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

TL;DR

This paper investigates the second homotopy group of spaces of commuting elements in compact Lie groups, providing explicit computations and exploring their topological and algebraic structures, with applications to classifying spaces and bundle structures.

Contribution

It offers new calculations of homotopy groups for spaces of commuting elements in Lie groups, linking these to Dynkin indices and Schur multipliers, and applies results to classifying spaces and bundle examples.

Findings

  ext{Hom}(\u2124^2,G) ext{ has } \u00a7 ext{Z} ext{ as its } \u03c0_2

The quotient map induces multiplication by the Dynkin index

Constructs non-trivial structures on trivial bundles over } \u221a^4

Abstract

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