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I would like to know what has been done in terms of specific calculations for the cohomology of principal bundle. For instance, it is known (see e.g. Greub, Halperin and Vanstone's "Connection, Curvature and Cohomology") that if $G$ is a reductive Lie group, the cohomology of a principal $G$-bundle over a manifold can be computed from the cohomology of the group and the differential forms on $M$. This can be extended to any finite dimentional topological group (see e.g. Félix, Halperin and Thomas's "Rational Homotopy Theory"). But what of the other situations ? Has something been done for some infinite-dimentional groups ? For dicrete groups ? What are good and / or classical references for this kind of things ?

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    $\begingroup$ Things have been done for finite groups. I suppose work of Bredon from the 60s is a good reference. If you start googling for equivariant cohomology you´ll come across the reference immediately; everyone cites it. For compact Lie groups check out Peter May´s work, especially the notes from the Alaska conference, which are dedicated to Gaunce Lewis iirc. More recent references will probably focus on spectra more than on spaces, so maybe will be less interesting to you. If you want to go more general than compact Lie groups then I have no idea. $\endgroup$
    – David White
    Commented May 17, 2013 at 15:33
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    $\begingroup$ David, thanks for the plug, but in this case I think it is not warranted. I've done a lot about equivariant bundles, as in the Alaska notes (dedicated to Bob Piacenza by the way, Gaunce was still alive and well then) and I'm writing an expository book (with Bob Bruner and Mike Catanzaro) about classical characteristic classes, but Samuel's question is non-equivariant: his G is the structure group of the bundle, whereas in the Alaska notes we considered (G,\PI)-bundles, where G is the ambient group of equivariance and \PI is the structural group. Samuel wants something geometric, I suspect. $\endgroup$
    – Peter May
    Commented May 18, 2013 at 3:30

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