Torsion in the space of commuting elements in a Lie group

TL;DR

This paper investigates the presence of torsion in the homology of spaces of commuting elements in compact Lie groups, revealing connections to Weyl group order and providing new homotopy decompositions.

Contribution

It establishes criteria for p-torsion in homology of commuting element spaces and introduces a novel homotopy decomposition method.

Findings

p-torsion occurs iff p divides Weyl group order for certain groups

Always 2-torsion in homology for simply-connected simple groups

Computed top homology of the commuting element space

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.