Stable Mapping Class Group
One can embed the surface of genus g and 1 boundary component into by attaching an additional hole on the end (i.e., gluing together and ), 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 conjectures). The integral (not just rational) cohomology ring was computed in 2002 by Madsen and Weiss, proving Mumford's conjecture.
Read more about this topic: Mapping Class Group
Famous quotes containing the words stable, class and/or group:
“One of the oddest features of western Christianized culture is its ready acceptance of the myth of the stable family and the happy marriage. We have been taught to accept the myth not as an heroic ideal, something good, brave, and nearly impossible to fulfil, but as the very fibre of normal life. Given most families and most marriages, the belief seems admirable but foolhardy.”
—Jonathan Raban (b. 1942)
“A revolution is not a dinner party, or writing an essay, or painting a picture, or doing embroidery; it cannot be so refined, so leisurely and gentle, so temperate, kind, courteous, restrained and magnanimous. A revolution is an insurrection, an act of violence by which one class overthrows another.”
—Mao Zedong (1893–1976)
“Jury—A group of twelve men who, having lied to the judge about their hearing, health, and business engagements, have failed to fool him.”
—H.L. (Henry Lewis)