Stable moduli spaces of high-dimensional manifolds

Abstract

We prove an analogue of the Madsen–Weiss theorem for high-dimensional manifolds. In particular, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of g copies of Sⁿ×Sⁿ, in the limit $g\to \infty$. 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 $\mathcal{N}(P)$ of those null-bordisms of a fixed (2n–1)-dimensional manifold P which are (n–1)-connected relative to P. We determine the homology of $\mathcal{N}(P)$ after stabilisation using certain self-bordisms of P. The stable homology is identified with that of an infinite loop space.