Group cohomology

1. Feb 18, 2014

electroweak

In Dijkgraaf and Witten's paper "Topological Gauge Theory and Group Cohomology" it is claimed that...

Why are either of these statements (the Lie group case or the finite case) true?

2. Feb 18, 2014

electroweak

Note: for discrete G, BG is the Eilenberg-Maclane space K(G,1). Perhaps this will help with the finite case.

3. Feb 18, 2014

jgens

One argument that works: Notice there is a fibration G→EG→BG so using the Leray-Serre spectral sequence given information about H*(G;Q) and H*(EG;Q) one can hopefully determine something about H*(BG;Q). Since EG is contractible this gives us one piece of the puzzle and since G has the homotopy type of a finite CW-complex with some difficulty one can actually show H*(G;Q) is an exterior algebra with generators of odd degree. Using our spectral sequence it then turns out H*(BG;Q) is a polynomial algebra with generators of even degree and the desired result follows. This might be overkill, but it works at least!

Edit: I wrote the above for coefficients in Q, but the same argument should work for R. Essentially the important fact is that over Q one can ignore torsion and the same obviously holds for R.