Nowak, Sławomir On the relationships between shape properties of subcompacta of \(S^ n\) and homotopy properties of their complements. (English) Zbl 0633.55009 Fundam. Math. 128, 47-60 (1987). 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 \({\mathcal S}\). Cited in 1 ReviewCited in 4 Documents MSC: 55P55 Shape theory 57N25 Shapes (aspects of topological manifolds) 55P25 Spanier-Whitehead duality 55P42 Stable homotopy theory, spectra Keywords:stable homotopy category; stable weak homotopy category; stable shape category; subcompacta of \(S^ n\); Spanier-Whitehead duality; approximatively 1-connected continuum PDF BibTeX XML Cite \textit{S. Nowak}, Fundam. Math. 128, 47--60 (1987; Zbl 0633.55009) Full Text: DOI EuDML