Surfaces in a background space and the homology of mapping class groups

Abstract

In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, Sg,n(X,). This is the space consisting of triples, (Fg,n,�,f), where Fg,n is a Riemann surface of genus g and n-boundary components, � is a parameterization of the boundary, @Fg,n, and f : Fg,n! X is a continuous map that satisfies a boundary condition . We prove three theorems about these spaces. Our main theorem is the identification of the stable homology type of the space S1,n(X;), defined to be the limit as the genus g gets large, of the spaces Sg,n(X;). Our result about this stable topology is a parameterized version of the theorem of Madsen and Weiss proving a generalization of the Mumford conjecture on the stable cohomology of mapping class groups. Our second result describes a stable range in which the homology of Sg,n(X;) is isomorphic to the stable homology. Finally we prove a stability theorem about the homology of mapping class groups with certain families of twisted coefficients. The second and third theorems are generalizations of stability theorems of Harer and Ivanov.