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 $S^n \times S^n$.   Combined with previous work of the authors, we obtain an analogue of the Madsen--Weiss theorem for any simply-connected manifold of dimension $2n \geq 6$. 
Soren Galatius, Oscar Randal-Williams
math.AT math.GT