Borel, continuous and smooth cohomology of Lie groups

From: John Baez (baez_at_galaxy.ucr.edu)
Date: 08/27/04 

Date: Fri, 27 Aug 2004 18:30:03 +0000 (UTC)

If G is a Lie group and A is an abelian Lie group,
we can define "Borel", "continuous" and "smooth" versions
of group cohomology H^n(G,A) by taking the usual
chain complex for group cohomology and adding the
requirement that the n-cocycles f: G^n -> A be Borel
measurable, continuous or smooth functions.

I would like to know H^3(G,A) for all three versions
when G is a compact Lie group and A is R or U(1), with
G acting trivially on A.

In Daniel Baker's paper "Differential characters and Borel
cohomology", he seems to say that

H^n_{Borel}(G,U(1)) = H^n(g,R)

where H^n(g,R) is the cohomology of the Lie algebra of a
compact Lie group G with coefficients in the trivial representation
on R. So, for example, we'd get R as our cohomology group when
G is compact simple and n = 3.

In Jim Stasheff's paper "Continuous cohomology of groups
and classifying spaces", he says that

H^n_{continuous}(G,R) = 0

whenever G is a compact Lie group.

I still need to read Hochschild and Mostow's paper "Cohomology of
Lie groups", which is earlier than these others... but the library
closed before I got to it!

I guess the cases I'm really most curious about are

H^3_{continuous}(G,U(1))

and

H^3_{smooth}(G,U(1))

I have a feeling these vanish at least when G is compact simple,
but I don't actually know this!

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