, Volume 271, Issue 1-2, pp 33-44
Date: 18 Mar 2011

On the fundamental group of \({{\rm Hom}({\mathbb Z}^k,G)}\)

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Let G be a compact Lie group. Consider the variety \({{\rm Hom}({\mathbb Z}^{k},G)}\) of representations of \({{\mathbb Z}^k}\) into G. We can see this as a based space by taking as base point the trivial representation . The goal of this paper is to prove that \({\pi_1({\rm Hom}({\mathbb Z}^k,G))}\) is naturally isomorphic to π 1(G) k .

The second author has been partially supported by NSF Grant DMS-0856143 and NSF RTG Grant DMS-0602191. The third author has been partially supported by the NSF Grant DMS-0706878, NSF Career Award 0952106 and the Alfred P. Sloan Foundation.