Duality on noncompact manifolds and complements of topological knots

Abstract

Let Σ \Sigma be the image of a topological embedding of S n − 2 {S^{n - 2}} into S n {S^n} . In this paper the homotopy groups of the complement S n − Σ {S^n} - \Sigma are studied. In contrast with the situation in the smooth and piecewise-linear categories, it is shown that the first nonstandard homotopy group of the complement of such a topological knot can occur in any dimension in the range 1 through n − 2 n - 2 . If the first nonstandard homotopy group of the complement occurs above the middle dimension, then the end of S n − Σ {S^n} - \Sigma must have a nontrivial homotopy group in the dual dimension. The complement has the proper homotopy type of S 1 × R n − 1 {S^1} \times {\mathbb {R}^{n - 1}} if both the complement and the end of the complement have standard homotopy groups in every dimension below the middle dimension. A new form of duality for noncompact manifolds is developed. The duality theorem is the main technical tool used in the paper.