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:
“If it is to be done well, child-rearing requires, more than most activities of life, a good deal of decentering from one’s own needs and perspectives. Such decentering is relatively easy when a society is stable and when there is an extended, supportive structure that the parent can depend upon.”
—David Elkind (20th century)
“There is ... a class of fancies, of exquisite delicacy, which are not thoughts, and to which, as yet, I have found it absolutely impossible to adapt language.... Now, so entire is my faith in the power of words, that at times, I have believed it possible to embody even the evanescence of fancies such as I have attempted to describe.”
—Edgar Allan Poe (1809–1849)
“The poet who speaks out of the deepest instincts of man will be heard. The poet who creates a myth beyond the power of man to realize is gagged at the peril of the group that binds him. He is the true revolutionary: he builds a new world.”
—Babette Deutsch (1895–1982)