# Homological stability for moduli spaces of high dimensional manifolds. II

### 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 $2n \geq 6$.

## Authors

Søren Galatius

Stanford University, Stanford, CA

University of Copenhagen, Copenhagen, Denmark Oscar Randal-Williams

Centre for Mathematical Sciences, Wilberforce Road, Cambridge, United Kingdom