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:
“And neigh like Boanerges—
Then—punctual as a Star
Stop—docile and omnipotent
At its own stable door—”
—Emily Dickinson (1830–1886)
“There might be a class of beings, human once, but now to humanity invisible, for whose scrutiny, and for whose refined appreciation of the beautiful, more especially than for our own, had been set in order by God the great landscape-garden of the whole earth.”
—Edgar Allan Poe (1809–1849)
“Instead of seeing society as a collection of clearly defined “interest groups,” society must be reconceptualized as a complex network of groups of interacting individuals whose membership and communication patterns are seldom confined to one such group alone.”
—Diana Crane (b. 1933)