## 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.