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Duality on Noncompact Manifolds and Complements of Topological Knots
Gerard A. Venema
Proceedings of the American Mathematical Society
Vol. 123, No. 10 (Oct., 1995), pp. 3251-3262
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2160689
Page Count: 12
You can always find the topics here!Topics: Topological theorems, Topology, Mathematical manifolds, Embeddings, Homomorphisms, Mathematical complements, Mathematical theorems, Mathematical duality, Integers, Polyhedrons
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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.
Proceedings of the American Mathematical Society © 1995 American Mathematical Society