On Hawaiian homology groups

Year 2024, Volume: 53 Issue: 6, 1607 - 1626, 28.12.2024

Professor Dr. , Hanıeh Mırebrahımı , Ameneh Babaee

https://doi.org/10.15672/hujms.1298585

Abstract

In this paper, we introduce a kind of homology which we call Hawaiian homology to study and classify pointed topological spaces. The Hawaiian homology group has advantages over Hawaiian groups. Moreover, the first Hawaiian homology group is isomorphic to the abelianization of the first Hawaiian group for path-connected and locally path-connected topological spaces. Since Hawaiian homology has concrete elements and abelian structure, its calculation is easier than that of the Hawaiian group. Thus we use Hawaiian homology groups to compare Hawaiian groups, and then we obtain some information about Hawaiian groups of some wild topological spaces.

Keywords

Hawaiian group, singular homology, archipelago space, Higmann-complete group, cotorsion group

Supporting Institution

Ferdowsi University of Mashhad

Project Number

2/57128

References

• [1] J.F. Adamas, Stable homotopy and generalised homology, The university of Chicago press, Chicago, 1974.
• [2] A. Babaee, B. Mashayekhy and H. Mirebrahimi, On Hawaiian groups of some topological spaces, Topology Appl. 159 (8), 2043–2051, 2012.
• [3] A. Babaee, B. Mashayekhy, H. Mirebrahimi and H. Torabi, On a van Kampen Theorem for Hawaiian groups, Topology Appl. 241, 252–262, 2018.
• [4] A. Babaee, B. Mashayekhy, H. Mirebrahimi, H. Torabi, M. Abdullahi Rashid and S.Z. Pashaei, On topological homotopy groups and its relation to Hawaiian groups, Hacettepe Journal of Mathematics, 49 (4), 1437–1449, 2020.
• [5] S. Balcerzyk, On factor groups of some subgroups of a complete direct sum of infinite cyclic groups, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 7, 141–142, 1959.
• [6] M.G. Barratt and J. Milnor, An example of anomalous singular homology, Proceedings of the American Mathematical Society, 13 (2), 293–297, 1962.
• [7] G.R. Conner, W. Hojka and M. Meilstrup, Archipelago groups, Proc. Amer. Math. Soc. 143, 4973–4988, 2015.
• [8] K. Eda, The first integral singular homology groups of one point unions, Quart. J. Math. Oxford Ser. 42(1), 443–456, 1991.
• [9] K. Eda, Free $\sigma$ products and noncommutatively slender groups, Journal of Algebra, 148, 243–263, 1992.
• [10] K. Eda and K. Kawamura, Homotopy and homology groups of the n-dimensional Hawaiian Earring, Fundamenta Mathematicae, 165 (1), 17–28, 2000.
• [11] L. Fuchs, Infinite Abelian Groups I, Academic Press, New York, 1970.
• [12] L. Fuchs, Abelian Groups, Springer-Varlage, New York, 2015.
• [13] R. Ghrist, Three examples of applied and computational homology, Nieuw Arch. Wiskd. 9(2), 122–125, 2008.
• [14] W. Herfort and W. Hojka, Cotorsion and wild homology, Israel J. Math. 221, 275-290, 2017.
• [15] 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.
• [16] U.H. Karimov and D. Repovš, On the homology of the Harmonic archipelago, Central European Journal of Mathematics, 10, 863–872, 2012.
• [17] B. Mashayekhy, H. Mirebrahimi, H. Torabi and A. Babaee, On small n-Hawaiian loops, Mediterranean Journal of Mathematics, 17, 202, 2020.
• [18] W.S. Massey, Singular Homology Theory, Springer-Verlag, New York, 1980.
• [19] J.R. Munkres, Elements of Algebraic topology, Addison-Wesley, Redwood City, 1984.
• [20] J.J. Rotman, An Introduction to Algebraic Topology, Springer, New York, 1988.
• [21] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
• [22] Z. Virk, Small loop spaces, opology Appl. 157, 451–455, 2010.