Homology, Homotopy and Applications

Volume 21 (2019)

Number 1

Higher homotopy commutativity in localized Lie groups and gauge groups

Pages: 107 – 128

DOI: https://dx.doi.org/10.4310/HHA.2019.v21.n1.a6

Authors

Sho Hasui (Institute of Mathematics, University of Tsukuba, Ibaraki, Japan)

Daisuke Kishimoto (Department of Mathematics, Kyoto University, Kyoto, Japan)

Mitsunobu Tsutaya (Faculty of Mathematics, Kyushu University, Fukuoka, Japan)

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.

Keywords

homotopy commutativity, Lie group, gauge group, $A_n$-space

2010 Mathematics Subject Classification

55P35

Received 6 March 2018

Received revised 13 June 2018

Accepted 1 June 2018

Published 29 August 2018