The purpose of this paper is to introduce $N_k(\ell )$-maps ($\text{◂,▸}1⩽k,\ell ⩽\infty$), 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 $\text{◂→▸}SU(m)\to SU(n)$ and $\text{◂→▸}SO(2m+1)\to SO(2n+1)$ are $p$-locally $N_k(\ell )$-maps.

中文翻译：

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拓扑群中的高同伦正态性

本文的目的是介绍${否}_k(\ell )$-地图（$\text{◂,▸}\mathrm{1个}⩽k,\ell ⩽\infty$)，它描述了更高的同伦常态，并研究了它们的基本性质和例子。一个${否}_k(\ell )$-map 是用更高的同伦条件定义的。证明同态是${否}_k(\ell )$-map 当且仅当在具有某些属性的纤维射影空间之间存在纤维映射。此外，a 的同伦商${否}_k(k)$-map 显示为$H$-space 如果它的 LS 类别不大于$k$. 作为一个应用程序，我们调查夹杂物何时$\text{◂→▸}苏(米)\to 苏(n)$和$\text{◂→▸}所以(\mathrm{2个}米+\mathrm{1个})\to 所以(\mathrm{2个}n+\mathrm{1个})$是$p$-本地${否}_k(\ell )$-地图。