Higher generation by abelian subgroups in 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 Gómez, 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.

13 pages, comments welcome!

References in corpus (7)

• A classifying space for commutativity in Lie groups
• Infinite loop spaces and nilpotent K-theory
• Colimits of abelian groups
• Classifying spaces for commutativity of low-dimensional Lie groups
• Spherical posets from commuting elements
• The spectrum for commutative complex $K$-theory
• Polyhedral products, flag complexes and monodromy representations