On the second homotopy group of spaces of commuting elements in Lie groups

Abstract

Let $G$ be a compact connected Lie group and $n⩾1$ 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 ${\mathbb{S}}^4$.

Final version accepted for publication (open access CC-BY) in Int. Math. Res. Not. IMRN

References in corpus (3)

• Homological stability for spaces of commuting elements in Lie groups
• The symmetric power and étale realisation functors commute
• Commuting matrices and Atiyah's Real K-theory

Cited by in corpus (1)

• Torsion in the space of commuting elements in a Lie group