Higher homotopy commutativity in localized Lie groups and gauge groups

TL;DR

This paper investigates the p-local higher homotopy commutativity of Lie groups and gauge groups, applying Sugawara's framework, and resolves a long-standing non-commutativity problem for G2 at p=5.

Contribution

It introduces the study of Sugawara's higher homotopy commutativity for Lie groups and applies it to gauge groups, also solving the G2 p=5 case.

Findings

Established p-local higher homotopy commutativity for certain Lie groups

Applied results to gauge groups in the same context

Resolved the G2 p=5 non-commutativity problem

Abstract

The first aim of this paper is to study the $p$-local higher homotopy commutativity of Lie groups in the sense of Sugawara. The second aim is to apply this result to the $p$-local higher homotopy commutativity of gauge groups. Although the higher homotopy commutativity of Lie groups in the sense of Williams is already known, the higher homotopy commutativity in the sense of Sugawara is necessary for this application. The third aim is to resolve the $5$-local higher homotopy non-commutativity problem of the exceptional Lie group $\mathrm{G}_2$, which has been open for a long time.