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)
“No government can help the destinies of people who insist in putting sectional and class consciousness ahead of general weal.”
—Franklin D. Roosevelt (1882–1945)
“The government of the United States at present is a foster-child of the special interests. It is not allowed to have a voice of its own. It is told at every move, “Don’t do that, You will interfere with our prosperity.” And when we ask: “where is our prosperity lodged?” a certain group of gentlemen say, “With us.””
—Woodrow Wilson (1856–1924)