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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Slicing theorems for $ n$-spheres in Euclidean $ (n+1)$-space


Author: Robert J. Daverman
Journal: Trans. Amer. Math. Soc. 166 (1972), 479-489
MSC: Primary 57A35
DOI: https://doi.org/10.1090/S0002-9947-1972-0295356-4
MathSciNet review: 0295356
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Abstract: This paper describes conditions on the intersection of an n-sphere $ \Sigma $ in Euclidean $ (n + 1)$-space $ {E^{n + 1}}$ with the horizontal hyperplanes of $ {E^{n + 1}}$ sufficient to determine that the sphere be nicely embedded. The results generally are pointed towards showing that the complement of $ \Sigma $ is 1-ULC (uniformly locally 1-connected) rather than towards establishing the stronger property that $ \Sigma $ is locally flat. For instance, the main theorem indicates that $ {E^{n + 1}} - \Sigma $ is 1-ULC provided each non-degenerate intersection of $ \Sigma $ and a horizontal hyperplane be an $ (n - 1)$-sphere bicollared both in that hyperplane and in $ \Sigma $ itself $ (n \ne 4)$.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0295356-4
Keywords: Horizontal hyperplane of Euclidean space, bicollared submanifold, locally flat submanifold, locally simply connected, homeomorphic approximation, topological embedding
Article copyright: © Copyright 1972 American Mathematical Society