Higher homotopy normalities in topological groups

TL;DR

This paper introduces the concept of $N_k(\ell)$-maps to describe higher homotopy normalities in topological groups, exploring their properties, examples, and conditions under which certain inclusions are $p$-locally $N_k(\ell)$-maps.

Contribution

It defines $N_k(\ell)$-maps with higher homotopical conditions and studies their properties, providing new insights into homotopy normalities in topological groups.

Findings

Homomorphisms are $N_k(\ell)$-maps iff fiberwise maps between projective spaces exist.

Homotopy quotients of $N_k(k)$-maps are $H$-spaces if LS category ≤ $k$.

Certain inclusions of special unitary and orthogonal groups are $p$-locally $N_k(\ell)$-maps.

Abstract

The purpose of this paper is to introduce $N_k(\ell)$-maps ($1\le k,\ell\le\infty$), which describe higher homotopy normalities, and to study their basic properties and examples. An $N_k(\ell)$-map is defined with higher homotopical conditions. It is shown that a homomorphism is an $N_k(\ell)$-map if and only if there exists fiberwise maps between fiberwise projective spaces with some properties. Also, the homotopy quotient of an $N_k(k)$-map is shown to be an $H$-space if its LS category is not greater than $k$. As an application, we investigate when the inclusions $\operatorname{SU}(m)\to\operatorname{SU}(n)$ and $\operatorname{SO}(2m+1)\to\operatorname{SO}(2n+1)$ are $p$-locally $N_k(\ell)$-maps.