# Harer stability

Harer stability states that the degree homology of the mapping class group is independent of and if is small compared to . More precisely, consider the following maps on classifying spaces. First, we construct a map by adjoining a disk to a given boundary component. Second, we can construct a map by gluing a torus with two boundary components along a given boundary component of our original Riemann surface. Harer's stability theorem asserts that both of these maps induce an isomorphism on for . In particular, it allows us to talk about the stable homology/cohomology of the moduli space of curves, as in Mumford's conjecture.

Jeffrey Herschel Giansiracusa 2005-06-27