Glasnik Matematicki, Vol. 42, No.1 (2007), 189-194.

THE STABLE SHAPE OF COMPACT SPACES WITH COUNTABLE COHOMOLOGY GROUPS

Slawomir Nowak

Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
e-mail: snowak@mimuw.edu.pl


Abstract.   Let X be a compact Hausdorff space and q in Z be an integer such that the integral cohomology groups Hn(X;Z) are countable for n > q and the stable cohomotopy groups πsn(X) of X are countable for nq. Then there exists a compact metrizable compact space Y with the same stable shape as X.

2000 Mathematics Subject Classification.   55P55, 55P40, 55N05, 55Q07.

Key words and phrases.   Stable cohomotopy groups, stable shape.


Full text (PDF) (free access)

DOI: 10.3336/gm.42.1.13


References:

  1. J. F. Adams, Stable Homotopy and Generalized Homology, University of Chicago Press, Chicago, 1974.
    MathSciNet

  2. K. Borsuk, Theory of Shape, PWN-Polish Scientific Publishers, Warszawa, 1975.
    MathSciNet

  3. D. S. Kahn, J. Kaminker and C. Schochet, Generalized homology theories on compact metric spaces, Michigan Math. J. 24 (1977), 203--224.
    MathSciNet     CrossRef

  4. S. Mardesic and J. Segal, Shape Theory, North-Holland Publishing Co., Amsterdam-New York-Oxford 1982.
    MathSciNet

  5. H. R. Margolis, Spectra and the Steenrod Algebra. Modules over the Steenrod algebra and the stable homotopy category, North-Holland Publishing Co., Amsterdam, 1983.
    MathSciNet

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

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

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

  9. 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

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

  11. R. M. Switzer, Algebraic Topology - Homotopy and Homology, Springer-Verlag, New York-Heidelberg, 1975.
    MathSciNet

  12. C. T. C. Wall, Finiteness conditions for CW-complexes, Ann. of Math. (2) 81 (1965), 59-69.
    MathSciNet     CrossRef

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