Higher generation by abelian subgroups in Lie groups

TL;DR

This paper characterizes abelian compact Lie groups by the vanishing of certain homotopy groups of a space associated with abelian subgroups, extending a discrete group analogy to Lie groups.

Contribution

It proves that a compact Lie group is abelian if and only if specific homotopy groups of the associated space vanish, providing a new topological criterion.

Findings

G is abelian iff π_i(E(2,G))=0 for i=1,2,4

Extends discrete group poset properties to Lie groups

Provides a topological characterization of abelian Lie groups

Abstract

To a compact Lie group $G$ one can associate a space $E(2,G)$ akin to the poset of cosets of abelian subgroups of a discrete group. The space $E(2,G)$ was introduced by Adem, F. Cohen and Torres-Giese, and subsequently studied by Adem and G\'omez, and other authors. In this short note, we prove that $G$ is abelian if and only if $\pi_i(E(2,G))=0$ for $i=1,2,4$. This is a Lie group analogue of the fact that the poset of cosets of abelian subgroups of a discrete group is simply--connected if and only if the group is abelian.