Splitting of the homology of the punctured mapping class group

Let (Formula presented.) be the mapping class group of the orientable surface (Formula presented.) of genus (Formula presented.) with one parametrized boundary curve and (Formula presented.) permutable punctures; when (Formula presented.) we omit it from the notation. Let (Formula presented.) be the braid group on (Formula presented.) strands of the surface (Formula presented.). We prove that (Formula presented.). The main ingredient is the computation of (Formula presented.) as a symplectic representation of (Formula presented.).