Stability phenomena in the topology of moduli spaces

Abstract

The recent proof by Madsen and Weiss of Mumford's conjecture on the stable cohomology of moduli spaces of Riemann surfaces, was a dramatic example of an important theorem about the topology of moduli spaces. In this article we give a survey of families of classifying spaces and moduli spaces where stability occur in their topologies. Such theorems have been proved in many situations in the history of topology and geometry, and the payoff has often been quite remarkable. In this paper we discuss classical theorems such as the Freudenthal suspension theorem, Bott periodicity, and Whitney's embedding theorems. We then discuss more modern examples such as those involving configuration spaces of points in manifolds, holomorphic curves in complex manifolds, gauge theoretic moduli spaces, the stable topology of general linear groups, and pseudoisotopies of manifolds. We then discuss the theorems regarding the moduli spaces of Riemann surfaces: Harer's theorem on the cohomology of moduli space, and the Madsen-Weiss theorem, which proves a generalization of Mumford's conjecture. We also describe Galatius's recent theorem on the stable cohomology of automorphisms of free groups. We end by speculating on the existence of general conditions in which one might expect these phenomena to occur.