On noncontractible compacta with trivial homology and homotopy groups
HTML articles powered by AMS MathViewer
- by Umed H. Karimov and Dušan Repovš PDF
- Proc. Amer. Math. Soc. 138 (2010), 1525-1531 Request permission
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.References
- J. F. Adams, Talk on Toda’s work, Homotopy theory and related topics (Kinosaki, 1988) Lecture Notes in Math., vol. 1418, Springer, Berlin, 1990, pp. 7–14. MR 1048172, DOI 10.1007/BFb0083689
- J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21–71. MR 198470, DOI 10.1016/0040-9383(66)90004-8
- Karol Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 0216473
- Glen E. Bredon, Sheaf theory, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR 1481706, DOI 10.1007/978-1-4612-0647-7
- Katsuya Eda, Umed H. Karimov, and Dušan Repovš, A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions, Topology Appl. 156 (2009), no. 3, 515–521. MR 2492298, DOI 10.1016/j.topol.2008.07.019
- A. Hatcher, Vector Bundles and K-Theory, lecture notes, Cornell University, Ithaca, NY, http://www.math.cornell.edu/$\sim$hatcher/VBKT/VBpage.html.
- Daniel S. Kahn, An example in Čech cohomology, Proc. Amer. Math. Soc. 16 (1965), 584. MR 179785, DOI 10.1090/S0002-9939-1965-0179785-7
- U. Kh. Karimov and D. Repovsh, Hawaiian groups of topological spaces, Uspekhi Mat. Nauk 61 (2006), no. 5(371), 185–186 (Russian); English transl., Russian Math. Surveys 61 (2006), no. 5, 987–989. MR 2328264, DOI 10.1070/RM2006v061n05ABEH004363
- Max Karoubi, $K$-theory, Grundlehren der Mathematischen Wissenschaften, Band 226, Springer-Verlag, Berlin-New York, 1978. An introduction. MR 0488029
- J. Krasinkiewicz, On a method of constructing ANR-sets. An application of inverse limits, Fund. Math. 92 (1976), no. 2, 95–112. MR 420546, DOI 10.4064/fm-92-2-95-112
- Sam B. Nadler Jr., Continuum theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR 1192552
- Larry Siebenmann, Chapman’s classification of shapes: a proof using collapsing, Manuscripta Math. 16 (1975), no. 4, 373–384. MR 431183, DOI 10.1007/BF01323466
- Joseph L. Taylor, A counterexample in shape theory, Bull. Amer. Math. Soc. 81 (1975), 629–632. MR 375328, DOI 10.1090/S0002-9904-1975-13768-2
- Hirosi Toda, On unstable homotopy of spheres and classical groups, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 1102–1105. MR 123329, DOI 10.1073/pnas.46.8.1102
- George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508, DOI 10.1007/978-1-4612-6318-0
Additional Information
- Umed H. Karimov
- Affiliation: Institute of Mathematics, Academy of Sciences of Tajikistan, Ul. Ainy 299A, Dushanbe 734063, Tajikistan
- Email: umedkarimov@gmail.com
- Dušan Repovš
- Affiliation: Faculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana, P.O. Box 2964, Ljubljana 1001, Slovenia
- MR Author ID: 147135
- ORCID: 0000-0002-6643-1271
- Email: dusan.repovs@guest.arnes.si
- 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
- Communicated by: Alexander N. Dranishnikov
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 1525-1531
- MSC (2010): Primary 54F15, 55N15; Secondary 54G20, 57M05
- DOI: https://doi.org/10.1090/S0002-9939-09-10217-4
- MathSciNet review: 2578548
Dedicated: Dedicated to the memory of Professor Evgenij Grigor’evich Sklyarenko (1935-2009)