Homological Stability for Diffeomorphism Groups of High Dimensional Handlebodies

In this paper we prove a homological stability theorem for the diffeomorphism groups of high dimensional manifolds with boundary, with respect to forming the boundary connected sum with the product $D^{p+1}\times S^{q}$ for $|q - p| < \min\{p, q\} - 2$. In a recent joint paper with Boris Botvinnik (see arXiv:1509.03359 ), we identify the homology of $colim_{g\to \infty}BDiff((D^{n+1}\times S^{n})^{\natural g}, \; D^{2n})$ with that of the infinite loopspace $Q_{0}BO(2n+1)\langle n\rangle_{+}$, in the case that $n \geq 4$. Combining this "stable homology" calculation with this paper's homological stability theorem enables one to compute the (co)homology groups of $BDiff((D^{n+1}\times S^{n})^{\natural g}, D^{2n})$ in degrees $k \leq \tfrac{1}{2}(g - 4)$. This leads to the determination of the characteristic classes in degrees $k \leq \tfrac{1}{2}(g - 4)$ for all smooth fibre-bundles with fibre diffeomorphic to $(D^{n+1}\times S^{n})^{\natural g}$.