Group Cohomology: Borel's Finite & Lie Group Cases

[electroweak]

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

For a compact Lie group we have the very useful property, due to Borel, that with real coefficients all odd cohomology vanishes: H^odd(BG; R) = 0. So the odd cohomology (and homology) consists completely of torsion. For finite groups an even stronger result holds: all cohomology is finite: H(BG; R) = 0.

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

 

[electroweak]

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

 

[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.