Stable moduli spaces of high dimensional manifolds

TL;DR

This paper extends the Madsen-Weiss theorem to high-dimensional manifolds, describing the stable characteristic classes of certain fiber bundles and analyzing the homology of related moduli spaces.

Contribution

It provides a high-dimensional analogue of Mumford's conjecture by explicitly describing the stable ring of characteristic classes for fiber bundles with connected sum fibers.

Findings

Rational stable characteristic classes form a polynomial ring.

Homology of moduli spaces stabilizes and matches that of an infinite loop space.

Explicit description of the stable homology of null-bordism moduli spaces.

Abstract

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.