#### 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 SStrSh_{n} be the full
subcategory of the stable strong shape category
SStrSh of pointed compacta whose objects are all pointed
subcompacta of *S*^{n} and let
*SO*_{n} be the full subcategory of the
stable homotopy category *S* whose objects are all open
subsets of *S*^{n}. In this paper it is shown
that there exists a contravariant additive functor
*D*_{n} : SStrSh_{n}
→
*SO*_{n}
such that
*D*_{n}(*X*) =
*S*^{n} \ *X*
for every subcompactum
*X* of *S*^{n} and
*D*_{n} : SStrSh_{n}(*X*,
*Y*) →
*SO*_{n}(*S*^{n} \ *Y*,
*S*^{n} \ *X*)
is an isomorphism of abelian
groups for all compacta *X*, *Y*
⊂
*S*^{n}. Moreover, if
*X* ⊂ *Y*
⊂
*S*^{n},
*j* : *S*^{n} \ *Y*
→
*S*^{n} \ *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.

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