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

A SUBSHAPE SPECTRUM FOR COMPACTA

Nikica Uglešić and Branko Červar

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.

DOI: 10.3336/gm.40.2.15

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