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 $2n \ge 6$.
"Homological stability for moduli spaces of high dimensional manifolds. II." Ann. of Math. (2) 186 (1) 127 - 204, July 2017. https://doi.org/10.4007/annals.2017.186.1.4