Harer stability

Harer stability states that the degree $d$ homology of the mapping class group $\Gamma_{g,n}$ is independent of $g$ and $n$ if $d$ is small compared to $g$. More precisely, consider the following maps on classifying spaces. First, we construct a map $B\Gamma_{g,b} \rightarrow B\Gamma_{g,b-1}$ by adjoining a disk to a given boundary component. Second, we can construct a map $B\Gamma_{g,b} \rightarrow B\Gamma_{g+1,b}$ 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 $H_{d}(-,\mathbb{Z})$ for $2d < g-1$. In particular, it allows us to talk about the stable homology/cohomology of the moduli space of curves, as in Mumford's conjecture.