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The Vietoris system in strong shape and strong homology

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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.
Twórcy
  • Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, Robert-Mayer-Strasse, 6-10 6000 Frankfurt, Germany
Bibliografia
  • [1] R. A. Alo and H. L. Shapiro, Normal Topological Spaces, Cambridge Tracts in Math. 65, Cambridge University Press, 1974.
  • [2] N. A. Berikashvili, Steenrod-Sitnikov homology theories on the category of compact spaces, Soviet Math. Dokl. 22 (1980), 544-547.
  • [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] F. W. Cathey and J. Segal, Strong shape theory and resolutions, Topology Appl. 15 (1983), 119-130.
  • [5] C. H. Dowker, Homotopy groups of relations, Ann. of Math. 56 (1952), 84-95.
  • [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.
  • [7] R. Engelking, General Topology, 2nd ed., Heldermann, Berlin 1989.
  • [8] B. Günther, Comparison of the coherent pro-homotopy theories of Edwards-Hastings, Lisica-Mardešić and Günther, Glas. Mat., to appear.
  • [9] B. Günther, Properties of normal embeddings concerning strong shape theory, II, Tsukuba J. Math., to appear.
  • [10] Ju. T. Lisica and S. Mardešić, Coherent prohomotopy and strong shape theory, Glas. Mat. 19 (39) (1984), 335-399.
  • [11] Ju. T. Lisica and S. Mardešić, Strong homology of inverse systems of spaces, I, Topology Appl. 19 (1985), 29-43.
  • [12] Ju. T. Lisica and S. Mardešić, Strong homology of inverse systems of spaces, II, ibid., 45-64.
  • [13] Ju. T. Lisica and S. Mardešić, Strong homology of inverse systems of spaces, III, ibid. 20 (1985), 29-37.
  • [14] S. Mardešić, Resolutions of spaces are strong expansions, Publ. Inst. Math. (Beograd) 49 (63) (1991), 179-188.
  • [15] S. Mardešić, Strong expansions and strong shape theory, Topology Appl. 38 (1991), 275-291.
  • [16] S. Mardešić and Z. Miminoshvili, The relative homeomorphism and the wedge axioms for strong homology, Glas. Mat., to appear.
  • [17] S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307 (1988), 725-744.
  • [18] S. Mardešić and J. Segal, Shape Theory, Math. Library 26, North-Holland, 1982.
  • [19] S. Mardešić and T. Watanabe, Strong homology and dimension, Topology Appl. 29 (1988), 185-205.
  • [20] W. S. Massey, Homology and Cohomology Theory, Pure and Appl. Math. 46, Marcel Dekker, 1978.
  • [21] E. Michael, A note on closed maps and compact sets, Israel J. Math. 2 (1964), 173-176.
  • [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).
  • [23] T. Porter, Čech homotopy, I, J. London Math. Soc. (2) 6 (1973), 429-436.
  • [24] T. Porter, Čech homotopy, II, ibid., 662-675.
  • [25] T. Porter, Čech homotopy, III, Bull. London Math. Soc. 6 (1974), 307-311.
  • [26] E. H. Spanier, Algebraic Topology, McGraw-Hill, 1966.
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Bibliografia
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