, 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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Abstract

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

http://static-content.springer.com/image/art%3A10.1007%2Fs00209-011-0850-6/MediaObjects/209_2011_850_Figa_HTML.gif
. 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 .