Abstract
We calculate the singular homology and Čech cohomology groups of the Harmonic Archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda’s proof that the first singular homology groups of these spaces are isomorphic.
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References
Balcerzyk S., On factor groups of some subgroups of a complete direct sum of infinite cyclic groups, Bull. Acad. Polon. Sci., 1959, 7, 141–142
Bogley W.A., Sieradski A.J., Universal path spaces, preprint available at http://people.oregonstate.edu/_bogleyw/research/ups.pdf
Conner G., Some interesting open problems in low-dimensional wild topology, In: Workshop on Topology of Wild Spaces and Fractals, Strobl, July 4–8, 2011, abstract available at http://dmg.tuwien.ac.at/dorfer/wild_topology/abstracts.pdf
Curtis M.L., Fort M.K. Jr., Singular homology of one-dimensional spaces, Ann. of Math., 1959, 69, 309–313
Eda K., The singular homology groups of certain wild spaces (personal note, September 2011)
Eda K., Kawamura K., The singular homology of the Hawaiian earring, J. London Math. Soc., 2000, 62(1), 305–310
Fuchs L., Infinite Abelian Groups. I, Pure Appl. Math., 36, Academic Press, New York-London, 1970
Griffiths H.B., The fundamental group of two spaces with a common point, Q. J. Math., 1954, 5, 175–190
Harlap A.E., Local homology and cohomology, homological dimension, and generalized manifolds, Mat. Sb. (N.S.), 1975, 96(138), 347–373
Hatcher A., Algebraic Topology, Cambridge University Press, Cambridge, 2002
Meilstrup M., Archipelago groups, In: Workshop on Topology of Wild Spaces and Fractals, Strobl, July 4–8, 2011, abstract available at http://dmg.tuwien.ac.at/dorfer/wild_topology/abstracts.pdf
Spanier E.H., Algebraic Topology, McGraw-Hill, New York-Toronto, 1966
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Karimov, U.H., Repovš, D. On the homology of the Harmonic Archipelago. centr.eur.j.math. 10, 863–872 (2012). https://doi.org/10.2478/s11533-012-0038-2
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DOI: https://doi.org/10.2478/s11533-012-0038-2