The paper is devoted to study the structure of Hawaiian groups of some topological spaces. We present some behaviors of Hawaiian groups with respect to product spaces, weak join spaces, cone spaces, covering spaces and locally trivial bundles. In particular, we determine the structure of the n-dimensional Hawaiian group of the m-dimensional Hawaiian earring space, for all 1 ≤ m ≤ n.
By generalizing the whisker topology on the nth homotopy group of pointed space (X, x 0), denoted by π wh n (X, x 0), we show that π wh n (X, x 0) is a topological group if n ≥ 2. Also, we present some necessary and sufficient conditions for π wh n (X, x 0) to be discrete, Hausdorff and indiscrete. Then we prove that L n (X, x 0) the natural epimorphic image of the Hawaiian group H n (X, x 0) is equal to the set of all classes of convergent sequences to the identity in π wh n (X, x 0). As a consequence, we show that L n (X, x 0) ∼ = L n (Y, y 0) if π wh n (X, x 0) ∼ = π wh n (Y, y 0), but the converse does not hold in general, except for some conditions. Also, we show that on some classes of spaces such as semilocally n-simply connected spaces and n-Hawaiian like spaces, the whisker topology and the topology induced by the compact-open topology of n-loop space coincide. Finally, we show that n-SLT paths can transfer π wh n and hence L n isomorphically along its points.
In this paper, we study some properties of homotopical closeness for paths. We define the quasi-small loop group as the subgroup of all classes of loops that are homotopically close to null-homotopic loops, denoted by π 1 q s ( X , x ) $\pi_1^{qs} (X, x)$ for a pointed space (X, x). Then we prove that, unlike the small loop group, the quasi-small loop group π 1 q s ( X , x ) $\pi_1^{qs}(X, x)$ does not depend on the base point, and that it is a normal subgroup containing π 1 s g ( X , x ) $\pi_1^{sg}(X, x)$ , the small generated subgroup of the fundamental group. Also, we show that a space X is homotopically path Hausdorff if and only if π 1 q s ( X , x ) $\pi_1^{qs} (X, x)$ is trivial. Finally, as consequences, we give some relationships between the quasi-small loop group and the quasi-topological fundamental group.
No abstract
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.