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:
“In short, no association or alliance can be happy or stable without me. People can’t long tolerate a ruler, nor can a master his servant, a maid her mistress, a teacher his pupil, a friend his friend nor a wife her husband, a landlord his tenant, a soldier his comrade nor a party-goer his companion, unless they sometimes have illusions about each other, make use of flattery, and have the sense to turn a blind eye and sweeten life for themselves with the honey of folly.”
—Desiderius Erasmus (c. 1466–1536)
“Alas for the cripple Practice when it seeks to come up with the bird Theory, which flies before it. Try your design on the best school. The scholars are of all ages and temperaments and capacities. It is difficult to class them, some are too young, some are slow, some perverse. Each requires so much consideration, that the morning hope of the teacher, of a day of love and progress, is often closed at evening by despair.”
—Ralph Waldo Emerson (1803–1882)
“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)