On spaces of commuting elements in Lie groups

Abstract

The main purpose of this paper is to introduce a method to stabilize certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group $G$, namely $Hom({\mathbb{Z}}^n,G)$. We show that this stabilized space of homomorphisms decomposes after suspending once with summands which can be reassembled, in a sense to be made precise below, into the individual spaces $Hom({\mathbb{Z}}^n,G)$ after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilized space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilized space also allows the description of the additive reduced homology of the individual spaces $Hom({\mathbb{Z}}^n,G)$, with the order of the Weyl group inverted.

27 pages, with an appendix by Vic Reiner

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• On the fundamental group of Hom(Z^k,G)
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Cited by in corpus (8)

• A survey on spaces of homomorphisms to Lie groups
• Poincare series of character varieties for nilpotent groups
• Homological stability for spaces of commuting elements in Lie groups
• Hilbert-Poincare series for spaces of commuting elements in Lie groups
• Cosimplicial Groups and Spaces of Homomorphisms
• Differentiable stratified groupoids and a de Rham theorem for inertia spaces
• Commuting matrices and Atiyah's Real K-theory
• Cohomology of the spaces of commuting elements in Lie groups of rank two