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, S_{g,n} (X, \gamma). This is the space consisting of triples, (F_{g,n}, \phi, f), where F_{g,n} is a Riemann surface of genus g and n-boundary components, \phi is a parameterization of the boundary, and f : F_{g,n} \to X is a continuous map that satisfies a boundary condition \gamma. We prove three theorems about these spaces. Our main theorem is the identification of the stable homology type of the space S_{\infty, n}(X; \gamma), defined to be the limit as the genus g gets large, of the spaces S_{g,n} (X; \gamma). 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 S_{g,n} (X; \gamma) 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.