Homological stability for moduli spaces of high dimensional manifolds. II

TL;DR

This paper establishes a homological stability theorem for moduli spaces of high-dimensional manifolds, extending previous results and providing an analogue of the Madsen–Weiss theorem for simply-connected manifolds of dimension at least 6.

Contribution

It proves a new homological stability theorem for moduli spaces of high-dimensional manifolds with handle attachments, generalizing prior work and connecting to the Madsen–Weiss theorem.

Findings

Homological stability holds for moduli spaces after stabilisation with $S^n imes S^n$.

An analogue of the Madsen–Weiss theorem is established for simply-connected manifolds of dimension ≥ 6.

The results apply to manifolds of dimension $2n$ with handle attachments of index ≥ $n$.

Abstract

We prove a homological stability theorem for moduli spaces of manifolds of dimension $2n$, for attaching handles of index at least $n$, after these manifolds have been stabilised by countably many copies of $S^n \times S^n$. Combined with previous work of the authors, we obtain an analogue of the Madsen--Weiss theorem for any simply-connected manifold of dimension $2n \geq 6$.