Homotopy equivalence of nearby Lagrangians and the Serre spectral sequence
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Abstract
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 profinite this implies homotopy equivalence.
Mathematics Subject Classification
53D12 53D401 Introduction
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 [7], Fukaya et al. 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.
Theorem 1
If \(L\subset T^*N\) is a closed exact Lagrangian submanifold with vanishing Maslov class, then the map \(L \rightarrow N\) is a homology equivalence and induces an isomorphism of fundamental groups.
Remark 1
Note that, the theorem implies (by applying it to finite covers) that if the fundamental group of N is profinite then \(L\rightarrow N\) 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 L with any local coefficient system of vector spaces over some field \(\mathbb {F}\) (and with a relative pin structure when needed).
The main technical part of the construction is carried out in Sect. 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 welldefined 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 Sect. 4 to prove that, for g selfindexing, 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 Sect. 3.

\(\pi ^{1}(X_0) \subset \cdots \subset \pi ^{1}(X_j) \subset \pi ^{1}(X_{j+1}) \subset \cdots \subset E\), where \(X_j\) denotes the jskeleton. This defines the Serre spectral sequence (see Hatcher [10]).

Similar, except \(X_j\) is not the jskeleton, 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.

Selfindexing Morse function—where the critical value equals the Morse index.

Any Morse function with distinct critical value for each critical point.
The final piece to proving Theorem 1 is essentially to establish a version of Poincaré duality fiberwise. A heuristic description of why this might be a useful property is as follows; the fibers represent the relative difference \(L \rightarrow N\). However if the fibers also behave as a manifold—then this is homologically supposed to look like a fiberbundle, 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 Sect. 2 we briefly describe Lagrangian intersection Floer homology of two exact Lagrangians K and L and how to apply local coefficients. In Sect. 3 we define the fiberwise (over N) intersection Floer homology of any Lagrangian L with itself using the idea of the bunches described above. We also prove that this fiberwise Floer homology defines a graded local system on N, and satisfies other natural properties that we will need—most importantly the PoincarT duality mentioned above. In Sect. 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 Sect. 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 Sect. 6 we prove Theorem 1 starting with the simply connected case not requiring Sect. 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.
Remark 2
In the paper [2] with Abouzaid we use the same large scale perturbations of L above together with some additional structure to prove the new result that any exact Lagrangian is in fact simple homotopy equivalent to the base.
2 Lagrangian intersection Floer homology and local coefficients
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 1, which means we need to understand the fiberwise 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.
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 \(T_qN\) 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 \(T_zL\). 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 \(\psi : L\rightarrow \mathbb {R}\) of the map \(L\rightarrow S^1 \cong \mathbb {R}/\mathbb {Z}\) 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 \(L_1\).
Remark 3
Everything in this paper except Sect. 6 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.

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. \(u(S^1\cap \mathbb {H}) \subset L\).

\(\mathbb {F}_2=\mathbb {Z}/2\), 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. \(\deg (z_{1})=\deg (z_{1})+1\).
Remark 4
Note that the sign conventions in [15] 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 C[1] we will mean the shift of the chain complex C with the negative differential.
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.
Corollary 1
3 Fiberwise intersection Floer homology

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: \(HF_*(L,q,V^m;C^\dagger ) \cong HF_{n*}(L,q,(V^m)^\perp ;C)^\dagger \).
Remark 5
It is a consequence of vanishing Maslov class that the contribution of the orientation line (or dualizing sheaf) of L over a point \(q\in N\) 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 [15] for the signs.
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 Sect. 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 zerosection and \(L_t\) is close to dg so only when dg is close to 0 do they intersect (see Fig. 1). For small \(t>0\) we will call the intersection points close to q the bunch of intersection points associated to q.
Lemma 1

Precisely one of the two points \(u(\pm 1)\) is in the cotangent ball \(T^*B_R(q)\).

The maximal distance of the upper boundary of u to df is \(\delta \).

