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• W3087367966 abstract "Let $G$ be a compact connected Lie group and $ngeqslant 1$ an integer. Consider the space of ordered commuting $n$-tuples in $G$, $Hom(mathbb{Z}^n,G)$, and its quotient under the adjoint action, $Rep(mathbb{Z}^n,G):=Hom(mathbb{Z}^n,G)/G$. In this article we study and in many cases compute the homotopy groups $pi_2(Hom(mathbb{Z}^n,G))$. For $G$ simply--connected and simple we show that $pi_2(Hom(mathbb{Z}^2,G))cong mathbb{Z}$ and $pi_2(Rep(mathbb{Z}^2,G))cong mathbb{Z}$, and that on these groups the quotient map $Hom(mathbb{Z}^2,G)to Rep(mathbb{Z}^2,G)$ induces multiplication by the Dynkin index of $G$. More generally we show that if $G$ is simple and $Hom(mathbb{Z}^2,G)_{1}subseteq Hom(mathbb{Z}^2,G)$ is the path--component of the trivial homomorphism, then $H_2(Hom(mathbb{Z}^2,G)_{1};mathbb{Z})$ is an extension of the Schur multiplier of $pi_1(G)^2$ by $mathbb{Z}$. We apply our computations to prove that if $B_{com}G_{1}$ is the classifying space for commutativity at the identity component, then $pi_4(B_{com}G_{1})cong mathbb{Z}oplus mathbb{Z}$, and we construct examples of non-trivial transitionally commutative structures on the trivial principal $G$-bundle over the sphere $mathbb{S}^{4}$." @default.
• W3087367966 created "2020-09-25" @default.
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• W3087367966 date "2020-09-18" @default.
• W3087367966 modified "2023-09-23" @default.
• W3087367966 title "On the second homotopy group of spaces of commuting elements in Lie groups" @default.
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