A Auxiliary results

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Abstract

Let be a symplectic manifold, a coisotropic submanifold, and a compact oriented (real) surface. I define a natural Maslov index for each continuous map that sends every connected component of to some isotropic leaf of . This index is real valued and generalizes the usual Lagrangian Maslov index. The idea is to use the linear holonomy of the isotropic foliation of to compensate for the loss of boundary data in the case . The definition is based on the Salamon-Zehnder (mean) Maslov index of a path of linear symplectic automorphisms. I prove a lower bound on the number of leafwise fixed points of a Hamiltonian diffeomorphism, if is geometrically bounded and is closed, regular (i.e. ”fibering”), and monotone. As an application, we obtain a presymplectic non-embedding result. I also prove a coisotropic version of the Audin conjecture.

A Maslov Map for Coisotropic Submanifolds]A Maslov Map for Coisotropic Submanifolds, Leaf-wise Fixed Points and Presymplectic Non-Embeddings

1 Motivation and main results

This article is concerned with the following two problems. Let be a symplectic manifold and a coisotropic submanifold. A leafwise fixed point of a map is by definition a point such that lies in the isotropic leaf through . We denote by the set of such points.

Problem A: Find conditions on under which is non-empty and give a lower bound on .

Note that in the case the set equals the set of usual fixed points. In the other extreme case, in which is Lagrangian, we have .

To formulate the second problem, let be a real vector space and a skew-symmetric form on . We denote . A presymplectic form on a manifold is a closed two-form of constant corank. We say that a presymplectic manifold embeds into another presymplectic manifold iff there exists an embedding such that . The next problem generalizes the symplectic and Lagrangian non-embedding problems:

Problem B: Find conditions on and under which does not embed into .

In [Zi], I gave some solution to problem A, imposing the conditions that is regular and the Hofer distance of and the identity is small enough. In the present article, the second condition is replaced by the assumption that is monotone. The paper [Zi] also contains some solution to problem B, assuming that is non-degenerate and aspherical. In the present article the latter condition is replaced by monotonicity of .

To define monotonicity for a coisotropic submanifold , I introduce a natural Maslov map for , which equals the usual Maslov index in the case , and twice the first Chern class of in the case .

1.1 Definition of the Maslov map

Let be a symplectic manifold (without boundary), a coisotropic submanifold, and a topological manifold. We denote by the set of connected components of and by the set of isotropic leaves of . We define

 C(X,M;N,ω):={u∈C(X,M)∣∣∀Y∈C(∂X)∃F∈Nω:u(Y)⊆F}.

Let . We call an -admissible homotopy iff for every there exists such that , for every , . We denote by the corresponding set of all -admissible homotopy classes of maps from to .

We denote by the class of all compact oriented (real) topological surfaces (possibly with boundary and disconnected). Let . The Maslov map introduced in this article is a map

 mΣ,N:=mΣ,ω,N:[Σ,M;N,ω]→R. (1)

Its definition involves the following four steps. A more direct, but less natural definition is given on page 1.2.

The Salamon-Zehnder Maslov index

Let be a symplectic vector space. We denote by the group of linear symplectic automorphisms of . We define the Salamon-Zehnder Maslov index

 mω:C([0,1],Autω)→R (2)

as follows. We define the winding map by , where is any path such that , for every . We denote by the Salamon-Zehnder map (see Proposition 42 below). Let . We define .

The Maslov map for pairs of flat transports

Let be a topological manifold. We denote by the fundamental groupoid of . This is a topological groupoid. Its set of objects is and its set of morphisms consists of all homotopy classes (with fixed end-points) of continuous paths in .

For two vector spaces and we denote by the set of all isomorphisms from to . Let be a vector bundle. We denote by the general linear groupoid of . This is a topological groupoid. Its set of objects is and its set of morphisms consists of all triples , where and .

By a flat (linear) transport we mean a (continuous) representation of on , i.e. a morphism of topological groupoids from to that covers the identity on . Such a associates to every homotopy class of paths an isomorphism . It is equivariant with respect to concatenation of paths. We denote by the set of all flat transports on .

We call regular iff , for every satisfying . Note that if is a smooth manifold and is a smooth vector bundle then the parallel transport of a smooth flat connection on is a flat transport.

For symplectic vector spaces and we denote by the set of linear isomorphisms such that . Let be a topological manifold and be a symplectic vector bundle over . We define to be the subgroupoid of consisting of all such that . We call a transport symplectic iff , and denote by the set of all such ’s.