The maximal distance of the lower boundary of u to the zerosection is \(\delta \).
Proof
The assumptions imply that for small \(\delta >0\) the one point of \(u(\pm 1)\) that lies inside \(T^*B_R(q)\) is in fact inside \(T^*B_{R/2}(q)\). Indeed, there is a positive distance from the closed annulus \(\overline{B_R(q)B_{R/2}(q)} \subset N \subset T^*N\) to df. So we may choose \(\delta \) to be smaller than half this distance.
It follows by standard monotonicity (Lemma 2) that u has area at least a for some small \(a>0\).
Lemma 2
Let M be any open symplectic manifold with a compatible almost complex structure J. Then for any compact subset \(C\subset M\) and an open neighborhood U around C there is a lower bound on the area of any nonconstant connected pseudoholomorphic curve passing through C defined on an open domain and with proper image in U.
Proof
This was proven (but not phrased like this) in [9].
Lemma 3
The fiberwise intersection Floer homology is welldefined and independent of the choices up to a chain homotopy equivalence, which is unique up to chain homotopy.
Proof
Initially we consider g as fixed. For small enough Hamiltonian perturbation we can assume that all the intersection points of \(K_t\) and \(L_t'\) in the bunch have action in \((g(q)a/3,g(q)+a/3)\) and that \(L_t'\) also lies \(\delta \) close to dg. Lemma 1 was used in the definition above for the original J. However, for small perturbations of J we can assume that any pseudoholomorphic disc with precisely one of the points \(u(\pm 1)\) in the bunch and boundaries on \(K_t\) and \(L_t'\) has symplectic area larger than 2a / 3, 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 \(\partial _{\mid }^2 = 0\) 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. [15]). If all the perturbations in the path are small enough the bound in Lemma 1 is valid also for the associated continuation map. Hence this map restricts to a chain map on the fiberwise 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 g and note that this is a contractible choice, so for any two choices there is a path \(g_s,s\in I\) between them, and since changing g slightly changes \(K_t\) and \(L_t\) by a slight perturbation we can cut I into small pieces and get a sequence of chain homotopy equivalences (each as above for small t) relating the two chain complexes. Since the path \(g_s\) 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 \(E^m \rightarrow N\) be the Grassmann bundle with fibers \(E^m_q\) the m dimensional linear subspaces of \(T_qN\). Hence \(E^0=E^n=N\).
Lemma 4
Proof
Lemma 5
Remark 6
Notice here that the continuity and uniqueness of the map on homology is equivalent to: on the space of choices \((q,V^m,v)\) we have two fiberbundle structures given by projections to \(E^m\) (with fiber \(S^{m1}\)—the choice of v) and to \(E^{m1}\) (with fiber \(S^{nm}\)—since v is a choice of unit vector in the orthogonal complement of \(V^{m1}\)). Now, the construction defines a canonical global isomorphism of local systems between the pull backs of the two local systems of fiberwise Floer homologies on \(E^m\) and \(E^{m1}\) to the common fiber bundle.
Proof
For any \(\delta >0\) there is an \(\epsilon >0\) small so that for \(0<t<\epsilon \) and \(s\in [\epsilon ,\epsilon ]\) all of the Lagrangians \(L_t^s\) are within a \(\delta \)neighborhood of \(dg_0\) and \(K_t^s\) within a \(\delta \)neighborhood of the zerosection. Hence using Lemma 1 on \(g_0\) (and some \(R>0\)) provides an \(a>0\) (for our fixed J) which we can use for this family of Lagrangian pairs. By making \(\epsilon \) even smaller we get that the critical action interval of intersection points of \(L_t^s\cap K_t^s\) in \(T^*B_R(q)\) is again smaller than 2a / 3, and thus for any such pair (s, t) the Floer homology complex, say \(SCF_*\) (“S” for singularity), of all the intersection points inside \(T^*B_R(q)\) is welldefined (using a sufficiently small perturbation). So, as above, this “singularity Floer homology”, say \(SHF_*\), defines a graded local system on the space \((s,t)\in [\epsilon ,\epsilon ]\,\times \,]0,\epsilon ]\).
For \(s=\epsilon \) and t sufficiently small we see that \(L_t^s\cap K_t^s \cap T^*B_R(q) = \varnothing \) and so \(SHF_*\) must be the trivial local system.
In the above lemma there are essentially two different isomorphisms for fixed \(V^{m1}\subset V^m\)—one for v and one for \(v\). However, when dealing with a birthdeath bifurcation, which of these is involved is uniquely determined by how the two critical points cancel.
Corollary 2
Let \(f :N \rightarrow \mathbb {R}\) be a function such that \(f^{1}[a,b]\) has precisely two critical points \(q_0\) and \(q_1\) in its interior. Assume also that these are nondegenerate and that there is a unique gradient trajectory between them so that they cancel in Morse homology. Assume \(q_0\) is the one with the lower index and denote by \(V^{m1} \subset T_{q_0}N\) and \(V^{m}\subset T_{q_1}N\) the negative eigenspaces of the Hessian of f at the points.
Proof
By picking orientations of the unstable manifolds (corresponds to orientations of \(V^m\) and \(V^{m1}\)) 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 \(V^{m1}\oplus \mathbb {R}[v]=V^m\) is orientation preserving or not.
Lemma 6
Proof
Now the fact that this differential squares to 0 gives that \(H^m_{v_1,v_2}\) is a chain homotopy equivalence from \(F^{m1}_{v_2}\circ F^m_{v_1}\) to \(F^{m1}_{v_1}\circ F^m_{v_2}\).
Lemma 7
Proof
For the proofs in Sect. 6 the most important consequence of this section is the following “0dimensional” Poincare duality for the fiberwise Floer homology.
Corollary 3
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 \(g:N \rightarrow \mathbb {R}\). So, we start by describing the Morse function we are going to use in more detail.

