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 $B{\Gamma }_{g,b}\to B{\Gamma }_{g,b-1}$ by adjoining a disk to a given boundary component.
• Second, we can construct a map $B{\Gamma }_{g,b}\to B{\Gamma }_{g+1,b}$ 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.