Glasnik Matematicki, Vol. 40, No.2 (2005), 347-384.

A SUBSHAPE SPECTRUM FOR COMPACTA

Nikica Uglešić and Branko Červar

University of Split, Department of Mathematics, 21 000 Split, Teslina 12/III, Croatia
e-mail: uglesic@pmfst.hr
e-mail: brankoch@pmfst.hr


Abstract.   A sequence of categories and functors between them are constructed. They form a subshape spectrum for compacta in the following sense: Each of these categories classifies compact ANR's just as the homotopy category does; the classification of compacta by the "finest" of these categories coincides with the shape type classification; moreover, the finest category contains a subcategory which is isomorphic to the shape category; there exists a functor of the shape category to each of these categories, as well as of a "finer" category to a "coarser" one; the functors commute according to the indices.

Further, a few applications of the "subshape spectrum theory" are demonstrated. It is shown that the S*-equivalence (a uniformization of the Mardešić S-equivalence) and the q*-equivalence (a uniformization of the Borsuk quasi-equivalence) admit the category characterizations within the subshape spectrum, and that the q*-equivalence implies (but is not equivalent to) the S*-equivalence.

2000 Mathematics Subject Classification.   55P55, 18A32.

Key words and phrases.   Compactum, ANR, inverse sequence, limit, shape type, quasi-equivalence, S-equivalence.


Full text (PDF) (free access)

DOI: 10.3336/gm.40.2.15


References:

  1. K. Borsuk, Theory of Shape, Monografie Matematyczne 59, Polish Scientific Publishers, Warszawa, 1975.

  2. K. Borsuk, Some quantitative properties of shapes, Fund. Math. 93 (1976), 197-212.

  3. K. R. Goodearl and T. B. Rushing, Direct limit groups and the Keesling-Mardešić shape fibration, Pacific J. Math. 86 (1980), 471-476.

  4. H. Herlich and G. E. Strecker, Category Theory, Allyn and Bacon Inc., Boston, 1973.

  5. J. Keesling and S. Mardesic}, A shape fibration with fibers of different shape, Pacific J. Math. 84 (1979), 319-331.

  6. S. Mardesic, Comparing fibres in a shape fibration, Glasnik Mat. 13(33) (1978), 317-333.

  7. S. Mardesic and N. Uglesic, A category whose isomorphisms induce an equivalence relation coarser than shape, Top. Appl. (to appear).

  8. S. Mardesic and J. Segal, Shape Theory, North Holland, Amsterdam, 1982.

  9. N. Uglesic, A note on the Borsuk quasi-equivalence, submitted.

Glasnik Matematicki Home Page

closePristupaÄŤnostrefresh

Ako želite spremiti trajne postavke, kliknite Spremi, ako ne - vaše će se postavke poništiti kad zatvorite preglednik.