Let be an oriented closed curve (i.e. topological real one-manifold), a symplectic vector bundle over , and be such that or is regular. We define the number as follows. Namely, we choose a path such that and the map has degree one. We define by , and

 mω(Φ,Φ′):=mωz(0)(Ψ). (3)

By Lemma 7 below this number is well-defined.

The coisotropic Maslov map for bundles

Let be a symplectic vector space and be a subspace. We denote by its symplectic complement. Assume that is coisotropic. We denote by its linear symplectic quotient, and for we define

 ΦW:Wω→(ΦW)ω,ΦW(v+Wω):=Φv+(ΦW)ω.

Let be a vector bundle over , a subbundle and . We say that leaves invariant iff , for every . Let be a symplectic vector bundle over . We define to be the set of all pairs , where is an -coisotropic subbundle, and . Let be a coisotropic subbundle, and be a transport that leaves invariant. We define by .

Theorem 1

(Coisotropic Maslov map for bundles) Let be connected and such that , and let be a symplectic vector bundle over . Then there exists a unique map with the following properties.

1. (Boundary) For every regular transport and every we have .

2. (Invariant subbundle) Let be an -coisotropic subbundle, and . If leaves invariant then .

For the proof of this theorem, the idea is to define

 mΣ,E,ω(W,Φ):=m∂Σ,ω|∂Σ(Ψ,Φ0),

where is a lift of , and is a regular transport. In order to show that this does not depend on the choice of , the following result is crucial. Namely, let be a symplectic vector space, a coisotropic subspace, and be such that . Then . (See Proposition 28 below.) The proof of this identity is based on the existence of a path , such that , leaves three fixed subspaces of invariant, and the map is constant.

Let be a connected surface satisfying . We define to be the class of all quadruples , where is a symplectic vector bundle over and . We define

 mΣ:EΣ→R,mΣ(E,ω,W,Φ):=mΣ,E,ω(W,Φ),

where is the unique map satisfying the conditions of Theorem 1.

Definition of mΣ,ω,N

We now define the map (1) as follows. Assume first that is connected. If then we define . Assume now that . We denote by the linear holonomy of the isotropic foliation of (see (38) below). We define the map by

 ˜mΣ,ω,N(u):=mΣ(u∗(TM,ω),u|∗∂Σ(TN,holN,ω)). (4)

It follows from Theorem 24(iii) below that this map is invariant under -admissible homotopies. For a general we define by .

Definition 2

Let be a symplectic manifold, a coisotropic submanifold, and . We define the Maslov map to be the map induced by .

As an example, let be the unit disk, , the standard structure , , and the inclusion . Then . For more examples see the subsection on page 1.4 about the Gaio-Salamon Maslov index.

The map may be viewed as a mean Maslov index. Analogously to the definition of the Conley-Zehnder index there should also be a natural integer valued map with the same domain.

The regular case

Let be a compact topological manifold. We call a map a weakly -admissible homotopy iff for every and there exists such that , for every . We denote by the corresponding set of homotopy classes. We call regular iff its isotropic leaf relation is a closed subset and a submanifold of . Assume now that is regular. Then it follows from Theorem 19(iv,vi) below that the Maslov map takes on integer values and is invariant under weak homotopies. If is also orientable then by Theorem 19(vi) takes on even values.

1.2 More elementary description

In more elementary, but less natural terms, the map is given as follows. Let be a symplectic vector space, and a coisotropic subspace. We define the framed coisotropic Grassmannian to be the manifold consisting of all pairs , where is a coisotropic subspace and .

Let be a path such that . We choose a path satisfying and . (It follows from Lemma 11 below that such a path exists.) We define . (It follows from Theorem 9(ii) below that this number does not depend on the choice of .)

Let now and be as above. For simplicity, assume that . We denote . We choose a symplectic trivialization , and define . We define the path as follows. Let . We set . Furthermore, we define to be the linear holonomy of the isotropic foliation of along the path . We set . The Maslov index of is now given by

 ˜mD,ω,N(u)=mω0(W,Φ).

1.3 Leaf-wise fixed points, presymplectic embeddings and minimal Maslov numbers

Leaf-wise fixed points

Assume that is regular. We define the minimal Maslov number

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We call monotone iff there exists a constant such that for every we have .

We denote by the group of Hamiltonian diffeomorphisms on . For every the pair is called non-degenerate iff the following holds. For we denote by the canonical projection. Let be an isotropic leaf, and a path. Assume that , and let be a vector. Then implies that

 holω,Nxprx(0)v≠prx(1)dφ(x(0))v. (5)

In the case this condition means that for every , is not an eigenvalue of . Furthermore, in the case that is Lagrangian the condition means that for every connected component we have , i.e. and intersect transversely.

