Wild spheres in that are locally flat modulo tame Cantor sets

Author:
Robert J. Daverman

Journal:
Trans. Amer. Math. Soc. **206** (1975), 347-359

MSC:
Primary 57A45

DOI:
https://doi.org/10.1090/S0002-9947-1975-0375329-6

MathSciNet review:
0375329

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Abstract: Kirby has given an elementary geometric proof showing that if an -sphere in Euclidean -space is locally flat modulo a Cantor set that is tame relative to both and , then is locally flat. In this paper we illustrate the sharpness of the result by describing a wild -sphere in such that is locally flat modulo a Cantor set and is tame relative to . These examples then are used to contrast certain properties of embedded spheres in higher dimensions with related properties of spheres in .

Rather obviously, as Kirby points out in [11], his result cannot be weak-ened by dismissing the restriction that the Cantor set be tame relative to . It is well known that a sphere in containing a wild (relative to ) Cantor set must be wild. Consequently the only variation on his work that merits consideration is the one mentioned above.

The phenomenon we intend to describe also occurs in -space. Alexander's horned sphere [1] is wild but is locally flat modulo a tame Cantor set. In fact, at one spot methods used here parallel those used to construct that example. However, other properties of -space are strikingly dissimilar to what can be derived from the higher dimensional examples constructed here, for, as discussed in §2, natural analogues to some important results concerning locally flat embeddings in are false.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0375329-6

Keywords:
Locally flat embedding,
wild embedding,
tame Cantor set,
defining sequence for Cantor set,
crumpled -cube,
cellular,
taming set,
-taming set

Article copyright:
© Copyright 1975
American Mathematical Society