The flux homomorphism and central extensions of diffeomorphism groups

TL;DR

This paper studies the flux homomorphism on a group of symplectomorphisms of a disk, constructs a central extension, and explores its relation to known invariants and cocycles in symplectic geometry.

Contribution

It introduces the flux extension for symplectomorphism groups and determines its Euler class, linking it to existing cocycles and invariants.

Findings

Determined the Euler class of the flux extension.

Connected the flux extension to the Ismagilov-Losik-Michor cocycle.

Related the flux extension to the Calabi invariant.

Abstract

Let $D$ be a 2-dimensional closed unit disk and $\rm{Symp}(D,0)_{\rm{rel}}$ the group of symplectomorphisms preserving the origin and the boundary $\partial D$ pointwise. We consider the $\mathbb{R}$-valued flux homomorphism on $\rm{Symp}(D,0)_{\rm{rel}}$ and define the central $\mathbb{R}$-extension called the $\mathbb{R}$-valued flux extension. We determine the Euler class of this extension and investigate the relation between the extension, the group $2$-cocycle defined by Ismagilov, Losik, and Michor, and the Calabi invariant of $D$.