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Embedding phenomena based upon decomposition theory: locally spherical but wild codimension one spheres


Author: Robert J. Daverman
Journal: Proc. Amer. Math. Soc. 90 (1984), 139-144
MSC: Primary 57N50; Secondary 54B15, 57M30, 57N15, 57N45
DOI: https://doi.org/10.1090/S0002-9939-1984-0722432-5
MathSciNet review: 722432
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Abstract: For $ n \geqslant 7$ we describe an $ (n - 1)$-sphere $ \Sigma $ wildly embedded in the $ n$-sphere yet every point of $ \Sigma $ has arbitrarily small neighborhoods bounded by flat $ (n - 1)$-spheres, each intersecting $ \Sigma $ in an $ (n - 2)$-sphere. Not only do these examples for large $ n$ run counter to what can occur when $ n = 3$, they also illustrate the sharpness of high-dimensional taming theorems developed by Cannon and Harrold and Seebeck. Furthermore, despite their wildness, they have mapping cylinder neighborhoods, which both run counter to what is possible when $ n = 3$ and also partially illustrate the sharpness of another high-dimensional taming theorem due to Bryant and Lacher.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0722432-5
Keywords: Wild embedding, locally flat, codimension one sphere, locally spherical, locally weakly flat, homology cell, upper semicontinuous decomposition
Article copyright: © Copyright 1984 American Mathematical Society

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