Homological Stability for Diffeomorphism Groups of High Dimensional Handlebodies

Abstract

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 $coli{m}_{g\to \infty }BDiff((D^{n+1}\times S^n{)}^{♮g},D^{2n})$ with that of the infinite loopspace $Q_{0}BO(2n+1)⟨n{⟩}_{+}$, in the case that $n\ge 4$. Combining this "stable homology" calculation with this paper's homological stability theorem enables one to compute the (co)homology groups of $BDiff((D^{n+1}\times S^n{)}^{♮g},D^{2n})$ in degrees $k\le \frac{1}{2}(g-4)$. This leads to the determination of the characteristic classes in degrees $k\le \frac{1}{2}(g-4)$ for all smooth fibre-bundles with fibre diffeomorphic to $(D^{n+1}\times S^n{)}^{♮g}$.

30 pages, streamlined some of the proofs and the exposition, mathematics is the same as the previous version