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

Let Sigma(1)(g)(,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 -> G(rel) -> G -> Diff(+)(S-1)-> 1, we construct a central R-extension of Diff(+)(S-1). We also show that the second cohomology class corresponding to the central R-extension is equal to the Euler class of Diff(+)(S-1) up to non-zero constant multiple.