Higher homotopy normalities in topological groups

Abstract

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