Generalized Conley-Zehnder index

TL;DR

This paper introduces new methods to compute the Conley-Zehnder index and provides an axiomatic characterization of its generalization by Robbin and Salamon, applicable to continuous symplectic matrix paths.

Contribution

It offers an explicit computation method and an axiomatic framework for the generalized Conley-Zehnder index, extending previous definitions.

Findings

Provides explicit computation techniques for the generalized index.

Establishes an axiomatic characterization of the index.

Connects the index to paths of Lagrangians and regular crossings.

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.