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, Br_n, as the summand. As an application of this method, we also prove there is no cohomological obstruction to lifting the ``standard" embedding Br_{2g+2} into the mapping class group of a surface of genus g and two boundary components to a group homomorphism between diffeomorphism groups.