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In this paper, we compute explicitly both the $K$-theory and integralcohomology rings of the space of commuting elements in $SU(2)$ via the$K$-theory of its desingularization. We also briefly discuss the differentbehavior of its cohomology with complex and $\mathbb{Z}_2$ coefficients in thecontext of representation stability and FI-modules.

Cohomology and $K$-theory rings of the space of commuting elements in $SU(2)$