Let $\mathrm{Hom}(\mathbb{Z}^m,G)$ denote the space of commuting $m$-tuples in a Lie group $G$. This space is identified with the based moduli space of flat bundles over a torus, so it is an important object not only in topology but also in geometry and physics. I will talk about torsion in the homology of $\mathrm{Hom}(\mathbb{Z}^m,G)$. We prove that for $m\geq 2$, $\mathrm{Hom}(\mathbb{Z}^m,SU(n))$ has $p$-torsion in homology if and only if $p\leq n$. The proof includes a new homotopy decomposition of $\mathrm{Hom}(\mathbb{Z}^m,G)$ in terms of a homotopy colimit. This talk is based on the joint work with Daisuke Kishimoto.

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