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Duality on noncompact manifolds and complements of topological knots

Author: Gerard A. Venema
Journal: Proc. Amer. Math. Soc. 123 (1995), 3251-3262
MSC: Primary 57M30; Secondary 55M05, 55Q05
MathSciNet review: 1307570
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Abstract: Let $ \Sigma $ be the image of a topological embedding of $ {S^{n - 2}}$ into $ {S^n}$. In this paper the homotopy groups of the complement $ {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$. If the first nonstandard homotopy group of the complement occurs above the middle dimension, then the end of $ {S^n} - \Sigma $ must have a nontrivial homotopy group in the dual dimension. The complement has the proper homotopy type of $ {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.

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Keywords: Knot complement, homotopy groups, duality
Article copyright: © Copyright 1995 American Mathematical Society

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