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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On noncontractible compacta with trivial homology and homotopy groups


Authors: Umed H. Karimov and Dusan Repovs
Journal: Proc. Amer. Math. Soc. 138 (2010), 1525-1531
MSC (2010): Primary 54F15, 55N15; Secondary 54G20, 57M05
Published electronically: December 8, 2009
MathSciNet review: 2578548
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Abstract: We construct an example of a Peano continuum $ X$ such that: (i) $ X$ is a one-point compactification of a polyhedron; (ii) $ X$ is weakly homotopy equivalent to a point (i.e. $ \pi_n(X)$ is trivial for all $ n \geq 0$); (iii) $ X$ is noncontractible; and (iv) $ X$ is homologically and cohomologically locally connected (i.e. $ X$ is an HLC and $ clc$ space). We also prove that all classical homology groups (singular, Čech, and Borel-Moore), all classical cohomology groups (singular and Čech), and all finite-dimensional Hawaiian groups of $ X$ are trivial.


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Additional Information

Umed H. Karimov
Affiliation: Institute of Mathematics, Academy of Sciences of Tajikistan, Ul. Ainy 299A, Dushanbe 734063, Tajikistan
Email: umedkarimov@gmail.com

Dusan Repovs
Affiliation: Faculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana, P.O. Box 2964, Ljubljana 1001, Slovenia
Email: dusan.repovs@guest.arnes.si

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10217-4
PII: S 0002-9939(09)10217-4
Keywords: Noncontractible compactum, weak homotopy equivalence, reduced complex $\widetilde {K}_{\mathcal {C}}$-theory, admissible spectrum, Peano continuum, infinite-dimensional Hawaiian earrings, Hawaiian group
Received by editor(s): November 25, 2008
Received by editor(s) in revised form: August 7, 2009, and September 18, 2009
Published electronically: December 8, 2009
Dedicated: Dedicated to the memory of Professor Evgenij Grigor’evich Sklyarenko (1935-2009)
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2009 American Mathematical Society