Homological Stability for Diffeomorphism Groups of High Dimensional Handlebodies

TL;DR

This paper establishes homological stability for diffeomorphism groups of high-dimensional handlebodies with boundary, enabling computation of their homology and characteristic classes in stable ranges.

Contribution

It proves a new homological stability theorem for these groups and combines it with prior stable homology results to compute their (co)homology and characteristic classes.

Findings

Homological stability holds for diffeomorphism groups with boundary connected sum operations.

Stable homology groups are identified with those of an infinite loop space.

Characteristic classes are computed in stable degrees for certain fiber bundles.

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 $colim_{g\to \infty}BDiff((D^{n+1}\times S^{n})^{\natural g}, \; D^{2n})$ 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 $(D^{n+1}\times S^{n})^{\natural g}$.