Classifying spaces for commutativity of low-dimensional Lie groups

TL;DR

This paper computes the cohomology rings, Steenrod algebra action, homotopy type, and low-dimensional homotopy groups of classifying spaces for commutativity of certain low-dimensional Lie groups, advancing understanding of their topological structures.

Contribution

It provides explicit calculations of cohomology rings and homotopy types for the classifying spaces for commutativity of O(2), SU(2), and U(2), which were previously not fully understood.

Findings

Computed integral and mod 2 cohomology rings of B_com G for G=O(2), SU(2), U(2)

Determined the Steenrod algebra action on these cohomology rings

Identified the homotopy type of E_com G and some low-dimensional homotopy groups

Abstract

For each of the groups $G = O(2), SU(2), U(2)$, we compute the integral and $\mathbb{F}_2$-cohomology rings of $B_\text{com} G$ (the classifying space for commutativity of $G$), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of $E_\text{com} G$ (the homotopy fiber of the inclusion $B_\text{com} G \to BG$), and some low-dimensional homotopy groups of $B_\text{com} G$.