ITES2001 Fourth Italian-Spanish conference on General Topology and its applications Bressanone, 27-30 June 2001

Products in the strong shape category

Sibe Mardešić University of Zagreb, Croatia

Direct products are defined in arbitrary categories and are unique, whenever they exist. In the category H(Top) of topological spaces and homotopy classes of mappings, for any two spaces X, Y their product exists and is given by the Cartesian product X×Y and by the homotopy classes [π_{X}] and [π_{Y}] of the canonical projections π_{X}: X ×Y → X, π_{Y} : X ×Y → Y, respectively. Since shape theory is a modification of homotopy theory, it is natural to ask whether in the shape category Sh(Top) the Cartesian product X × Y and the shape morphisms S[π_{X}], S[π_{y}], induced by [π_{X}] and [π_{Y}], form the direct product of X and Y ? In 1974 J.E. Keesling [1] exhibited a simple (non-compact) space X ⊆ ℝ^{2} such that X × X and the two shape morphisms S[π_{X}]: X × X → X do not form a product in Sh(Top). However, he gave a positive answer in the case when X and Y are compact Hausdorff spaces. In 1977 Y. Kodama proved that the answer is positive also in the case when X is an FANR (fundamental absolute neighorhood retract) and Y is a paracompact space [2].

The main aim of the present talk is to announce analogous positive results in strong shape [4], i.e. for the strong shape category SSh(Top) and the strong shape functor \overline{S}: H(Top) → SSh(Top) (for definitions see [3]).

Theorem 1. If X and Y are compact Hausdorff spaces, then X × Y, \overline{S}[π_{X}] and \overline{S}[π_{Y}] form the product of X and Y in SSh(Top).

Theorem 2. If X is an FANR and Y is a finite-dimensional space (more general, a finitistic space), then X × Y, \overline{S}[π_{X}] and \overline{S}[π_{Y}] form the product of X and Y in SSh(Top).

References

[1] J. E. Keesling, Products in the shape category and some applications, Symp. Math. Istituto Nazionale di Alta Matematica 16 (1973), Academic Press, New York, 1974, 133–142.

[2] Y. Kodama, On product of shape and a question of Sher, Pacific J. Math. 72 (1977), 115–134.

[3] S. Mardešić, Strong shape and homology, Springer Monographs in Mathematics, Springer, Berlin 2000.

[4] S. Mardešić, Strong expansions of products and products in strong shape, (submitted).

Mathematics Subject Classification 54B35, 54C56, 55P55.

Key words and phrases: inverse limit, strong expansion, homotopy expansion, direct product, shape, strong shape, finitistic space.