Topological components of spaces of commuting elements in connected nilpotent Lie groups

TL;DR

This paper investigates the topological structure of spaces of commuting elements in connected nilpotent Lie groups, identifying conditions for path-connectedness and describing homotopy types of components.

Contribution

It provides a necessary and sufficient condition on the fundamental group for the path-connectedness of these spaces and characterizes their homotopy types for specific groups.

Findings

Hom(a0Z^k,G) is path-connected under certain fundamental group conditions.

Hom(a0Z^k,G) is not path-connected for reduced upper unitriangular and generalized Heisenberg groups.

Hom(a0Z^k,G) components' homotopy types relate to Stiefel manifolds and maximal tori.

Abstract

We study the homotopy type of spaces of commuting elements in connected nilpotent Lie groups, via almost commuting elements in their Lie algebras. We give a necessary and sufficient condition on the fundamental group of such a Lie group $G$ to ensure $\mathrm{Hom}(\mathbb{Z}^k,G)$ is path-connected. In particular for the reduced upper unitriangular groups and the reduced generalized Heisenberg groups, $\mathrm{Hom}(\mathbb{Z}^k,G)$ is not path-connected, and we compute the homotopy type of its path-connected components in terms of Stiefel manifolds and the maximal torus of $G$.