Higher homotopy normalities in topological groups

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 $\mathrm{SU}(m)\to \mathrm{SU}(n)$ and $\mathrm{SO}(2m+1)\to \mathrm{SO}(2n+1)$ are $p$-locally $N_k(\ell )$-maps.

22 pages, 4 figures

References in corpus (5)

• Finiteness of $A_n$-equivalence types of gauge groups
• Samelson products in p-regular SO(2n) and its homotopy normality
• Homotopy Normal Maps
• Mapping spaces from projective spaces
• Higher homotopy commutativity in localized Lie groups and gauge groups