On topological homotopy groups and relation to Hawaiian groups

Year 2020, Volume: 49 Issue: 4, 1437 - 1449, 06.08.2020

Ameneh Babaee Behrooz Mashayekhy Hanieh Mirebrahimi Hamid Torabi Mahdi Abdullahi Rashid Seyyed Zeynal Pashaei

https://doi.org/10.15672/hujms.565367

Cited By: 2

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.

Keywords

Whisker topology, Hawaiian group, Harmonic archipelago, n-dimensional Hawaiian earring

References

• [1] M. Abdullahi Rashid, N. Jamali, B. Mashayekhy, S.Z. Pashaei and H. Torabi, On subgroup topologies on fundamental groups, doi:10.15672/hujms.464056.
• [2] A. Arhangegel’skii and M. Tkachenko, Topological Groups and Related Structures, Atlantis Press, Amsterdam, 2008.
• [3] A. Babaee, B. Mashayekhy and H. Mirebrahimi, On Hawaiian groups of some topological spaces, Topology Appl. 159 (8), 2043–2051, 2012.
• [4] W.A. Bogley and A.J. Sieradski, Universal path spaces, http://people. oregonstate.edu/~bogleyw/research/ups.pdf.
• [5] N. Brodskiy, J. Dydak, B. Labuz and A. Mitra, Covering maps for locally path connected spaces, Fund. Math. 218, 13–46, 2012.
• [6] N. Brodskiy, J. Dydak, B. Labuz and A. Mitra, Topological and uniform structures on universal covering spaces, arXiv:1206.0071, 2012.
• [7] G.R. Conner and J. Lamoreaux, On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane, Fund. Math. 187, 95–110, 2005.
• [8] G.R. Conner, W. Hojka and M. Meilstrup, Archipelago groups, Proc. Amer. Math. Soc. 143, 4973–4988, 2015.
• [9] K. Eda and K. Kawamura, Homotopy and homology groups of the n-dimensional Hawaiian Earring, Fund. Math. 165 (1), 17–28, 2000.
• [10] P. Fabel, Multiplication is discontinuous in the Hawaiian Earring droup (with the Quotient Topology), Bull. Pol. Acad. Sci. Math. 59 (1), 77–83, 2011.
• [11] H. Fischer and A. Zastrow, Generalized universal coverings and the shape group, Fund. Math. 197, 167–196, 2007.
• [12] L. Fuchs, Infinite Abelian Groups I, Academic Press, New York, 1970.
• [13] F.H. Ghane, Z. Hamed, B. Mashayekhy, and H. Mirebrahimi, Topological homotopy groups, Bull. Belg. Math. Soc. Simon Stevin, 15 (3), 455–464, 2008.
• [14] F.H. Ghane, Z. Hamed, B. Mashayekhy and H. Mirebrahimi, On topological homotopy groups of n-Hawaiian like spaces, Topology Proc. 36, 255–266, 2010.
• [15] Herfort and Hojka, Cotorsion and wild homology, Israel J. Math. 221, 275–290, 2017.
• [16] N. Jamali, B. Mashayekhy, H. Torabi, S.Z. Pashaei and M. Abdullahi Rashid, On topologized fundamental groups with small loop transfer viewpoints, Acta Math. Vietnamica, 43, 1–27, 2018.
• [17] U.H. Karimov and D. Repovš, Hawaiian groups of topological spaces (Russian), Uspekhi. Mat. Nauk. 61 (5), 185–186, 2006; transl. in Russian Math. Surv. 61 (5), 987–989, 2006.
• [18] U.H. Karimov and D. Repovš, On the homology of the Harmonic archipelago, Central European J. Math. 10, 863–872, 2012.
• [19] S.Z. Pashaei, B. Mashayekhy, H. Torabi and M. Abdullahi Rashid, Small loop transfer spaces with respect to subgroups of fundamental groups, Topology Appl. 232, 242–255, 2017.
• [20] H. Passandideh, F.H. Ghane and Z. Hamed, On the homotopy groups of separable metric spaces, Topology Appl. 158, 1607–1614, 2011.
• [21] E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
• [22] B. Zimmermann-Huisgen, On Fuchs’ problem 76, J. Reine Angew. Math. 309, 86–91, 1979.