Topology Seminar: Thomas Church

  • Date: 03/20/2013
Thomas Church- Stanford University

University of British Columbia


Stability in the unstable cohomology of mapping class groups, SL_n(Z), and Aut(F_n)

For each of the sequences of groups in the title, the k-th rational cohomology is independent of n in a linear range n >= c*k, and this "stable cohomology" has been explicitly computed in each case. In contrast, very little is known about the unstable cohomology, which lies outside this range.

I will explain a conjecture on a new kind of stability in the unstable cohomology of these groups, in a range near the "top dimension" (the virtual cohomological dimension). For SL_n(Z) the conjecture implies that the unstable cohomology actually vanishes in that range. One key ingredient is a version of Poincare duality for these groups based on the topology of the curve complex and the algebra of modular symbols. Based on joint work with Benson Farb and Andrew Putman.
Other Information: 

ESB 4127