Samelson products in p-regular SO(2n) and its homotopy normality

TL;DR

This paper investigates the Samelson products in p-regular SO(2n) groups, determining their triviality or nontriviality, and applies these results to analyze the homotopy normality of certain group inclusions.

Contribution

It completes the classification of Samelson product triviality in p-regular simple Lie groups and explores the homotopy normality of SO(2n-1) into SO(2n).

Findings

Determined triviality/nontriviality of Samelson products in p-regular SO(2n).

Established homotopy normality of SO(2n-1) into SO(2n) at any prime p.

Completed the list of Samelson product behaviors in p-regular simple Lie groups.

Abstract

A Lie group is called $p$-regular if it has the $p$-local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into $p$-regular $\mathrm{SO}(2n)_{(p)}$ is determined, which completes the list of (non)triviality of such Samelson products in $p$-regular simple Lie groups. As an application, we determine the homotopy normality of the inclusion $\mathrm{SO}(2n-1)\to\mathrm{SO}(2n)$ in the sense of James at any prime $p$.