Braid groups and discrete diffeomorphisms of the punctured disk

Abstract: We show that the group cohomology of the diffeomorphisms of the disk with $n$ punctures has the cohomology of the braid group of $n$ strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding $\mathrm{Br}{2g+2}\hookrightarrow \mathrm{Mod}{g,2}$ to a group homomorphism between diffeomorphism groups.