On $\check{H}^n$-bubbles in n-dimensional compacta

• Autorzy: Umed H. Karimov , Dušan Repovš
• Języki publikacji: EN

Abstrakty

EN

A topological space X is called an $\check{H}^n$-bubble (n is a natural number, $\check{H}^n$ is Čech cohomology with integer coefficients) if its n-dimensional cohomology $\check{H}^n(X)$ is nontrivial and the n-dimensional cohomology of every proper subspace is trivial. The main results of our paper are: (1) Any compact metrizable $\check{H}^n$-bubble is locally connected; (2) There exists a 2-dimensional 2-acyclic compact metrizable ANR which does not contain any $\check{H}^2$-bubbles; and (3) Every n-acyclic finite-dimensional $L\check{H}^n$-trivial metrizable compactum contains an $\check{H}^n$-bubble.

Słowa kluczowe

EN

locally connected spaces; low-dimensional compacta; Čech cohomology; bubble; slender group

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1998
• Tom: 75
• Numer: 1
• Strony: 39-51

Daty

wydano

1998

otrzymano

1996-02-23

poprawiono

1997-03-11

Twórcy

autor

Umed H. Karimov

• Institute of Mathematics, Tadžik Academy of Sciences, Ul. Akademičeskaya 10, Dušanbe, 734013 Tadžikistan

autor

Dušan Repovš

• Institute for Mathematics, Physics and Mechanics, University of Ljubljana, P.O. Box 2964, 1001 Ljubljana, Slovenia

Bibliografia

• [1] P. S. Aleksandrov, Dimensionstheorie. Ein Beitrag zur Geometrie der abgeschlossenen Mengen, Math. Ann. 106 (1932), 161-238.
• [2] K. Borsuk, Theory of Retracts, Monograf. Mat. 44, PWN, Warszawa, 1967.
• [3] G. E. Bredon, Sheaf Theory, 2nd ed., Springer, New York, 1997.
• [4] R. Engelking, General Topology, Heldermann, Berlin, 1989.
• [5] D. B. Fuks and V. A. Rokhlin, Introductory Course in Topology: Geometric Chapters, Nauka, Moscow, 1977 (in Russian).
• [6] R. Godement, Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1958.
• [7] A. E. Harlap, Local homology and cohomology, homological dimension and generalized manifolds, Mat. Sb. 96 (1975), 347-373 (in Russian); English transl.: Math. USSR-Sb. 25 (1975), 323-349.
• [8] U. H. Karimov, On the generalized homotopy axiom, Izv. Akad. Nauk Tadžik. SSR Otdel. Fiz.-Mat. Khim. i Geol. Nauk 71 (1979), 83-84 (in Russian).
• [9] W. Kuperberg, On certain homological properties of finite-dimensional compacta. Carries, minimal carries and bubbles, Fund. Math. 83 (1973), 7-23.
• [10] K. Kuratowski, Topology, Vol. 2, Academic Press, New York, 1968.
• [11] S. Mardešić and J. Segal, Shape Theory: The Inverse System Approach, North-Holland, Amsterdam 1982.
• [12] W. J. R. Mitchell, Homology manifolds, inverse systems and cohomological local connectedness, J. London Math. Soc. (2) 19 (1979), 348-358.
• [13] E. Sąsiada, Proof that every countable and reduced torsion-free abelian group is slender, Bull. Acad. Polon. Sci. 7 (1959), 143-144.