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

Daverman, Robert J.

doi:10.1090/S0002-9939-1984-0722432-5

Embedding phenomena based upon decomposition theory: locally spherical but wild codimension one spheres
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by Robert J. Daverman PDF

Proc. Amer. Math. Soc. 90 (1984), 139-144 Request permission

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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