Higher Maslov Indices

TL;DR

This paper introduces new Maslov-type indices for contact and symplectic groups, generalizing classical indices and exploring their properties and applications in symplectic topology.

Contribution

It defines two new classes of Maslov-type indices for contactomorphism and symplectomorphism groups, extending the classical Maslov index framework.

Findings

Defined indices as homotopy group maps to integer quotients

Constructed indices as maps to homotopy groups of homogeneous spaces

Explored properties and potential applications of these indices

Abstract

We define Maslov--type indices associated to contact and symplectic transformation groups. There are two such families of indices. The first class of indices are maps from the homotopy groups of the contactomorphism or symplectomorphism group to a quotient of the integers. These are based on a generalization of the Maslov index. The second class of indices are maps from the homotopy groups of the space of contact structures or the space of cohomologous symplectic forms to the homotopy groups of a simple homogeneous space. We provide a detailed construction and describe some properties of these indices and their applications.