Subject: answer to Baez's problem From: "pjz" Date: Fri, 31 Dec 2004 16:09:05 +0800 (CST) Dear John Baez: Following are relations between various cohomologies you are interested in: 1.For Lie group and smooth coefficients, smooth and continuous group cohomologies coincide. 2.Continuous group cohomology is the same as Borel group cohomology if the group is locally compact and the coefficients are vector spaces. 3.The algebraic group cohomology you mentioned is the group cohomology of the group made discrete (forgetting the topology) as I understand and thus is the cohomology of the classifying space of the group made discrete. 4.Topological group extension of locally compact groups is Borel split, thus short exact sequence of coefficients which are locally compact leads to a Bockstein type long exact sequence of cohomology. 5.The continuous cohomology of a compact Lie group with trivial coefficient R is trivial and thus 2 combined with 4 gives the desired isomorphism H^n_{Borel}(G,U(1)) = H^{n+1}(BG,Z) However this is not true in general for continuous cohomology since, for connected and 1-connected compact Lie groups G, H^n_{cont}(G,A) = H^n_{cont}(G,\tilde{A}) , n>0 where \tilde{A} is the universal covering of the topological abelian group A. Hope this is helpful to you. Best wishes Jianzhong Pan