On Topological Homotopy Groups and Relation to Hawaiian Groups

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.

References in corpus (1)

• On the homology of the Harmonic Archipelago