map changing from the first basis to the second describes a unique element in 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 .
This is smooth in and , and thus it induces a smooth map from any Lagrangian submanifold to , by sending to the number defined by . The induced map on or is known as the Maslov class. For a Lagrangian submanifold with vanishing Maslov class a grading (defined in [MR1765826]) is a lift of the map defined above. From now on we assume that and are two exact Lagrangians with vanishing Maslov classes and gradings and . Notice that when one has an isotopy of Lagrangians then a grading on “parallel transports” to a unique grading on each .
Everything in this paper except Section LABEL:sec:consequences can be carried out in the general case (with modified grading), but we assume vanishing Maslov classes already here to make the exposition more clear.
Now let be a transverse intersection point of and . Since the space of linear Lagrangians in which are transverse to is contractible there is a path unique up to homotopy in this space from to . This path lifts using the determinant construction above to a path from to some other real number . We now define the grading of (dependent on the order before ) by the formula
This is an integer since , and each represents a half turn around . Notice that with this convention it is an easy exercise to see that pushing the zero-section off itself using a Morse function makes the intersection points between and (in that order) have grading equal to the Morse index of (here the grading on is induced from by the obvious isotopy). This grading also satisfies:
However, when the order is implied from context we will simply write .
Now assume that we have two transverse intersection points . Let be the upper half plane in and consider the space of maps such that
maps to ,
maps the lower edge to – i.e. , and
maps the upper edge to – i.e. .
and let be the subspace of pseudo-holomorphic maps. For generic (usually achieved by a small perturbation) this subspace is a manifold of dimension . Indeed, this is the Fredholm index of the linearization of the operator.
The space has an 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 the quotient is for generic a manifold of dimension - we refer to these points as rigid discs. For transverse to and generic (which we assume for the rest of this section) Floer defined the chain complex:
, but we will describe more general coefficients later,
the grading is given by , and
counts the number of rigid discs between the intersection points going down in degree - I.e. .
By the assumptions the space when 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:
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 - as in Morse homology. Floer also proved that this homology is invariant under Hamiltonian isotopy of either or .
Now consider any local coefficient system of vector spaces on (or ), and define the chain complex
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 (or ). The proof that easily extends to this case since the boundary path in 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 LABEL:sec:consequences. This was first observed by Damian in [MR2914855] and Abouzaid in [Abou1].
When and are not transverse one defines this by perturbing one of them by a Hamiltonian flow. Floer proved that if with the same grading then
In fact, by using a small Morse function to push off itself (and changing ) Floer proved that
Here denotes Morse complex of . Floer proved this by proving that the gradient trajectories of 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
when is a local system on either of the two copies of , implying that
To define the intersection Floer homology with coefficients not 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 [MR1200162]) which is compatible with gluing. To achieve this we need to choose relative pin structures on and (see e.g. [MR2441780] or [MR2553465]). We will use the conventions from [MR2441780] and define
where denotes the infinite cyclic group generated by the orientations (representing generators with opposite signs) of a certain line as defined in [MR2441780] (sections 12b and 12f). We note that in the case of and we can canonically identify the two possible generators of with orientations on the negative eigen-space of the Hessian of . 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
to each rigid disc as above. To define the differential we now sum the latter maps tensored with the induced maps on the local system from before. The relative pin structures also allow us to associate a compatible orientation on the 1-manifolds in the proof of , which means that that proof extends to this case. Even the Hamiltonian invariance generalizes.
Note that the sign conventions in [MR2441780] are such that if one changes the grading of a Lagrangian by adding 1 to the lift then all the signs on the differentials change, which means that by we will mean the shift of the chain complex with the negative differential.
Floer’s proof extends to signs (given the same relative pin structure on both copies of ) in the sense that the signs equal the signs in the Morse complex. So, his proof immediately generalizes to show the following corollary.
Let be a local system on . If is 2-torsion or is equipped with a relative pin structure then
3. Fiber-wise intersection Floer homology
Let be any point, and let be an -dimensional linear subspace. In this section we define the fiber-wise self-intersection Floer homology
of a graded exact Lagrangian (with relative pin structure when necessary). Here is a field or more generally a local coefficient system of -vector spaces on (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 with as a Morse critical point with unstable manifold tangent to . 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: canonically defines a graded local system on the Grassmann bundle of choices .
