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

DUALITY BETWEEN STABLE STRONG SHAPE MORPHISMS AND STABLE HOMOTOPY CLASSES

Qamil Haxhibeqiri and Slawomir Nowak

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


Abstract.   Let SStrShn be the full subcategory of the stable strong shape category SStrSh of pointed compacta whose objects are all pointed subcompacta of Sn and let SOn 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 Dn : SStrShn SOn such that Dn(X) = Sn \ X for every subcompactum X of Sn and Dn : SStrShn(X, Y) SOn(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 SStrShn(X, Y) is induced by the inclusion of X into Y then Dn(α) = {j}.

1991 Mathematics Subject Classification.   55P55, 55P25.

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


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