The Flux Homomorphism on a Surface with Boundary and Central Extensions of Diffeomorphism Groups

Abstract

Let ${\Sigma }_{g,1}^1$ be a genus $g$ compact oriented surface with one boundary component and one marked point $x$. Let $G$ be the identity component of the group of symplectomorphisms that preserve the marked point $x$. By using the flux homomorphism and the short exact sequence $1\to G_{\text{rel}}\to G\to {\text{Diff}}_{+}\left(S^1\right)\to 1$, we construct a central $\mathrm{ℝ}$-extension of ${\text{Diff}}_{+}\left(S^1\right)$. We also show that the second cohomology class corresponding to the central $\mathrm{ℝ}$-extension is equal to the Euler class of ${\text{Diff}}_{+}\left(S^1\right)$ up to non-zero constant multiple.