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:
“My whole working philosophy is that the only stable happiness for mankind is that it shall live married in blessed union to woman-kind—intimacy, physical and psychical between a man and his wife. I wish to add that my state of bliss is by no means perfect.”
—D.H. (David Herbert)
“Criminals are never very amusing. It’s because they’re failures. Those who make real money aren’t counted as criminals. This is a class distinction, not an ethical problem.”
—Orson Welles (1915–1985)
“We begin with friendships, and all our youth is a reconnoitering and recruiting of the holy fraternity they shall combine for the salvation of men. But so the remoter stars seem a nebula of united light, yet there is no group which a telescope will not resolve; and the dearest friends are separated by impassable gulfs.”
—Ralph Waldo Emerson (1803–1882)