Wild spheres in $E^{n}$ that are locally flat modulo tame Cantor sets
Author:
Robert J. Daverman
Journal:
Trans. Amer. Math. Soc. 206 (1975), 347359
MSC:
Primary 57A45
DOI:
https://doi.org/10.1090/S00029947197503753296
MathSciNet review:
0375329
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
Abstract: Kirby has given an elementary geometric proof showing that if an $(n  1)$sphere $\Sigma$ in Euclidean $n$space ${E^n}$ is locally flat modulo a Cantor set that is tame relative to both $\Sigma$ and ${E^n}$, then $\Sigma$ is locally flat. In this paper we illustrate the sharpness of the result by describing a wild $(n  1)$sphere $\Sigma$ in ${E^n}$ such that $\Sigma$ is locally flat modulo a Cantor set $C$ and $C$ is tame relative to ${E^n}$. These examples then are used to contrast certain properties of embedded spheres in higher dimensions with related properties of spheres in ${E^3}$. Rather obviously, as Kirby points out in [11], his result cannot be weakened by dismissing the restriction that the Cantor set be tame relative to ${E^n}$. It is well known that a sphere in ${E^n}$ containing a wild (relative to ${E^n}$) 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 $3$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 $3$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 ${E^3}$ are false.

J. W. Alexander, An example of simply connected surface bounding a region which is not simply connected, Proc. Nat. Acad. Sci. U. S. A. 10 (1924), 810.
L. Antoine, Sur l’homéomorphic de deux figures et de leurs voisinages, J. Math. Pures Appl. 86 (1921), 221325.
 R. H. Bing, A wild surface each of whose arcs is tame, Duke Math. J. 28 (1961), 1–15. MR 123302
 William A. Blankinship, Generalization of a construction of Antoine, Ann. of Math. (2) 53 (1951), 276–297. MR 40659, DOI https://doi.org/10.2307/1969543
 C. E. Burgess, Properties of certain types of wild surfaces in $E^{3}$, Amer. J. Math. 86 (1964), 325–338. MR 163295, DOI https://doi.org/10.2307/2373168
 C. E. Burgess, Criteria for a $2$sphere in $S^{3}$ to be tame modulo two points, Michigan Math. J. 14 (1967), 321–330. MR 216481
 J. W. Cannon, Characterization of taming sets on $2$spheres, Trans. Amer. Math. Soc. 147 (1970), 289–299. MR 257996, DOI https://doi.org/10.1090/S00029947197002579966
 James W. Cannon, $^{\ast } $taming sets for crumpled cubes. I. Basic properties, Trans. Amer. Math. Soc. 161 (1971), 429–440. MR 282353, DOI https://doi.org/10.1090/S00029947197102823537
 D. G. DeGryse and R. P. Osborne, A wild Cantor set in $E^{n}$ with simply connected complement, Fund. Math. 86 (1974), 9–27. MR 375323, DOI https://doi.org/10.4064/fm861927
 P. H. Doyle and J. G. Hocking, Some results on tame disks and spheres in $E^{3}$, Proc. Amer. Math. Soc. 11 (1960), 832–836. MR 126839, DOI https://doi.org/10.1090/S00029939196001268392
 Robion C. Kirby, On the set of nonlocally flat points of a submanifold of codimension one, Ann. of Math. (2) 88 (1968), 281–290. MR 236900, DOI https://doi.org/10.2307/1970575
 D. R. McMillan Jr., A criterion for cellularity in a manifold, Ann. of Math. (2) 79 (1964), 327–337. MR 161320, DOI https://doi.org/10.2307/1970548
 R. P. Osborne, Embedding Cantor sets in a manifold. II. An extension theorem for homeomorphisms on Cantor sets, Fund. Math. 65 (1969), 147–151. MR 247620, DOI https://doi.org/10.4064/fm652147151
 R. P. Osborne, Embedding Cantor sets in a manifold. III. Approximating spheres, Fund. Math. 90 (1975/76), no. 3, 253–259. MR 405429, DOI https://doi.org/10.4064/fm903253259
 C. L. Seebeck III, Collaring and $(n1)$manifold in an $n$manifold, Trans. Amer. Math. Soc. 148 (1970), 63–68. MR 258045, DOI https://doi.org/10.1090/S00029947197002580456
 W. R. Alford and R. B. Sher, Defining sequences for compact $0$dimensional decompositions of $E^{n}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 209–212 (English, with Russian summary). MR 254824
Retrieve articles in Transactions of the American Mathematical Society with MSC: 57A45
Retrieve articles in all journals with MSC: 57A45
Additional Information
Keywords:
Locally flat embedding,
wild embedding,
tame Cantor set,
defining sequence for Cantor set,
crumpled <IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$n$">cube,
cellular,
taming set,
<IMG WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$\ast$">taming set
Article copyright:
© Copyright 1975
American Mathematical Society