Morse shifting: (sign dependent on a choice of an orientation of ).
Poincare duality: .
Here the latter is vector space dual. However, 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 . This latter local system is trivial if is orientable with respect to .
It is a consequence of vanishing Maslov class that the contribution of the orientation line (or dualizing sheaf) of over a point is trivial. This implies that for this version of Poincare duality we actually do not need to tensor with this orientation line. However, to avoid a lengthy sign discussion we simply state it as above and refer to [MR2441780] for the signs.
In Section 2 we fixed a Riemannian structure on inducing an almost complex structure on . Let
be a smooth function which has as a non-degenerate critical point, and whose Hessian has negative eigenspace equal to . This is easily constructed using a normal neighborhood of , and it is a contractible choice.
which we will consider for very small . Here is a Lagrangian, but means that we shift a point to . This is the same as the Hamiltonian time 1 flow using the Hamiltonian . So, and are both Hamiltonian isotopic to . Fix a primitive for the restrictions of , then
will be used as primitives for on and respectively.
Using the canonical identification we can transport and the gradings to corresponding structures on and . The intersection Floer homology with these structures can be defined as in Section 2. However, for small we get that the intersections of and are close to critical points of . Indeed, is close to the zero-section and is close to so only when is close to do they intersect (see Figure 1). For small we will call the intersection points close to the bunch of intersection points associated to .
The action of an intersection point is given by the difference of the primitives:
where and is the solution to . Since is bounded this means that the critical action values will for small cluster around the critical values of . More importantly, the action values of the bunch associated to cluster around . 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 . However, we formulate it using any function with any isolated singularity at since we will need this later.
Let be any function such that is the only critical point in the closure of the ball . Then there exist a and an such that: if is a pseudo-holomorphic disc satisfying:
Precisely one of the two points is in the cotangent ball .
The maximal distance of the upper boundary of to is .
The maximal distance of the lower boundary of to the zero-section is .
then the symplectic area of is larger than . Furthermore, neither of the points is in the set
The assumptions imply that for small the one point of that lies inside is in fact inside . Indeed, there is a positive distance from the closed annulus to . So we may choose to be smaller than half this distance.
Consider the co-dimension 2 sub-manifold given by
for . The manifold is compact without boundary, and it is disjoint from and the zero section. By the above choice of it is also disjoint from when is as described in the lemma. In fact, the upper part of the boundary can only pass over points of (here “over” means with larger value) and the lower part of the boundary passes under. Hence the assumptions imply that the algebraic intersection of and is .
It follows by standard monotonicity (Lemma 3.3) that has area at least for some small . ∎
Let be any open symplectic manifold with a compatible almost complex structure . Then for any compact subset and an open neighborhood around there is a lower bound on the area of any non-constant connected pseudo-holomorphic curve passing through defined on an open domain and with proper image in .
This was proven (but not phrased like this) in [MR809718]. ∎
Now let and be as in Lemma 3.2 for with some isolating from other critical points. We now use these to define the fiber-wise intersection Floer homology. Firstly, pick so small that any disc with upper boundary on and lower boundary on satisfies the distance bound in the lemma, but also such that the critical actions of the bunch associated to lies in the interval . Then define the fiber-wise intersection Floer homology:
Here is a small Hamiltonian perturbation of using a small Hamiltonian , and is a generic small perturbation of . The differential is the restriction of the differential discussed in Section 2.
The fiber-wise intersection Floer homology is well-defined and independent of the choices up to a chain homotopy equivalence, which is unique up to chain homotopy.
Initially we consider as fixed. For small enough Hamiltonian perturbation we can assume that all the intersection points of and in the bunch have action in and that also lies close to . Lemma 3.2 was used in the definition above for the original . However, for small perturbations of we can assume that any pseudo-holomorphic disc with precisely one of the points in the bunch and boundaries on and has symplectic area larger than , which is still more than the entire interval of critical action spanned by these critical points - hence there are no interactions from outside the bunch. More concisely, the usual proof that works unchanged since there can be no breaking on this subset of generators which involves points outside of the bunch.
