Stable Moduli Spaces of High Dimensional Handlebodies

TL;DR

This paper investigates the homology of moduli spaces of high-dimensional handlebodies, establishing an isomorphism with a well-understood infinite loop space in certain dimensions, thus enabling homology computations.

Contribution

It constructs a map linking the moduli space of handlebodies to a known infinite loop space and proves it induces an isomorphism on homology in high dimensions, extending previous theorems.

Findings

Homology groups of handlebody moduli spaces can be computed in a stable range.

The main theorem generalizes the Madsen-Weiss and Galatius-Randal-Williams results.

Established an isomorphism on integral homology for high-dimensional handlebodies.

Abstract

We study the moduli space of handlebodies diffeomorphic to $(D^{n+1}\times S^{n})^{\natural g}$, i.e. the classifying space $BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n})$ of the group of diffeomorphisms that restrict to the identity near a $2n$-dimensional disk embedded in the boundary, $\partial(D^{n+1}\times S^n)^{\natural g}$. We construct a map $colim_{g\to\infty}BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n}) \longrightarrow Q_{0}BO(2n+1)\langle n \rangle_{+}$ and prove that it induces an isomorphism on integral homology in the case that $2n+1 \geq 9$. Above, $BO(2n+1)\langle n \rangle$ denotes the $n$-connective cover of $BO(2n+1)$. The (co)homology of the space $Q_{0}BO(2n+1)\langle n \rangle_{+}$ is well understood and so our results enable one to compute the homology groups $H_{k}(BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n}))$ in a range of degrees when $k << g$. Our main theorem can be viewed as an analogue of the Madsen-Weiss theorem for the moduli spaces of surfaces and the recent theorem of Galatius and Randal-Williams for the moduli spaces of manifolds of dimension $2n \geq 6$.