Subject: Borel, continuous and smooth group cohomology From: "John Baez" Date: Tue, 31 Aug 2004 08:17:58 -0700 (PDT) To: dmd1@lehigh.edu Dear Topologists - 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 H3(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. While I'm at it, I should make sure I know the usual "algebraic" group cohomology H3(G,A) in these cases. In Daniel Baker's paper "Differential characters and Borel cohomology", he seems to say that H^n_{Borel}(G,U(1)) = H^{n+1}(BG,Z) where H^{n+1}(BG,Z) is the singular cohomology of the classifying space of the Lie group G with integral coefficients. This would handle this case. He also says that someone named Wigner proved H^n_{Borel}(G,R) = 0 In Jim Stasheff's paper "Continuous cohomology of groups and classifying spaces", he says that H^n_{continuous}(G,R) = 0 I guess the cases I'm really most curious about are H3_{continuous}(G,U(1)) and H3_{smooth}(G,U(1)) I remember someone saying these vanish when G is compact simple, but I don't see why this is true! I would like to use a Bockstein exact sequence H^k_{continuous}(G,R) -> H^k_{continuous}(G,U(1)) -> H^{k+1}_{continuous}(G,Z) -> H^{k+1}_{continuous}(G,R) together with the vanishing of the two ends, to show that H^k_{continuous}(G,U(1)) is isomorphic to H^{k+1}_{continuous}(G,Z) and hence zero. But, Mostow's paper "Cohomology of Topological Groups and Solvmanifolds" only gives a Bockstein exact sequence when the short exact sequence of coefficient groups has a continuous section, which 0 -> Z -> R -> U(1) -> 0 very much does NOT! So, I don't know how to tackle H3_{continuous}(G,U(1)) What is it??? I must admit I'm also curious about H3_{algebraic}(G,U(1)). Baker describes a way to get elements in here from elements of H4(BG,Z), but he doesn't seem to come right out and say they're the same.