Duality on noncompact manifolds and complements of topological knots

Venema, Gerard A.

doi:10.1090/S0002-9939-1995-1307570-8

Duality on noncompact manifolds and complements of topological knots
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by Gerard A. Venema PDF

Proc. Amer. Math. Soc. 123 (1995), 3251-3262 Request permission

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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