The space of commuting n-tuples in SU(2)

Abstract

Let $Y := \operatorname{Hom}(\mathbb{Z}^n, \operatorname{SU}(2))$ denote the space of commuting $n$-tuples in $\operatorname{SU}(2)$. We determine the homotopy type of the suspension $\Sigma Y$, and compute the integral cohomology groups of $Y$ for all positive integers $n$.