On Topological Homotopy Groups and Relation to Hawaiian Groups
| Autor: | Ameneh Babaee, Hanieh Mirebrahimi, Behrooz Mashayekhy, Mahdi Abdullahi Rashid, Hamid Torabi, Seyyed Zeynal Pashaei |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
Statistics and Probability
55Q05 55Q20 55P65 55Q52 Matematik Homotopy group Algebra and Number Theory Group (mathematics) Image (category theory) Hausdorff space Whisker topology Hawaiian group Harmonic archipelago n-dimensional Hawaiian earring Space (mathematics) Topology Transfer (group theory) FOS: Mathematics Algebraic Topology (math.AT) Mathematics - Algebraic Topology Geometry and Topology Topological group Pointed space Mathematics Analysis |
| Zdroj: | Volume: 49, Issue: 4 1437-1449 Hacettepe Journal of Mathematics and Statistics |
| ISSN: | 2651-477X |
| DOI: | 10.48550/arxiv.2209.07195 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|