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The Vietoris system in strong shape and strong homology

Bernd Günther

Fundamenta Mathematicae (1992)

  • Volume: 141, Issue: 2, page 147-168
  • ISSN: 0016-2736

Abstract

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We show that the Vietoris system of a space is isomorphic to a strong expansion of that space in the Steenrod homotopy category, and from this we derive a simple description of strong homology. It is proved that in ZFC strong homology does not have compact supports, and that enforcing compact supports by taking limits leads to a homology functor that does not factor over the strong shape category. For compact Hausdorff spaces strong homology is proved to be isomorphic to Massey's homology.

How to cite

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Günther, Bernd. "The Vietoris system in strong shape and strong homology." Fundamenta Mathematicae 141.2 (1992): 147-168. <http://eudml.org/doc/211958>.

@article{Günther1992,
abstract = {We show that the Vietoris system of a space is isomorphic to a strong expansion of that space in the Steenrod homotopy category, and from this we derive a simple description of strong homology. It is proved that in ZFC strong homology does not have compact supports, and that enforcing compact supports by taking limits leads to a homology functor that does not factor over the strong shape category. For compact Hausdorff spaces strong homology is proved to be isomorphic to Massey's homology.},
author = {Günther, Bernd},
journal = {Fundamenta Mathematicae},
keywords = {vietoris nerve; Steenrod homotopy category; strong shape theory; strong homology; compact supports; Vietoris system; strong shape; strong ANR-expansion},
language = {eng},
number = {2},
pages = {147-168},
title = {The Vietoris system in strong shape and strong homology},
url = {http://eudml.org/doc/211958},
volume = {141},
year = {1992},
}

TY - JOUR
AU - Günther, Bernd
TI - The Vietoris system in strong shape and strong homology
JO - Fundamenta Mathematicae
PY - 1992
VL - 141
IS - 2
SP - 147
EP - 168
AB - We show that the Vietoris system of a space is isomorphic to a strong expansion of that space in the Steenrod homotopy category, and from this we derive a simple description of strong homology. It is proved that in ZFC strong homology does not have compact supports, and that enforcing compact supports by taking limits leads to a homology functor that does not factor over the strong shape category. For compact Hausdorff spaces strong homology is proved to be isomorphic to Massey's homology.
LA - eng
KW - vietoris nerve; Steenrod homotopy category; strong shape theory; strong homology; compact supports; Vietoris system; strong shape; strong ANR-expansion
UR - http://eudml.org/doc/211958
ER -

References

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  1. [1] R. A. Alo and H. L. Shapiro, Normal Topological Spaces, Cambridge Tracts in Math. 65, Cambridge University Press, 1974. 
  2. [2] N. A. Berikashvili, Steenrod-Sitnikov homology theories on the category of compact spaces, Soviet Math. Dokl. 22 (1980), 544-547. Zbl0482.55007
  3. [3] N. A. Berikashvili, On the axiomatics of Steenrod-Sitnikov homology theory on the category of compact Hausdorff spaces, Proc. Steklov Inst. Math. 4 (1984), 25-39. 
  4. [4] F. W. Cathey and J. Segal, Strong shape theory and resolutions, Topology Appl. 15 (1983), 119-130. Zbl0505.55012
  5. [5] C. H. Dowker, Homotopy groups of relations, Ann. of Math. 56 (1952), 84-95. Zbl0046.40402
  6. [6] D. A. Edwards and H. M. Hastings, Čech and Steenrod Homotopy Theories with Applications to Geometric Topology, Lecture Notes in Math. 542, Springer, 1976. Zbl0334.55001
  7. [7] R. Engelking, General Topology, 2nd ed., Heldermann, Berlin 1989. 
  8. [8] B. Günther, Comparison of the coherent pro-homotopy theories of Edwards-Hastings, Lisica-Mardešić and Günther, Glas. Mat., to appear. Zbl0795.55010
  9. [9] B. Günther, Properties of normal embeddings concerning strong shape theory, II, Tsukuba J. Math., to appear. Zbl0808.54017
  10. [10] Ju. T. Lisica and S. Mardešić, Coherent prohomotopy and strong shape theory, Glas. Mat. 19 (39) (1984), 335-399. Zbl0553.55009
  11. [11] Ju. T. Lisica and S. Mardešić, Strong homology of inverse systems of spaces, I, Topology Appl. 19 (1985), 29-43. Zbl0574.55005
  12. [12] Ju. T. Lisica and S. Mardešić, Strong homology of inverse systems of spaces, II, ibid., 45-64. 
  13. [13] Ju. T. Lisica and S. Mardešić, Strong homology of inverse systems of spaces, III, ibid. 20 (1985), 29-37. Zbl0574.55005
  14. [14] S. Mardešić, Resolutions of spaces are strong expansions, Publ. Inst. Math. (Beograd) 49 (63) (1991), 179-188. Zbl0774.54015
  15. [15] S. Mardešić, Strong expansions and strong shape theory, Topology Appl. 38 (1991), 275-291. Zbl0715.55008
  16. [16] S. Mardešić and Z. Miminoshvili, The relative homeomorphism and the wedge axioms for strong homology, Glas. Mat., to appear. Zbl0794.55004
  17. [17] S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307 (1988), 725-744. Zbl0648.55007
  18. [18] S. Mardešić and J. Segal, Shape Theory, Math. Library 26, North-Holland, 1982. 
  19. [19] S. Mardešić and T. Watanabe, Strong homology and dimension, Topology Appl. 29 (1988), 185-205. Zbl0648.55006
  20. [20] W. S. Massey, Homology and Cohomology Theory, Pure and Appl. Math. 46, Marcel Dekker, 1978. 
  21. [21] E. Michael, A note on closed maps and compact sets, Israel J. Math. 2 (1964), 173-176. Zbl0136.19303
  22. [22] Z. Miminoshvili, On the sequences of exact and half-exact homologies of arbitrary spaces, Soobshch. Akad. Nauk Gruzin. SSR 113 (1) (1984), 41-44 (in Russian). Zbl0546.55006
  23. [23] T. Porter, Čech homotopy, I, J. London Math. Soc. (2) 6 (1973), 429-436. 
  24. [24] T. Porter, Čech homotopy, II, ibid., 662-675. 
  25. [25] T. Porter, Čech homotopy, III, Bull. London Math. Soc. 6 (1974), 307-311. 
  26. [26] E. H. Spanier, Algebraic Topology, McGraw-Hill, 1966. 

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