Glasnik Matematicki, Vol. 36, No.2 (2001), 297-310.

DUALITY BETWEEN STABLE STRONG SHAPE MORPHISMS AND STABLE HOMOTOPY CLASSES

Qamil Haxhibeqiri and Slawomir Nowak

Abstract.   Let SStrSh_n be the full subcategory of the stable strong shape category SStrSh of pointed compacta whose objects are all pointed subcompacta of Sⁿ and let SO_n be the full subcategory of the stable homotopy category S whose objects are all open subsets of Sⁿ. In this paper it is shown that there exists a contravariant additive functor D_n : SStrSh_n → SO_n such that D_n(X) = Sⁿ \ X for every subcompactum X of Sⁿ and D_n : SStrSh_n(X, Y) → SO_n(Sⁿ \ Y, Sⁿ \ X) is an isomorphism of abelian groups for all compacta X, Y ⊂ Sⁿ. Moreover, if X ⊂ Y ⊂ Sⁿ, j : Sⁿ \ Y → Sⁿ \ X is an inclusion and α ∈ SStrSh_n(X, Y) is induced by the inclusion of X into Y then D_n(α) = {j}.

1991 Mathematics Subject Classification.   55P55, 55P25.

Key words and phrases.   Stable strong shape, stable homotopy, proper map, proper homotopy.