Generalized Conley-Zehnder index

Abstract

The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. We give new ways to compute this index. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space $(W,\bar{\Omega})$, having chosen a given reference Lagrangian $V$. Paths of symplectic endomorphisms of $(\R^{2n},\Omega_0)$ are viewed as paths of Lagrangians defined by their graphs in $(W=\R^{2n}\oplus \R^{2n},\bar{\Omega}=\Omega_0\oplus -\Omega_0)$ and the reference Lagrangian is the diagonal. Robbin and Salamon give properties of this generalized Conley-Zehnder index and an explicit formula when the path has only regular crossings. We give here an axiomatic characterization of this generalized Conley-Zehnder index. We also give an explicit way to compute it for any continuous path of symplectic matrices.