For a topological space and we denote by the -th -Betti number of .

Theorem 3

Let be a (geometrically) bounded symplectic manifold, a closed monotone regular coisotropic submanifold and . If is non-degenerate then

 |Fix(φ,N)|≥∑i=dimN−m(N)+2,…,m(N)−2bi(N,Z2). (6)

This theorem generalizes a result for the case , which is due to P. Albers [Al].

Examples

A big class of examples is given as follows. Let and be closed symplectic manifolds and a closed Lagrangian submanifold. We define . Then is a closed regular coisotropic submanifold of .

Let . We define to be the closed surface obtained from by collapsing each boundary circle to a point. By straight-forward arguments the map , , is well-defined and a bijection. Furthermore, . This follows from Theorem 19(ii,viii) below. It follows that is the greatest common divisor of twice the minimal Chern number of and .

Assume that there exists such that for every , and , for every satisfying . Then is monotone, and hence satisfies the conditions of Theorem 3.

As a concrete example, let , , , be the Fubini-Studi form , the torus with the standard form , and the standard Lagrangian subtorus. Let be such that is non-degenerate. Then applying Theorem 3 we obtain choose ”.

Idea of proof of Theorem 3

The idea is to find a Lagrangian embedding of into a suitable symplectic manifold, and then apply the Main Theorem in [Al]. Since is regular, the set of isotropic leaves carries canonical smooth and symplectic structures and . We define

 ˜M:=M×Nω,˜ω:=ω⊕(−ωN), (7) ιN:N→˜M,ιN(x):=(x,Nx),˜N:=ιN(N). (8)

Then is an embedding of into that is Lagrangian with respect to the symplectic form on . In order for the hypotheses of Albers’ result to be satisfied, the inequality is crucial. It follows from Theorem 19(x) and Propositions 61 below.

Application: presymplectic non-embeddings

Let be a symplectic manifold. We denote by the contraction with the first Chern class of , and by the (spherical) minimal Chern number. Let be a regular presymplectic manifold. This means that the isotropic leaf relation of is a closed subset and a submanifold of . For we denote by the greatest common divisor of and . (Our convention is that , for , and .) We define . The proof of the following result is based on Theorem 3.

Theorem 4

Assume that is connected and bounded, every compact subset of is Hamiltonianly displaceable, is connected and closed, there exists an index such that , for some fiber every loop is contractible in , , and the following condition is satisfied.

1. There exists a constant such that on and on .

Then does not embed into .

Note that the condition is critical in the sense that in the case there is no presymplectic embedding of any open non-empty subset of into , whereas in the case for every point there exists an open neighbourhood that embeds presymplectically into .

The next result gives a criterion under which condition (i) in Theorem 4 holds and becomes simpler.

Proposition 5

Let be a connected symplectic manifold and a regular presymplectic manifold, such that some isotropic fiber is simply-connected, , and embeds into . Then . Furthermore, if is spherically monotone then condition (i) of Theorem 4 holds.

It follows from Theorem 4 and Proposition 5 that does not embed into , provided that and some conditions on and some conditions on are satisfied. (The point here is that there are no further assumptions involving both and .)

As an example, let and be positive integers, a closed symplectic manifold and a closed smooth fiber bundle with simply connected fibers, such that and there exists such that , where denotes the dimension of the fibers. We define and denote by the Fubini-Studi form on and by the standard symplectic form on . It follows from Theorem 4 that does not embed into .

More concretely, let be a positive integer and . Then does not embed into .

Coisotropic Audin conjecture

Recall that a topological space is called aspherical iff , for every . Furthermore, a manifold is called spin iff it is orientable and its second Stiefel-Whitney number vanishes.

Theorem 6

Let be a symplectic manifold that is convex at infinity, and a coisotropic submanifold that is closed, regular, aspherical, spin, and displaceable. Then .

In the Lagrangian case this result is due to K. Fukaya [Fu]. It generalizes a conjecture by Audin about the minimal Maslov number of a Lagrangian submanifold of diffeomorphic to the torus . The idea of proof of Theorem 6 is to reduce to the Lagrangian case using the construction (7,8), and then to apply Fukaya’s result.

1.4 Related work

Oh’s Maslov index

Let be an -compatible almost complex structure, assume that is gradable and equipped with a grading in the sense of [Oh], and that . In this situation, Y.-G. Oh defined a Maslov index , see Definition 3.3. in [Oh]. If is a smooth map then . Note that is defined on a larger set of maps than (after restriction to ), but requires as an additional datum. Observe also that the definition of does not involve the choice of any -compatible almost complex structure on .

