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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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  • 1. J. F. Adams, Talk on Toda's work, in: Homotopy Theory and Related Topics, Lect. Notes Math. 1418, Springer-Verlag, Berlin, 1990, pp. 7-14. MR 1048172 (91c:55022)
  • 2. J. F. Adams, On the groups J(X). IV, Topology 5 (1966), 21-71. MR 0198470 (33:6628)
  • 3. K. Borsuk, Theory of Retracts, Monogr. Mat. 44, PWN, Warsaw, 1967. MR 0216473 (35:7306)
  • 4. G. E. Bredon, Sheaf Theory, 2nd ed., Graduate Texts Math., 170, Springer-Verlag, Berlin, 1997. MR 1481706 (98g:55005)
  • 5. K. Eda, U. H. Karimov, D. Repovš, A nonaspherical cell-like $ 2$-dimensional simply connected continuum and related constructions, Topology Appl. 156 (2009), 515-521. MR 2492298
  • 6. A. Hatcher, Vector Bundles and K-Theory, lecture notes, Cornell University, Ithaca, NY,$ \sim$hatcher/VBKT/VBpage.html.
  • 7. D. S. Kahn, An example in Čech cohomology, Proc. Amer. Math. Soc. 16 (1965), 584. MR 0179785 (31:4027)
  • 8. U. H. Karimov, D. Repovš, Hawaiian groups of topological spaces, Uspehi. Math. Nauk. 61:5 (2007), 185-186. (in Russian); English transl. in Russian Math. Surv. 61:5 (2006), 987-989. MR 2328264 (2008d:55009)
  • 9. M. Karoubi, K-Theory, Springer-Verlag, Berlin, 1978. MR 0488029 (58:7605)
  • 10. J. Krasinkiewicz, On a method of constructing ANR-sets. An application of inverse limits, Fund. Math. 92 (1976), 95-112. MR 0420546 (54:8560)
  • 11. S. B. Nadler, Jr., Continuum Theory. An Introduction, Monogr. and Textbooks in Pure and Appl. Math., 158, Marcel Dekker, Inc., New York, 1992. MR 1192552 (93m:54002)
  • 12. L. C. Siebenmann, Chapman's classification of shapes: a proof using collapsing, Manuscr. Math. 16 (1975), 373-384. MR 0431183 (55:4185)
  • 13. J. L. Taylor, A counterexample in shape theory, Bull. Amer. Math. Soc. 81 (1975), 629-632. MR 0375328 (51:11523)
  • 14. H. Toda, On unstable homotopy of spheres and classical groups, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 1102-1105. MR 0123329 (23:A657)
  • 15. G. W. Whitehead, Elements of Homotopy Theory, Springer-Verlag, Berlin, 1978. MR 516508 (80b:55001)

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Additional Information

Umed H. Karimov
Affiliation: Institute of Mathematics, Academy of Sciences of Tajikistan, Ul. Ainy 299A, Dushanbe 734063, Tajikistan

Dusan Repovs
Affiliation: Faculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana, P.O. Box 2964, Ljubljana 1001, Slovenia

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

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