Let @math be a symplectic manifold, @math a coisotropic submanifold, and @math a compact oriented (real) surface. I define a natural Maslov index for each continuous map @math that sends every connected component of @math to some isotropic leaf of @math . 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 @math to compensate for the loss of boundary data in the case codimension @math . 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 @math is geometrically bounded and @math 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.
For @math the map @math was introduced by R. Gaio and D. A. Salamon in @cite_8 . (More precisely, their definition relies on a choice of an @math -compatible almost complex structure @math on @math and a unitary trivialization of @math .)
@cite_8
2018165282
A graph is planar if it can be embedded in the plane, or in the sphere, so that no two edges cross at an interior point. A planar graph together with a particular embedding is called a map. There is a rich theory of counting maps, started by Tutte in the 1960's. However, in this paper we are interested in counting graphs as combinatorial objects, regardless of how many nonequivalent topological embeddings they may have. As we are going to see, this makes the counting considerably more difficult.