The Dixmier-Douady class, the action homomorphism, and group cocycles on the symplectomorphism group

TL;DR

This paper constructs specific cocycles on the symplectomorphism group of a symplectic manifold to explore their relationship with the Dixmier-Douady class and Weinstein's action homomorphism, deepening understanding of symplectic topology.

Contribution

It introduces new two- and three-cocycles on the symplectomorphism group and clarifies their connection to key invariants like the Dixmier-Douady class and action homomorphism.

Findings

Constructed explicit two- and three-cocycles.

Established a relationship between cocycles and the Dixmier-Douady class.

Linked Weinstein's action homomorphism to symplectic fibrations.

Abstract

Let $X$ be a one-connected and integral symplectic manifold. In this paper, we construct and study a two-cocycle and three-cocycle on the symplectomorphism group of $X$. In particular, by using these cocycles, we clarify the relationship between Weinstein's action homomorphism and the universal Dixmier-Douady class of flat symplectic fibrations.