Duality on Noncompact Manifolds and Complements of Topological Knots
نویسنده
چکیده
A b str act . Let Σ be the image of a topological embedding of Sn−2 into Sn. In this paper the homotopy groups of the complement Sn−Σ 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 Sn − Σ must have a nontrivial homotopy group in the dual dimension. The complement has the proper homotopy type of S1 × Rn−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.
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