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

Chi-Kwong Fok

TL;DR

This work computes the integral $K$-theory and cohomology rings of the space of commuting elements in $SU(2)$ by analyzing a desingularization modeled as a blowup $G/T\times_{\Gamma} T^n$. It provides an explicit presentation of the $K$-theory ring $K^*(\operatorname{Hom}(\mathbb{Z}^n, SU(2)))$ via generators like $x_{ij}, w_i, u, v_i$ and their relations, and proves that the integral Chern character yields a ring isomorphism with the cohomology ring. The paper also determines the full additive structure of $K^*(\operatorname{Hom}(\mathbb{Z}^n, SU(2)))$ and $H^*(\operatorname{Hom}(\mathbb{Z}^n, SU(2)); \mathbb{Z})$, aligning with known results in special cases and establishing an explicit FI-module framework. Moreover, it reveals a dichotomy: cohomology with complex coefficients exhibits representation stability as an FI-module, while cohomology with $\mathbb{Z}_2$-coefficients does not, illustrating subtle torsion phenomena in these moduli spaces.

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.

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

TL;DR

This work computes the integral -theory and cohomology rings of the space of commuting elements in by analyzing a desingularization modeled as a blowup . It provides an explicit presentation of the -theory ring via generators like and their relations, and proves that the integral Chern character yields a ring isomorphism with the cohomology ring. The paper also determines the full additive structure of and , aligning with known results in special cases and establishing an explicit FI-module framework. Moreover, it reveals a dichotomy: cohomology with complex coefficients exhibits representation stability as an FI-module, while cohomology with -coefficients does not, illustrating subtle torsion phenomena in these moduli spaces.

Abstract

In this paper, we compute explicitly both the -theory and integral cohomology rings of the space of commuting elements in via the -theory of its desingularization. We also briefly discuss the different behavior of its cohomology with complex and coefficients in the context of representation stability and FI-modules.
Paper Structure (5 sections, 15 theorems, 72 equations)

This paper contains 5 sections, 15 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. The $K$-theory of the blowup
  3. The cohomology of the blowup
  4. The $K$-theory and cohomology of $\text{Hom}(\mathbb{Z}^n, SU(2))$
  5. Cohomology of $\text{Hom}(\mathbb{Z}^n, SU(2))$ as an FI-module

Key Result

Proposition 2.2

The integral cohomology groups of $G/T\times_\Gamma T^n$ are given by The odd $K$-theory group $K^{-1}(G/T\times_\Gamma T^n)$ is isomorphic to $\mathbb{Z}^{2^{n-1}}$, while the even $K$-theory group $K^0(G/T\times_\Gamma T^n)$ is isomorphic to $\mathbb{Z}^{2^{n-1}}\oplus M_n$ where $M_n$ is a finite abelian group of order $2^{2^n}$.

Theorems & Definitions (45)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Corollary 2.7
  • proof
  • ...and 35 more