It is standard to construct continuation maps for intersection Floer homology using generic paths of perturbation data (see e.g. [MR2441780]). If all the perturbations in the path are small enough the bound in Lemma 3.2 is valid also for the associated continuation map. Hence this map restricts to a chain map on the fiber-wise Floer complexes. Furthermore, since generic homotopies of such paths induce chain homotopies of these continuation maps it follows that for small enough perturbations the continuation maps are chain homotopy equivalences which are unique up to homotopy.
Now, we consider the choice of and note that this is a contractible choice, so for any two choices there is a path between them, and since changing slightly changes and by a slight perturbation we can cut into small pieces and get a sequence of chain homotopy equivalences (each as above for small ) relating the two chain complexes. Since the path is unique up to homotopy, we can relate any such two choices by a homotopy of paths, which when cut into pieces can be used to define a chain homotopy between the two sequences of chain homotopy equivalences. ∎
Let be the Grassmann bundle with fibers the dimensional linear sub-spaces of . Hence .
The fiber-wise intersection Floer homology
naturally defines a graded local system on the choices .
Let be a smooth family of functions parametrised by such that is a Morse critical point with the negative eigenspace of the Hessian equal to . This can be constructed explicitly using exponential maps and bump functions. By compactness of we can find an such that for each the critical point of at is unique in the closure of . By compactness we can find a and an as in Lemma 3.2 which works for the entire family simultaneously, and again we can find so small that all the fiber-wise Floer homologies are well-defined (each after a perturbation). Since changing slightly perturbs and slightly it follows from Lemma 3.4 that the homologies of the complexes
are locally defined up to unique isomorphism for , and hence defines a local system on . ∎
Let be a unit vector let be the orthogonal complement of . There is a chain homotopy equivalence (after choosing small perturbations)
unique up to chain homotopy. Moreover, the induced map on homology is continuous for varying and (and hence ).
Notice here that the continuity and uniqueness of the map on homology is equivalent to: on the space of choices we have two fiber-bundle structures given by projections to (with fiber - the choice of ) and to (with fiber - since is a choice of unit vector in the orthogonal complement of ). Now, the construction defines a canonical global isomorphism of local systems between the pull backs of the two local systems of fiber-wise Floer homologies on and to the common fiber bundle.
Fix . We will work in a normal coordinate chart around which identify the derivatives with and such that is mapped to . We will denote the image of the span of the first of these by , which is the orthogonal complement to the span of inside . Let be a family of functions for which in these coordinates is given by
This has two critical points in the chart when and none when . Defining the Lagrangians as above using instead of and some small provides smooth families and of Lagrangians.
For any there is an small so that for and all of the Lagrangians are within a -neighborhood of and within a -neighborhood of the zero-section. Hence using Lemma 3.2 on (and some ) provides an (for our fixed ) which we can use for this family of Lagrangian pairs. By making even smaller we get that the critical action interval of intersection points of in is again smaller than , and thus for any such pair the Floer homology complex, say (“S” for singularity), of all the intersection points inside is well-defined (using a sufficiently small perturbation). So, as above, this “singularity Floer homology”, say , defines a graded local system on the space .
For and sufficiently small we see that and so must be the trivial local system.
For the function has two Morse-critical points and close to with . Since the critical action values will cluster around these values we see that for small we have a block form differential on :
Here and are the differentials in the fiber-wise Floer homology chain complexes and at the points and respectively. By the sign convention discussed in Remark 2.2 it follows that
is a chain homotopy equivalence. Both points are very close to , so we may replace and with using continuation maps unique up to chain homotopy. We are using the local chart to identify the two instances of and here. The last statement in the lemma follows from the perturbation invariance from the previous lemma. Indeed, for any small change in and the perturbation invariance implies that the map induced on homology is locally constant in any local trivializations of the local systems. ∎
In the above lemma there are essentially two different isomorphisms for fixed - one for and one for . However, when dealing with a birth-death bifurcation, which of these is involved is uniquely determined by how the two critical points cancel.
Let be a function such that has precisely two critical points and in its interior. Assume also that these are non-degenerate and that there is a unique gradient trajectory between them so that they cancel in Morse homology. Assume is the one with the lower index and denote by and the negative eigenspaces of the Hessian of at the points.