The Gaio-Salamon Maslov index

Let be a Hamiltonian -manifold. This means that is a symplectic manifold, and is a connected Lie group acting on in a Hamiltonian way, with moment map . Assume that acts freely on . Let . We define the map as follows. Let . We choose a representative of , a symplectic vector space of dimension , a trivialization , and points , for every . We define by defining to be the unique solution of , for every and .

We define , where for every we denote by the differential of the action of . By a standard homotopy argument, this number does not depend on the choices of and . By Lemma 45 below the maps and agree.

For the map was introduced by R. Gaio and D. A. Salamon in [GS]. (More precisely, their definition relies on a choice of an -compatible almost complex structure on and a unitary trivialization of .)

Work by M. Entov and L. Polterovich and by V. L. Ginzburg

Let now be a closed (spherically) monotone symplectic manifold and a torus acting on in a Hamiltonian way, with moment map . Then by Theorem 1.7 in the article [EP] by M. Entov and L. Polterovich the pre-image of the special element of under is strongly (i.e. symplectically) non-displaceable.

Assume that the action of on is free. Then by Lemma 46 below is a closed, monotone regular coisotropic submanifold. Hence if is non-zero for some then it follows from Theorem 3 that is not leafwise displaceable (and hence not displaceable). Thus in this case we obtain a stronger statement than in Theorem 1.7 in [EP], provided that also .

In his recent paper [Gi] (Theorem 1.5) V. L. Ginzburg proved an upper bound on the minimal Maslov number of a closed, stable, displaceable coisotropic submanifold.

1.5 Organization and Acknowledgments

Organization of the article

In Section 2 it is shown that the Maslov map for pairs of flat transports is well-defined, and Theorem 1 is proved. Section 3 contains the proofs of the other results of Section 1. They are based on Theorem 19, which summarizes the main properties of the Maslov map. Section 4 is devoted to the proof of this theorem, using a similar result for the coisotropic Maslov index for bundles (Theorem 24). The appendix contains some results about the Salamon-Zehnder map, the Gaio-Salamon Maslov index, the relation with the mixed action-Maslov index, the linear holonomy of a foliation, and some topological results.

Acknowledgments

I would like to thank Yael Karshon for her continuous support and enlightening discussions, Masrour Zoghi and Dietmar Salamon for useful comments, Shengda Hu for making me aware of Lemma 11, and Viktor L. Ginzburg for his interest in my work.

2 Proof of Theorem 1 (Coisotropic Maslov map for bundles)

The following lemma was used in Section 1.

Lemma 7

The number in (3) is well-defined, i.e. it does not depend on the choice of . Furthermore, if and are regular then .

The next Remark is used in the proof of Lemma 7.

Remark 8

Let be a topological space and a symplectic vector bundle over . Then the map is continuous. To see this, we choose a symplectic vector space of dimension . Let be a pair, where is an open subset and . By Proposition 42(i) we have , for every and . Since the map is continuous, the statement follows.

{proof}

[Proof of Lemma 7] To prove the first assertion, let and be two choices of a path as above. We choose a map such that , for every , and . We denote , and we define , for . We also define by . It follows that and hence , for every . By Remark 8 the map is continuous.

Claim 1

The map is constant.

{proof}

[Proof of Claim 1] Consider the case in which is regular. We choose a path such that , for . We fix . By assumption we have . Furthermore, the paths and are homotopic with fixed end-points. It follows that . Hence by Proposition 42(i) we have . The case in which is regular, is treated similarly. This proves Claim 1. Claim 1, the fact (for every ) and continuity of imply that . Hence is well-defined.

The second assertion of the lemma follows directly from the definition of the Maslov index of a path of automorphisms of a symplectic vector space. This proves Lemma 7.

For the proof of Theorem 1 we need the following. Let be a topological manifold and . We define the equivalence relation on by iff there exists such that , and , for every and . We equip with the compact open topology and with the quotient topology. Then is a topological groupoid. We call admissible iff it contains the constant paths, and the following conditions hold. If and then . Furthermore, if are such that then the concatenation lies in . Assume that is admissible, and let be a topological vector bundle. A flat transport on along a morphism of topological groupoids that descends to the identity on . We denote by the set of such ’s. Let be another topological manifold and . Then the pullback is again admissible. For we define the pullback by .

Let be a topological manifold, a symplectic vector bundle over , and . We call a lift of iff for every we have and . Let be a closed curve. We denote by the canonical projection, and for , we define