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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 49 Issue: 4

On topological homotopy groups and relation to Hawaiian groups

Ameneh Babaee(Ferdowsi University of Mashhad)0000-0001-5243-0641
Behrooz Mashayekhy(Ferdowsi University of Mashhad)0000-0001-5243-0641
Hanieh Mirebrahimi(Ferdowsi University of Mashhad)0000-0001-5243-0641
Hamid Torabi(Ferdowsi University of Mashhad)0000-0001-5243-0641
Mahdi Abdullahi Rashid(Ferdowsi University of Mashhad)0000-0001-5243-0641
Seyyed Zeynal Pashaei(Ferdowsi University of Mashhad)0000-0001-5243-0641
Pages
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Publication Date
2020-08-06
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Finally, we show that $n$-SLT paths can transfer $\pi_n^{wh}$ and hence $L_n$ isomorphically along its points.

📌 3 Key Takeaways

🎯
Objective & Hypothesis: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 $n \ge 2$.
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Key Findings & Data:Also, we present some necessary and sufficient conditions for $\pi_n^{wh}(X,x_0)$ to be discrete, Hausdorff and indiscrete.
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Conclusion & Impact:Finally, we show that $n$-SLT paths can transfer $\pi_n^{wh}$ and hence $L_n$ isomorphically along its points.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Whisker topology · Hawaiian group · Harmonic archipelago

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.

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Cite This Article

Babaee, A., Mashayekhy, B., Mirebrahimi, H., Torabi, H., Rashid, M. A., Pashaei, S. Z. (2020). On topological homotopy groups and relation to Hawaiian groups. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 4. https://doi.org/10.15672/hujms.565367

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