Stable moduli spaces of high-dimensional handlebodies

We study the moduli space of handlebodies diffeomorphic to (Dn+1 x S-n)(right angle g), that is, the classifying space BDiff((Dn+1 x S-n)(right angle g), D-2n) of the group of diffeomorphisms that restrict to the identity near an embedded disk D-2n subset of (Dn+1 x S-n)(right angle g), D-2n. We prove that there is a natural map colim(g ->infinity)BDiff((Dn+1 x S-n)(right angle g), D-2n) -> Q(0)BO(2n +1) < n >(+), which induces an isomorphism in integral homology when n >= 4. Above, BO(2n +1) < n > denotes the n-connective cover of BO(2n +1).