Abelianization and Nielsen realization problem of the mapping class group of a handlebody

For the oriented 3-dimensional handlebody constructed from a 3-ball by attaching g 1-handles, it is shown that the natural surjection from the group of orientation preserving diffeomorphisms to the mapping class group has no section when g is at least 5. In order to prove the above result, we show the vanishing of the first homology group with the real coefficient of the mapping class group of the handlebody with genus g at least 3 and a distinguished disk on its boundary.