This data defines a chain homotopy equivalence
which is homotopic to parallel transport (continuation maps) composed with one of the two induced by the above lemma (determined by the cancellation).
Define and as above but now using . For very small we can argue as follows. Both of the chain complexes in the corollary are defined as different parts of the standard intersection Floer chain complex of and . Considering only those intersection points with action in we get a chain complex which precisely contains these two parts. The differential restricted to this complex is again on upper triangular form (as in the proof above):
Again and are the differentials in the fiber-wise Floer homology chain complexes and . By the assumptions we can deform inside through a single birth-death singularity to a situation with no critical points. This also deforms and , and we get induced continuation maps from Lemma 3.4. ∎
By picking orientations of the unstable manifolds (corresponds to orientations of and ) the sign of the differential in the usual Morse chain complex for in the above lemma is determined by the direction of given by the cancellation. Indeed, the sign is given by whether is orientation preserving or not.
The fiber-wise intersection Floer homology satisfies Morse shifting. I.e. for fixed we have
canonically defined by fixing a choice of orientation on . The isomorphisms for the two different orientations differ by a sign.
Any choice of ordered orthonormal basis for defines by Lemma 3.6 a sequence of chain homotopy equivalences and thus a chain of isomorphisms
which by the fact that these are locally maps of local systems is locally constant in the choice of such a basis and thus only dependent on the orientation class that the basis defines. Thus the only thing left to prove is that these two choices of isomorphisms differ by a sign.
We may assume that the dimension of is at least 2. This means that when considering the two isomorphisms we can factor through (notice this works even when and since we can go up using inverses). Hence we can determine the difference of the two maps by simply considering the difference using any two compositions of the maps in Equation (8). So, to see that the maps for the two different choices of orientations only differ by a sign (on the level of homology) we consider two orthogonal directions and denote the complement of by and by respectively. We also denote the common complement of the plane they span in by . Now consider the diagram
The fiber-wise intersection Floer homology satisfy Poincare duality
where denotes the fiber-wise dual local system over tensor the rank 1 local system of orientations on .
Firstly, if we replace by in the definition of and apply the Hamiltonian flow of for time then and are flowed to the two Lagrangians
This is more symmetric, and for small these can be used to define the Fiber-wise Floer homology. Indeed, it does not matter that we are using nor does the local Floer homology change when we apply the Hamiltonian isotopy (the bunching of all critical points near the intersection, and the bound in Lemma 3.2 is valid throughout the isotopy for sufficient small ). The primitives used on these can be chosen as:
If we now exchange for we get the exact same two Lagrangians from this construction, but in the opposite order. Hence we change the sign of the action and simultaneously the directions of the pseudo-holomorphic discs counted in the differential.
Since the Floer intersection chain complexes are given by a finite direct sum of fibers of the local system (which each may be infinite dimensional) the dual complex is a finite sum over the dual fibers of the local systems. Hence if we choose trivializations of for all we can use the fact that the Lagrangians are the same to identify
as vector spaces. The differential differs in signs by introducing the local system of orientations on since this “inversion” realizes Poincare duality of (see [MR2441780] section 12 for details on these signs and the Poincare duality). ∎
For the proofs in Section LABEL:sec:consequences the most important consequence of this section is the following “0-dimensional” Poincare duality for the fiber-wise Floer homology.
The shift and the Poincare duality properties imply that
depending on a choice of orientation of .
4. The spectral sequence
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 . 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 has dimension at least 6. Indeed, if the dimension is less than 6 then all the results follow from the same results for . So, we may pick a Morse function and a pseudo-gradient such that:
The pair is Morse-Smale,
the function is self-indexing (i.e. Morse index = critical value), and
if and are critical points of 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. [MR0190942]).
Let denote the critical points of . As in Section 3 let be an exact Lagrangians and define and . However, in this section we consider the global situation for this specific and do not focus on a specific critical point . We therefore (for small ) introduce the filtration on the entire complex:
given by restricting to all the intersection point with action less than . 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 for any . Since the action values of the intersection points bunch around the critical values of (all integers) for small this is true for small .
