Hacettepe Journal of Mathematics and Statistics 2651-477X 2651-477X Hacettepe University 10.15672/hujms.565367 Mathematical Sciences Matematik On topological homotopy groups and relation to Hawaiian groups https://orcid.org/0000-0001-5243-0641 Babaee Ameneh Ferdowsi University of Mashhad https://orcid.org/0000-0001-5243-0641 Mashayekhy Behrooz Ferdowsi University of Mashhad https://orcid.org/0000-0001-5243-0641 Mirebrahimi Hanieh Ferdowsi University of Mashhad https://orcid.org/0000-0001-5243-0641 Torabi Hamid Ferdowsi University of Mashhad https://orcid.org/0000-0001-5243-0641 Abdullahi Rashid Mahdi Ferdowsi University of Mashhad https://orcid.org/0000-0001-5243-0641 Pashaei Seyyed Zeynal Ferdowsi University of Mashhad 08 06 2020 49 4 1437 1449 05 14 2019 10 30 2019 Copyright © 2002, Hacettepe Journal of Mathematics and Statistics 2002 Hacettepe Journal of Mathematics and Statistics

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.

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