A 2-cocycle on a group of symplectomorphisms

TL;DR

This paper constructs a 2-cocycle on the group of symplectomorphisms for exact symplectic manifolds and explores conditions under which this cocycle is non-trivial, contributing to the understanding of symplectic group cohomology.

Contribution

It introduces a new 2-cocycle on symplectomorphism groups for exact symplectic manifolds and analyzes its non-triviality, advancing symplectic topology and group cohomology.

Findings

Constructed a 2-cocycle on symplectomorphism groups.

Identified cases where the cocycle is non-trivial.

Provides insights into the structure of symplectomorphism groups.

Abstract

For a symplectic manifold $(M,\om)$ with exact symplectic form we construct a 2-cocycle on the group of symplectomorphisms and indicate cases when this cocycle is not trivial.