This filtration defines a spectral sequence converging to the intersection Floer homology of with with coefficients in .
Page 1 of this spectral sequence is isomorphic as a bi-graded chain complex to .
Here denotes the Morse homology complex of using the pseudo-gradient with coefficients in the graded local system . Notice, that unlike the fiber bundle example above this may be non-trivial in negative -gradings.
Page one of such a spectral sequence has entry in bi-grading equal to the homology group of the quotient
For small the intersection points will cluster around the critical points and their action values will be close to the associated critical value (see Section 3). This critical value is the Morse index since is self indexing. The differential on each of the bunches around different critical points and with the same critical value cannot interact. Indeed, for small this would violate the energy bound from Lemma 3.2 on discs with one marked point sent to one bunch and the other to the other bunch. We thus get that the above quotient complex splits as a direct sum of the fiber-wise chain complexes from Section 3:
Here is the negative eigenspace of the Hessian of at . The homology of each of these are by Lemma 3.9 isomorphic and shifted by the Morse index (which by the self-indexing property equals ). By this and Lemma 3.5 we get
For each summand this isomorphism depends on a choice of orientation of the unstable manifold at the critical point, but that is as it should be (since Morse homology works that way). Indeed, to argue what the differential (on page 1) is, we need to be careful with orientations (comparing with a CW complex structure on one needs to pick orientations of each cell before we can define the degree of attaching maps).
The differential on page 1 of the spectral sequence is independent of for small . Indeed, since there can be no interactions between the individual bunches of critical points (associated to the same Morse index) there can be no handle slides for small .
Fix and critical for with adjacent Morse indices, i.e. . For very small we can pick a very small and change by a small perturbation such that
the critical value of becomes and
the critical value of becomes .
We can do this such that the change that this makes to and does not affect the identification above of page 1. Indeed for very small and there are no possible interactions between any of the bunches approximately on the same action level (i.e. no disc can go from one to the other) - even while we push the action level of some of them up or down a little bit (the area bound in Lemma 3.2 can be assumed to be much larger than ). It also does not affect the differential that we wish to identify. Indeed, any handle sliding is ruled out by the same argument.
Now by making even smaller (which again does not change the above identification) we can make sure that the clustering around the critical point values is such that
The intersection points in the bunch close to have action in the interval ,
The intersection points in the bunch close to have action in the interval , and
The intersection points in bunches close to all other critical points have action in the intervals .
This means that there are no critical action values close to and . The identification of the differential now follows by considering the chain complex defined by restricting to action between and . Indeed, this is either:
The birth-death situation we considered in Corollary 3.8 (if there is a single gradient trajectory between the associated critical points).
Or a situation where we can actually move the lower bunch up to the same height as the other and see that the differential on the fiber-wise homology has to be (by homotopy invariance). Indeed, if there are no gradient trajectories between the two critical points of we can by changing close to the unstable and stable manifolds move the critical points of in this way (see e.g. [MR0190942]).
The first point uses the isomorphisms we saw in Lemma 3.9 and Corollary 3.8, which has a sign depending on whether this cancellation is compatible with the chosen orientations on the unstable manifolds or not, which precisely is one way of defining the signs in . So this is the Morse complex differential with the local coefficient system . ∎
5. Local systems on the universal cover of
With the same assumptions as in Section 3 we will in this section define versions of the fiber-wise intersection Floer homology on the universal covering space of and prove compatibility with pull back and push forward maps. Then we will generalize Corollary 3.11 to dualizing the local systems on the universal covers.
Let be the universal covering space of . To this we have an associated universal covering .
Assume all the conditions of Lemma 3.2 - except assume that has both points mapping to instead of precisely one of them. Additionally assume that has energy less than . Then is homotopic in relative to to a map in .
Since the disc relative the points is homotopy equivalent to the interval relative its endpoints this is a question of what represents in
However, assuming it represents something non-trivial the disc will lift to have two endpoints in which are in two different components of the non-connected pre-image of the contractible sub-space . Hence as in the proof of Lemma 3.2 this intersects the pre-image of (from that proof) in . This implies that in fact intersects non-trivially, which gives a contradiction (if is chosen as in that proof). ∎
Define the covering space by the pull back diagram