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