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 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.

💡 Analysis

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.

📄 Content

arXiv:0908.1938v2 [math.AT] 30 Sep 2009 Stability phenomena in the topology of moduli spaces Ralph L. Cohen ∗ Dept. of Mathematics Stanford University Stanford, CA 94305 October 30, 2018 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 payoffhas 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. Contents 1 Classical stability theorems 4 1.1 The Freudenthal suspension theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Whitney’s Embedding Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Bott periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 ∗The author was partially supported by NSF grant DMS-0603713 1 2 Configuration spaces, permutations, and braids 8 2.1 Configurations of points in a manifold . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Symmetric groups and braid groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Holomorphic curves and gauge theory 11 3.1 Holomorphic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.1 Flat connections on Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . 14 3.2.2 Self dual connections on four-manifolds and the Atiyah-Jones Conjecture . . 16 4 General linear groups, Pseudoisotopies, and K-theory 18 4.1 The stable topology of general linear groups and algebraic K-theory . . . . . . . . . 18 4.2 Pseudoisotopies, and Waldhausen’s algebraic K-theory of spaces . . . . . . . . . . . 18 5 The moduli space of Riemann surfaces, mapping class groups, and the Mumford conjecture 20 5.1 Mapping class groups, moduli spaces, and Thom spaces . . . . . . . . . . . . . . . . 20 5.2 Automorphisms of free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6 Final Comments 29 Introduction In the last sixty years, the notions of classifying space and moduli space have played central roles in the development of topology and geometry. These are spaces that encode the basic topological or geometric structure to be studied, and therefore the topology of these spaces naturally have been a subject of intense interest. Probably the most fundamental among them are the moduli spaces of Riemann surfaces of genus g, Mg. In a dramatic application of algebraic topological methods to algebraic geometry, Madsen and Weiss recently proved a well known conjecture of Mumford regarding the stable cohomology of moduli space [42]. Namely, Mumford described a ring homomorphism from a graded polynomial algebra over the rationals, to the cohomology of moduli space with rational coefficients, Q[κ1, κ2, · · · κi, · · · ] −→H∗(Mg; Q), and conjectured that it is an isomorphism when the genus g is large with respect to the cohomological grading. Here κi is the Miller-Morita-Mumford canonical class, and has grading 2i. In [42] Madsen and Weiss described a homotopy theoretic model for the stable moduli space, M∞, and in so doing, not only proved Mumford’s conjecture, but also gave an implicit model for the stable cohomology of moduli space with any coefficients. Using this explicit model, Galatius [20] calculated this stable 2 cohomology explicitly, when the coefficients are Z/p for p any prime, and in so doing uncovered a vast amount of previously undetected torsion in the stable cohomology of moduli space. The Madsen-Weiss theorem can be viewed as one of the most recent examples of a stability theorem regarding the topology of classifying spaces or moduli spaces. The purpose of this paper is to give a survey of these types of theorems and their applications to a broad range of topics in topology and geometry. Stabi

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