Glasnik Matematicki, Vol. 49, No. 1 (2014), 195-220.

A SHAPE THEORETIC APPROACH TO GENERALIZED COHOMOLOGICAL DIMENSION WITH RESPECT TO TOPOLOGICAL SPACES

Takahisa Miyata

Department of Mathematics and Informatics, Graduate School of Human Development and Environment, Kobe University, Kobe 657-8501, Japan
e-mail: tmiyata@kobe-u.ac.jp


Abstract.   A. N. Dranishnikov introduced the notion of generalized cohomological dimension of compact metric spaces with respect to CW spectra. In this paper, taking an inverse system approach, we generalize this definition and obtain two types of generalized comological dimension with respect to general topological spaces, which are objects in the stable shape category. We characterize those two types of generalized cohomological dimension in terms of maps and obtain their fundamental properties. In particular, we obtain their relations to the integral cohomological dimension and the covering dimension. Moreover, we study the generalized cohomological dimensions of compact Hausdorff spaces with respect to the Kahn continuum and the Hawaiian earing.

2010 Mathematics Subject Classification.   55P55, 55P30.

Key words and phrases.   Shape theory, stable shape theory, generalized cohomological dimension, CW spectrum.


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DOI: 10.3336/gm.49.1.14


References:

  1. J. F. Adams, On the groups J(X) IV, Topology 5 (1966), 21-71.
    MathSciNet     CrossRef

  2. J. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, 1974.
    MathSciNet    

  3. P. S. Alexandroff, Zum allgemeinen Dimensionsproblem, Nachr. Göttingen (1928), 25-44.

  4. P. S. Alexandroff, Dimensionstheorie, Math. Ann. 106 (1932), 161-238.
    MathSciNet     CrossRef

  5. A. N. Dranishnikov, On a problem of P. S. Aleksandrov, Math. USSR-Sb. 63 (1989), 539-545.
    MathSciNet     CrossRef

  6. A. N. Dranishnikov, Generalized cohomological dimension of compact metric spaces, Tsukuba J. Math. 14 (1990), 247-262.
    MathSciNet    

  7. A. N. Dranishnikov and J. Dydak, Extension dimension and extension types, Tr. Mat. Inst. Steklova 212 (1996), 61-94.
    MathSciNet    

  8. J. Dydak, Cohomological dimension theory, in Handbook of geometric topology, R. J. Daverman (ed.) et al., North-Holland, Amsterdam, 2002, 423-470.
    MathSciNet    

  9. H. W. Henn, Duality in stable shape theory, Arch. Math. (Basel) 36 (1981), 327-341.
    MathSciNet     CrossRef

  10. P. J. Huber, Homotopical cohomology and Čech cohomology, Math. Ann. 144 (1961), 73-76.
    MathSciNet     CrossRef

  11. D. S. Kahn, An example in Čech cohomology, Proc. Amer. Math. Soc. 16 (1965), 584.
    MathSciNet     CrossRef

  12. E. L. Lima, The Spanier-Whitehead duality in new homotopy categories, Summa Brasil. Math. 4 (1959), 91-148.
    MathSciNet    

  13. S. Mardešić and J. Segal, Shape theory. The inverse system approach, North Holland, Amsterdam-New York, 1982.
    MathSciNet    

  14. T. Miyata, Generalized stable shape and duality, Topology Appl. 109 (2001), 75-88.
    MathSciNet     CrossRef

  15. T. Miyata, Approximate extension property of mappings, Top. Appl. 159 (2012), 921-932.
    MathSciNet     CrossRef

  16. T. Miyata and J. Segal, Generalized stable shape and the Whitehead theorem, Topology Appl. 63 (1995), 139-164.
    MathSciNet     CrossRef

  17. T. Miyata and J. Segal, Generalized stable shape and Brown's representation theorem, Topology Appl. 94 (1999), 275-305.
    MathSciNet     CrossRef

  18. S. Nowak, On the relationships between shape properties of subcompacta of Sn and homotopy properties of their complements, Fund. Math. 128 (1987), 47-60.
    MathSciNet    

  19. S. Nowak, On the stable homotopy types of the complements of subcompacta of a manifold, Bull. Polish Acad. Sci. Math. 35 (1987), 359-363.
    MathSciNet    

  20. S. Nowak, Stable cohomotopy groups of compact spaces, Fund. Math. 180 (2003), 99-137.
    MathSciNet     CrossRef

  21. S. Nowak, On stable cohomotopy groups of compact spaces, Topology Appl. 153 (2005), 464-476.
    MathSciNet     CrossRef

  22. S. Nowak, On stable cohomotopy groups of compact spaces II, Topology Appl. 158 (2011), 152-158.
    MathSciNet     CrossRef

  23. E. G. Skljarenko, On the definition of cohomology dimension, Soviet Math. Dokl. 6 (1965), 478-479.
    MathSciNet    

  24. I. A. Švedov, Dimensions and soft sheaves, in General topology and its relations to modern analysis and algebra. II. Proc. Second Prague Topological Symposium 1966, Prague, 1967, 347-348.

  25. R. M. Switzer, Algebraic topology-homotopy and homology, Springer-Verlag, New York-Heidelberg, 1975.
    MathSciNet    

  26. J. J. Walsh, Dimension, cohomological dimension, and cell-like mappings, in Shape theory and geometric topology, Proceedings, Dubrovnik, Springer, Berlin-New York, 1981, 105-118.
    MathSciNet    


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