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Let $(M,\omega)$ be a symplectic manifold, $N\subseteq M$ a coisotropicsubmanifold, and $\Sigma$ a compact oriented (real) surface. I define a naturalMaslov index for each continuous map $u:\Sigma\to M$ that sends every connectedcomponent of $\partial\Sigma$ to some isotropic leaf of $N$. This index is realvalued and generalizes the usual Lagrangian Maslov index. The idea is to usethe linear holonomy of the isotropic foliation of $N$ to compensate for theloss of boundary data in the case codimension $N<\dim M/2$. The definition isbased on the Salamon-Zehnder (mean) Maslov index of a path of linear symplecticautomorphisms. I prove a lower bound on the number of leafwise fixed points ofa Hamiltonian diffeomorphism, if $(M,\omega)$ is geometrically bounded and $N$is closed, regular (i.e. "fibering"), and monotone. As an application, weobtain a presymplectic non-embedding result. I also prove a coisotropic versionof the Audin conjecture.

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