Proceedings of the American Mathematical Society
Let be the image of a topological embedding of into. In this paper the homotopy groups of the complement. 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. If the first nonstandard homotopy group of the complement occurs above the middle dimension, then the end of must have a nontrivial homotopy group in the dual dimension. The complement has the proper homotopy type of 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.
Venema, Gerard A., "Duality on noncompact manifolds and complements of topological knots" (1995). University Faculty Publications and Creative Works. 551.