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

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Bernardo Villarreal ·

YOU? · · 2021 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2110.13109

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 $π_1(G)\oplusπ_1([G,G])$, 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 $G$.

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Mathematics Lie group Homotopy Commutative property Group (periodic table) Homotopy group Commutator Classifying space Combinatorics Pure mathematics Commutator subgroup Simply connected space Normal subgroup Physics Lie conformal algebra Quantum mechanics

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2110.13109
PDF: https://arxiv.org/pdf/2110.13109
OA Status: green
Related Works: 10
OpenAlex ID: https://openalex.org/W4286895308

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DOI: https://doi.org/10.48550/arxiv.2110.13109

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

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Type: preprint

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Language: en

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Publication year: 2021

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Publication date: 2021-10-25

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Authors: Bernardo Villarreal

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Landing page: https://arxiv.org/abs/2110.13109

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Concepts: Mathematics, Lie group, Homotopy, Commutative property, Group (periodic table), Homotopy group, Commutator, Classifying space, Combinatorics, Pure mathematics, Commutator subgroup, Simply connected space, Normal subgroup, Physics, Lie conformal algebra, Quantum mechanics

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