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

TL;DR

This paper explicitly computes the K-theory and cohomology rings of the space of commuting elements in SU(2), revealing differences in their behavior with various coefficients and implications for representation stability.

Contribution

It provides explicit calculations of the K-theory and integral cohomology rings of the commuting elements space in SU(2), using desingularization techniques.

Findings

Explicit K-theory and cohomology rings computed.

Differences observed between complex and Z_2 cohomology.

Insights into representation stability and FI-modules.

Abstract

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.