Computations in the unstable homology of moduli spaces of Riemann surfaces

Abstract

In this article we give a survey of homology computations for moduli spaces M m g,1 of Riemann surfaces with genus g 0, one boundary curve, and m 0 punctures. While rationally and stably this question has a satisfying answer by the Madsen-Weiss theorem, the unstable homology remains notoriously complicated. We discuss calculations with integral, mod-2, and rational coefficients. Furthermore, we determine, in most cases, explicit generators using homology operations.