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

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 π1(G)⊕π1([G,G])\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _1(G)\oplus \pi _1([G,G])$$\end{document}, 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)↪BG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(2,G)\hookrightarrow BG$$\end{document}. 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.