Article

Mathematische Zeitschrift

, Volume 271, Issue 1, pp 33-44

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

  • José Manuel GómezAffiliated withDepartment of Mathematics, University of British Columbia
  • , Alexandra PettetAffiliated withDepartment of Mathematics, University of MichiganMathematical Institute, University of Oxford Email author 
  • , Juan SoutoAffiliated withDepartment of Mathematics, University of Michigan

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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 .