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

Algebraic Topology;K-Theory and Homology