Stability phenomena in the topology of moduli spaces

TL;DR

This paper surveys various stability phenomena in the topology of moduli spaces and classifying spaces, highlighting classical and modern theorems, including recent breakthroughs like the Madsen-Weiss theorem, and discusses potential general conditions for such stability.

Contribution

It provides a comprehensive overview of stability theorems in topology, connecting classical results with recent advances like the Madsen-Weiss and Galatius theorems, and explores their implications.

Findings

Discussion of classical stability theorems such as Freudenthal suspension and Bott periodicity.

Coverage of modern stability results including configuration spaces and gauge theory.

Analysis of recent theorems like Madsen-Weiss and Galatius on moduli spaces and automorphisms.

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 stability theorem about the topology of moduli spaces. In this article we give a survey of families of classifying spaces and moduli spaces where "stability phenomena" occur in their topologies. Such stability 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 stability 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 stability theorems regarding the moduli spaces of Riemann surfaces: Harer's stability 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 stability phenomena to occur.