The pair is Morse–Smale,

the function g is selfindexing (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 pseudogradient trajectories connecting them or precisely 1.
This filtration defines a spectral sequence converging to the intersection Floer homology of L with L with coefficients in C.
Proposition 1
Page 1 of this spectral sequence is isomorphic as a bigraded chain complex to \(CM_{*_1}(g;HF_{*_2}(L,q,0;C))\).
Here \(CM_*(g;A)\) denotes the Morse homology complex of g using the pseudogradient X with coefficients in the graded local system A. Notice, that unlike the fiber bundle example above this may be nontrivial in negative \(*_2\)gradings.
Proof
The differential on page 1 of the spectral sequence is independent of t for small t. 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 t.

the critical value of \(q_i\) becomes \(p\delta '\) and

the critical value of \(q_j\) becomes \(p1+\delta '\).

The intersection points in the bunch close to \(q_j\) have action in the interval \(p1+[2\delta '/3,4\delta '/3]\),

The intersection points in the bunch close to \(q_i\) have action in the interval \(p[2\delta '/3,4\delta '/3]\), and

The intersection points in bunches close to all other critical points have action in the intervals \(\mathbb {N}+ [\delta '/3,\delta '/3] \subset \mathbb {R}\).

The birthdeath situation we considered in Corollary 2 (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 fiberwise homology has to be 0 (by homotopy invariance). Indeed, if there are no gradient trajectories between the two critical points of g we can by changing g close to the unstable and stable manifolds move the critical points of g in this way (see e.g. [12]).
5 Local systems on the universal cover of N
With the same assumptions as in Sect. 3 we will in this section define versions of the fiberwise intersection Floer homology on the universal covering space of N and prove compatibility with pull back and push forward maps. Then we will generalize Corollary 3 to dualizing the local systems on the universal covers.
Most of the results in this section are easy consequences of the following corollary to Lemma 1. However, the introduced language and notation will be convenient for the general proof of Theorem 1.
Let \(\pi _N:N'\rightarrow N\) be the universal covering space of N. To this we have an associated universal covering \(T^*N' \rightarrow T^*N\).
Corollary 4
Assume all the conditions of Lemma 1—except assume that u has both points \(u(\pm 1)\) mapping to \(T^*B_R(q)\) instead of precisely one of them. Additionally assume that u has energy less than a. Then u is homotopic in \(T^*N\) relative to \(\{\pm 1\} \subset D^2\) to a map in \(T^*B_R(q)\).
Proof
We have the following generalization of Floer’s result and the spectral sequence in Proposition 1.
Lemma 8
Proof
For any local system \(C'\) on \(L'\) we define (similar to the definition in Sect. 3) its dual \(C'^\dagger \) over \(L'\) to be the fiberwise dual vector space tensored with the local system defined by orientations on \(L'\). Notice that even if L is orientable \(\pi _{L*}(C'^\dagger )\) is not generally isomorphic to \((\pi _{L*}C')^\dagger \) if \(\pi _1(N)\) is not finite.
The above observations now makes it possible to generalize the Poincare duality from Corollary 3 to this dualization.
Corollary 5
Since these are trivial local systems we do not really need to fix \(q'\). However, the following proof is easier to mentally parse downstairs when \(q=\pi _N(q')\) is considered a fixed point.
6 Proof of Theorem 1
This section contains a proof of Theorem 1. So assume \(L\subset T^*N\) is an exact Lagrangian with vanishing Maslov class. In this section g is a function as in Sect. 4 such that Proposition 1 holds for this g.
As a warm up we start by giving a proof of homotopy equivalence in the case \(\pi _1(N)=1\) and connected L. This is similar to the argument given in [8]—except that instead of using the notion of the span of the homology we use the fiberwise Poincare duality in Corollary 3. This is also one reason we are able to get stronger results.
