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

2021/10/25 by Villarreal, Bernardo
#22E99 (Primary) 55Q05 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2110.13109

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 π₁(G)⊕π₁([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.

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