Editing Mapping class group (section)

== Stable mapping class group ==
{{Expand section|date=December 2009}}
One can embed the surface <math>\Sigma_{g,1}</math> of genus ''g'' and 1 boundary component into <math>\Sigma_{g+1,1}</math> by attaching an additional hole on the end (i.e., gluing together <math>\Sigma_{g,1}</math> and <math>\Sigma_{1,2}</math>), and thus automorphisms of the small surface fixing the boundary extend to the larger surface. Taking the [[direct limit]] of these groups and inclusions yields the '''stable mapping class group,''' whose rational cohomology ring was conjectured by [[David Mumford]] (one of conjectures called the [[Mumford conjecture (disambiguation)|Mumford conjecture]]s). The integral (not just rational) cohomology ring was computed in 2002 by [[Ib Madsen]] and [[Michael Weiss (mathematician)|Michael Weiss]], proving Mumford's conjecture.