Mathematische Zeitschrift

, Volume 271, Issue 1, pp 33–44

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


  • José Manuel Gómez
    • Department of MathematicsUniversity of British Columbia
    • Department of MathematicsUniversity of Michigan
    • Mathematical InstituteUniversity of Oxford
  • Juan Souto
    • Department of MathematicsUniversity of Michigan

DOI: 10.1007/s00209-011-0850-6

Cite this article as:
Gómez, J.M., Pettet, A. & Souto, J. Math. Z. (2012) 271: 33. doi:10.1007/s00209-011-0850-6


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 π1(G)k.
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© Springer-Verlag 2011