# zbMATH — the first resource for mathematics

On the relationships between shape properties of subcompacta of $$S^ n$$ and homotopy properties of their complements. (English) Zbl 0633.55009
Taking for the set of morphisms from X to Y the direct limit of the sets of all homotopy classes or all weak homotopy classes or all shaping between the n-fold suspension of X and Y we obtain (respectively) the stable homotopy category $${\mathcal S}$$ or the stable weak homotopy category $${\mathcal S}w{\mathcal O}_ n$$ of open subsets of $$S^ n$$ or the stable shape category $${\mathcal S}{\mathcal S}h_ n$$ of subcompacta of $$S^ n$$.
We prove that there exists an isomorphism $${\mathcal D}_ n: {\mathcal S}{\mathcal S}h_ n\to {\mathcal S}w{\mathcal O}_ n$$ such that $${\mathcal D}_ n(X)=S^ n\setminus X$$. If we limit ourselves to movable compacta, then $${\mathcal S}w{\mathcal O}_ n$$ can be replaced by a suitable full subcategory of $${\mathcal S}$$. These generalize the classical Spanier-Whitehead duality.
Applications to the ordinary shape theory are also given. In particular, if $$1<k\leq n$$ and $$X\subset S^ n$$ is an approximatively 1-connected continuum, then $$Sh(X)=Sh(S^ k)$$ iff $$S^ n\setminus X$$ and $$S^ n\setminus S^ k$$ are isomorphic in $${\mathcal S}$$.

##### MSC:
 55P55 Shape theory 57N25 Shapes (aspects of topological manifolds) 55P25 Spanier-Whitehead duality 55P42 Stable homotopy theory, spectra
Full Text: