Homological stability for moduli spaces of high dimensional manifolds. II

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 Sⁿ X Sⁿ. Combined with previous work of the authors, we obtain an analogue of the Madsen-Weiss theorem for any simply-connected manifold of dimension 2n ≥ 6.