On Topological Homotopy Groups and Relation to Hawaiian Groups

TL;DR

This paper explores the properties of topological homotopy groups with a generalized topology, establishing conditions for their topological features and relating them to Hawaiian groups and specific space classes.

Contribution

It introduces a generalized whisker topology on homotopy groups, analyzes their topological properties, and connects these groups to Hawaiian groups and space classes like semilocally n-simply connected spaces.

Findings

$ ext{pi}_n^{wh}(X, x_0)$ is a topological group for $n  2$

Conditions for discreteness, Hausdorffness, and indiscreteness of $ ext{pi}_n^{wh}(X, x_0)$

Equivalence of $L_n(X,x_0)$ and Hawaiian group classes under certain conditions

Abstract

By generalizing the whisker topology on the $n$th homotopy group of pointed space $(X, x_0)$, denoted by $\pi_n^{wh}(X, x_0)$, we show that $\pi_n^{wh}(X, x_0)$ is a topological group if $n \ge 2$. Also, we present some necessary and sufficient conditions for $\pi_n^{wh}(X,x_0)$ to be discrete, Hausdorff and indiscrete. Then we prove that $L_n(X,x_0)$ the natural epimorphic image of the Hawaiian group $\mathcal{H}_n(X, x_0)$ is equal to the set of all classes of convergent sequences to the identity in $\pi_n^{wh}(X, x_0)$. As a consequence, we show that $L_n(X, x_0) \cong L_n(Y, y_0)$ if $\pi_n^{wh}(X, x_0) \cong \pi_n^{wh}(Y, y_0)$, 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 $\pi_n^{wh}$ and hence $L_n$ isomorphically along its points.