Stable moduli spaces of high dimensional manifolds

We prove an analogue of the Madsen-Weiss theorem for high dimensional manifolds. For example, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of g copies of S^n x S^n, in the limit as g tends to infinity. Rationally it is a polynomial ring in certain explicit generators, giving a high dimensional analogue of Mumford's conjecture. More generally, we study a moduli space N(P) of those null-bordisms of a fixed (2n-1)-dimensional manifold P which are highly connected relative to P. We determine the homology of N(P) after stabilisation using certain self-bordisms of P. The stable homology is identified with that of a certain infinite loop space.