Torsion in the space of commuting elements in a Lie group

Abstract

Let $G$ be a compact connected Lie group, and let $\mathrm{Hom}(\mathbb{Z}^m,G)$ be the space of pairwise commuting $m$-tuples in $G$. We study the problem of which primes $p$ $\mathrm{Hom}(\mathbb{Z}^m,G)_1$, the connected component of $\mathrm{Hom}(\mathbb{Z}^m,G)$ containing the element $(1,\ldots,1)$, has $p$-torsion in homology. We will prove that $\mathrm{Hom}(\mathbb{Z}^m,G)_1$ for $m\ge 2$ has $p$-torsion in homology if and only if $p$ divides the order of the Weyl group of $G$ for $G=SU(n)$ and some exceptional groups. We will also compute the top homology of $\mathrm{Hom}(\mathbb{Z}^m,G)_1$ and show that $\mathrm{Hom}(\mathbb{Z}^m,G)_1$ always has 2-torsion in homology whenever $G$ is simply-connected and simple. Our computation is based on a new homotopy decomposition of $\mathrm{Hom}(\mathbb{Z}^m,G)_1$, which is of independent interest and enables us to connect torsion in homology to the combinatorics of the Weyl group.