Torsion in the space of commuting elements in a Lie group

Abstract

Let Hom(Z m , G) denote the space of commuting m-tuples in a Lie group G.In this paper, we study torsion in the homology of Hom(Z m , G). We describe Hom(Z m , G) as a homotopy colimit of a functor Fm defined by homogeneous spaces of G, and then by analyzing the functor Fm, we develop a method for detecting torsion in the homology of Hom(Z m , G) in terms of the extended Dynkin diagram of G. Using this method, we prove that for m ≥ 2, Hom(Z m , SU(n)) has p torsion in homology if and only if p ≤ n. We also compute some torsion in the homology of Hom(Z m , G) for G = Spin(2n) or exceptional. As an another application of the method, we determine the top homology of Hom(Z m , G) for any compact simply-connected simple Lie group G.