On H ˇ n -bubbles in n-dimensional compacta

Umed Karimov; Dušan Repovš

Colloquium Mathematicae (1998)

  • Volume: 75, Issue: 1, page 39-51
  • ISSN: 0010-1354

Abstract

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A topological space X is called an H ˇ n -bubble (n is a natural number, H ˇ n is Čech cohomology with integer coefficients) if its n-dimensional cohomology H ˇ 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 H ˇ n -bubble is locally connected; (2) There exists a 2-dimensional 2-acyclic compact metrizable ANR which does not contain any H ˇ 2 -bubbles; and (3) Every n-acyclic finite-dimensional L H ˇ n -trivial metrizable compactum contains an H ˇ n -bubble.

How to cite

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Karimov, Umed, and Repovš, Dušan. "On $\check{H}^n$-bubbles in n-dimensional compacta." Colloquium Mathematicae 75.1 (1998): 39-51. <http://eudml.org/doc/210528>.

@article{Karimov1998,
abstract = {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.},
author = {Karimov, Umed, Repovš, Dušan},
journal = {Colloquium Mathematicae},
keywords = {locally connected spaces; low-dimensional compacta; Čech cohomology; bubble; slender group; locally connected space; slender groups},
language = {eng},
number = {1},
pages = {39-51},
title = {On $\check\{H\}^n$-bubbles in n-dimensional compacta},
url = {http://eudml.org/doc/210528},
volume = {75},
year = {1998},
}

TY - JOUR
AU - Karimov, Umed
AU - Repovš, Dušan
TI - On $\check{H}^n$-bubbles in n-dimensional compacta
JO - Colloquium Mathematicae
PY - 1998
VL - 75
IS - 1
SP - 39
EP - 51
AB - 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.
LA - eng
KW - locally connected spaces; low-dimensional compacta; Čech cohomology; bubble; slender group; locally connected space; slender groups
UR - http://eudml.org/doc/210528
ER -

References

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  9. [9] W. Kuperberg, On certain homological properties of finite-dimensional compacta. Carries, minimal carries and bubbles, Fund. Math. 83 (1973), 7-23. Zbl0269.54020
  10. [10] K. Kuratowski, Topology, Vol. 2, Academic Press, New York, 1968. 
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  12. [12] W. J. R. Mitchell, Homology manifolds, inverse systems and cohomological local connectedness, J. London Math. Soc. (2) 19 (1979), 348-358. Zbl0395.57008
  13. [13] E. Sąsiada, Proof that every countable and reduced torsion-free abelian group is slender, Bull. Acad. Polon. Sci. 7 (1959), 143-144. Zbl0085.01702

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