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

Abstract

Let Σg,11 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→Grel→G→Diff+(S1)→1, we construct a central ℝ-extension of Diff+(S1). We also show that the second cohomology class corresponding to the central ℝ-extension is equal to the Euler class of Diff+(S1) up to non-zero constant multiple.