Perhaps this question is too general then I am sorry about this.

My question is the following.

Let $\pi$ be the fundamental group of a compact surface of genus $g$ (with if necessary $n$ punctures) and $G$ a Lie group. I make no assumptions on the Lie group but if it interesting to do so then it is ok.

Are there any results known about the cohomology of the representation variety $Hom(\pi,G)$ if it is not a manifold.

So I know the calculations for $Hom(\pi,G)/G$ by Atiyah and Bott where it has a smooth structure. But here I am only interested in the cohomology of the topological space that is in more general results.

  • 3
    $\begingroup$ If G is finite (and hence a Lie group of dimension 0), then the representation variety is discrete and its cardinality is determined by the degrees of irreps of G. arxiv.org/abs/1102.4353 This paper also handles compact G, but the results are in a different direction. $\endgroup$ Oct 10 '11 at 15:48
  • $\begingroup$ What do you mean about Hom$(\pi, G)/G$ having a smooth structure? And which work of Atiyah and Bott are you referring to? Their big paper about Yang-Mills theory? I've spent a lot of time with that paper and I don't really recall seeing anything explicit about the moduli space of representations. But maybe you mean the homotopy quotient (or a GIT quotient). That is certainly studied in Atiyah-Bott, but again I don't know what "smooth" would mean. $\endgroup$
    – Dan Ramras
    Oct 10 '11 at 23:46

It think it is difficult to say anything in general, but I would suggest starting by looking at the work of W. Goldman to get a feel for the subject.

For instance, even computing $H^0(\mathrm{Hom}(\pi,G); \mathbb{Z})$ is nontrivial when $G$ is nice. In his thesis, Goldman showed that $H^0(\mathrm{Hom}(\pi,\mathrm{PSL}_2(\mathbb{R})); \mathbb{Z}) \cong \mathbb{Z}^{4g-3}$, where there is a copy of $\mathbb{Z}$ for each possible euler class. See Goldman, Topological components of spaces of representations. Invent. Math. 93 (1988), no. 3, 557–607.


If $G=U(n)$ then I know a fair amount. Letting $n$ tend to infinity, Hom$(\pi_1 M^g, U)$ is homotopy equivalent to $U^{2g} \times BU$, so stably one can write down the cohomology using standard facts about the infinite unitary group. There is also a stability range for the inclusions Hom$(\pi_1 M^g, U(n))\to$ Hom$(\pi_1 M^g, U(n+1))$ (they are $(2n-2)$-connected maps). Roughly this range is the stability range for the unitary groups minus 2, if I remember correctly.

The way to prove these things is to follow Atiyah-Bott and think about the representation space as the space of flat connections modulo the based gauge group. If this is the sort of information you're looking for, and you'd like to know more details, I can say more. I think I wrote some notes a few years ago that go through this carefully. But since you mentioned Atiyah-Bott already, you may be looking in a different direction.

If you puncture the surface, the fundamental group becomes free. Then you might be interested in Tyler Lawson's paper about simultaneous similarity of unitary matrices (Math. Proc. Camb. Phil. Soc. 2008 or http://arxiv.org/abs/0809.0466 ). Or you might be interested in keeping track of some additional structure related to the punctures, in which case some of Tom Baird's work may be of interest to you.


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