Topology and Geometry Seminar

Title: H^3 and multi-extensions

Speaker: Ettore Aldrovandi

Abstract: Let B a group and A an abelian group. It is well known and classical that the extensions of B by A are classified (up to isomorphism) by the second cohomology group of B with coefficients in A. We ask what sort of extension problem (if any) is classified by the higher cohomology groups. We will focus on n=3, and the answer is known, and due to Grothendieck and his student H. X Sinh: the third cohomology group classifies extensions of B by A where the middle term is a group-like category, rather than an ordinary group. The extension sequence can be described in terms of ordinary groups, but it will have length 4, rather than 3. Group-like categories are used in topology and K-theory to construct deloopings and ultimately to model spectra. To motivate the interest in H^3, we will consider the case where B=SL(2,C), A=R, where the generator involves the volume computations in 3-hyperbolic space.