On the relationships between shape properties of subcompacta of \(S^ n\) and homotopy properties of their complements. (English) Zbl 0633.55009

Nowak, Sławomir

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 \({\mathcal S}\).

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