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Geometry-Topology Seminar

Thursday, February 9, 2017 - 4:30pm

Thomas Church

Stanford University / IAS


University of Pennsylvania


This is the first of two talks in the Geometry-Topology Seminar today, sponsored jointly with Temple, Bryn Mawr and Haverford.

 Borel proved that in low dimensions, the cohomology of a locally symmetric space can be represented not just by harmonic forms but by invariant forms.  This implies that the k-th rational cohomology of SL_n(Z) is independent of  n  in a linear range  n>= c*k , and tells us exactly what this "stable cohomology" is.  In contrast, very little is known about the unstable cohomology, in higher dimensions outside this range.

In this talk I will explain a conjecture on a new kind of stability in the unstable cohomology of arithmetic groups like SL_n(Z).  These conjectures deal with the "codimension-k" cohomology near the top dimension (the virtual cohomological dimension), and for SL_n(Z) they imply the cohomology vanishes there.  Although the full conjecture is still open, I will explain how we proved it for codimension-0 and codimension-1.  The key ingredient is a version of Poincare duality for these groups based on the algebra of modular symbols, and a new presentation for modular symbols.  Joint work with Benson Farb and Andrew Putman.