Surfaces in a background space and the homology of mapping class groups
  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. 
Ralph L. Cohen and Ib Madsen