On the second homotopy group of the classifying space for commutativity in Lie groups

TL;DR

This paper investigates the second homotopy group of the classifying space for commutativity in compact Lie groups, revealing its structure and implications for the connectivity of related spaces.

Contribution

It establishes a direct summand structure for the second homotopy group of the classifying space for commutativity, linking it to the fundamental groups of the Lie group and its commutator subgroup.

Findings

Second homotopy group contains a summand isomorphic to π₁(G)⊕π₁([G,G])

If E(2,G) is 2-connected, then [G,G] is simply-connected

Connects higher connectivity of E(2,G) with that of [G,G] for compact Lie groups

Abstract

In this note we show that the second homotopy group of $B(2,G)$, the classifying space for commutativity for a compact Lie group $G$, contains a direct summand isomorphic to $\pi_1(G)\oplus\pi_1([G,G])$, where $[G,G]$ is the commutator subgroup of $G$. It follows from a similar statement for $E(2,G)$, the homotopy fiber of the canonical inclusion $B(2,G)\hookrightarrow BG$. As a consequence of our main result we obtain that if $E(2,G)$ is 2-connected, then $[G,G]$ is simply-connected. This last result completes how the higher connectivity of $E(2,G)$ resembles the higher connectivity of $[G,G]$ for a compact Lie group $G$.