Now as noted in [8] this implies that the map \(L\rightarrow N\) is relatively orientable and relative spin, and that \(L\subset T^*N\) has a relative pin structure so that we can define the homologies with \(\mathbb {F}\) coefficients for any field. Now the exact same argument proves homology equivalence over any field, which implies homology equivalence over \(\mathbb {Z}\).
For the general proof of Theorem 1 we divide the argument into a few lemmas.
Lemma 9
Proof
Lemma 10
Proof
We can compute the Euler characteristic of \(HF_*(L,q,0;\mathbb {F})\) as \(p^2\). Indeed, for small t we see that the Lagrangians \(K_t=tL+dg\) and \(L_t=tL\) will be transverse to each other, and the parity of the Maslov index of an intersection can be computed using the orientation sign of the two layers of the lift associated with the intersection (see Fig. 2). If \(L'\) has \(p + k\) positive layers and k negative layers at \(q'\) we therefore get \((p + k)^2 + k^2\) even parity Maslov indices of intersection points and \(2(p + k)k\) odd parity Maslov indices of intersections points. Hence the Euler characteristic of the complex is \(p^2\). This implies together with Lemma 9 that in fact \(H_*(L';\mathbb {F}) \cong HF_0(L,q,0;\mathbb {F}) \cong \mathbb {F}^{p^2}\).
Now assume that \(p^2> 1\). Since L is connected, and \(L'\) is not, the map \(\pi _1(L) \rightarrow \pi _1(N)\) is not surjective—in fact \(\pi _1(N)/{{\mathrm{Im}}}(\pi _1(L))  = {{\mathrm{rank}}}(H_0(L'))=p^2\). This means that there is a covering space of N with \(p^2\) layers (associated to the image subgroup) where the lift of L has \(p^2\) components, but such a lift of N is finite and hence compact, and this contradicts the fact that we just proved that exact Lagrangians in such are connected. So, \(p=1\) and even \(L'\) is connected. We also conclude that the map \(L \rightarrow N\) has degree 1. Note, that degree is defined by passing to oriented covers in the case where L and N are nonorientable.
This means that the local systems \(HF_*(L,q,0;\mathbb {F})\) is free and of rank 1 with support in degree 0. Now assume for contradiction that it is not trivial. Then we get by the spectral sequence in Proposition 1 using \(C=\mathbb {F}\) that \(H_0(L;\mathbb {F})\cong 0\), which is a contradiction. We conclude that \(H_*(N,\mathbb {F}) \cong H_*(L,\mathbb {F})\), but since we have not proven naturality with respect to j we can only claim this as an abstract isomorphism. However, we have proven that the map j has degree 1, and a degree 1 map of closed manifolds is surjective with field coefficients, and so this abstract isomorphism shows (since the dimensions agree) that it is also injective.
Now as before all this implies existence of relative pin structure on L, and we can therefore run the parts of the argument needed using a general field \(\mathbb {F}\) to obtain homology equivalence.
Lemma 11
The map on fundamental groups induced by j is injective.
Proof
Assume for contradiction that it has a kernel \(\{1\} \ne \pi _1(L') \subset \pi _1(L)\), which as indicated by this notation is the fundamental group of the covering \(L'\). Let \(G\subset \pi _1(L')\) be a nontrivial cyclic subgroup of prime order (or order \(\infty \)) and \(\widetilde{L} \rightarrow L'\) its corresponding covering space. Now let \(C_G'\) denote the local coefficients over the field \(\mathbb {F}_{G }\) (with the convention \(\mathbb {F}_\infty =\mathbb {Q}\)) on \(L'\) corresponding to this covering, and define \(C_G=\pi _{L*}C_G'\). This is the local system on L corresponding to the covering space \(\widetilde{L}\rightarrow L'\rightarrow L\).
Notes
Acknowledgments
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.
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