Higher homotopy normalities in topological groups

Pub Date : 2023-02-17 DOI:10.1112/topo.12282

Mitsunobu Tsutaya

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Abstract

The purpose of this paper is to introduce $N_{k} (ℓ)$ -maps ( $1 ⩽ k, ℓ ⩽ \infty$ ), which describe higher homotopy normalities, and to study their basic properties and examples. An $N_{k} (ℓ)$ -map is defined with higher homotopical conditions. It is shown that a homomorphism is an $N_{k} (ℓ)$ -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 $SU (m) \to SU (n)$ and $SO (2 m + 1) \to SO (2 n + 1)$ are $p$ -locally $N_{k} (ℓ)$ -maps.

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拓扑群中的高同伦规范性

本文的目的是介绍Nk(ℓ)$N_ k（\ell）$-maps（1⩽k，ℓ⩽∞$1\leqslant k，\ell\leqslant\infty$），并研究了它们的基本性质和例子。An Nk(ℓ)$N_k（\ell）$-map是用更高的同位条件定义的。证明了同态是Nk(ℓ)$N_k（\ell）$-map当且仅当在具有某些性质的纤维状投影空间之间存在纤维状映射。此外，如果Nk（k）$N_k（k）$映射的LS范畴不大于k$k$，则其同伦商被证明是H$H$空间。作为一个应用，我们研究了当夹杂物SU（m）→SU（n）$\运算符名称｛SU｝（m）\rightarrow\运算符名称{SU}（n）$和SO（2m+1）→SO（2n+1）$\运算符名称｛SO｝（2m+1）\rightarrow\运算符名称{SO｝(ℓ)$N_k（\ell）$映射。

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