Mathematische Zeitschrift

, Volume 271, Issue 1, pp 33–44

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

  • José Manuel Gómez
  • Alexandra Pettet
  • Juan Souto
Article

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

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
https://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 π1(G)k.

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • José Manuel Gómez
    • 1
  • Alexandra Pettet
    • 2
    • 3
  • Juan Souto
    • 2
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Mathematical InstituteUniversity of OxfordOxfordUK