Wild high-dimensional Cantor fences in Rn, Part I

On a question of B.J. Baker and M. Laidacker concerning disjoint compacta in $\mathbb R^N$

“…Fix any homeomorphism f 0 : X 0 ∼ = K. Let F : S N −1 → R N be any embedding that extends f 0 . (The existence of such extension, even with the additional property of being piecewise-linear on S N −1 \ X 0 , is proved using "horn pulling method" which is introduced in the works of L. Antoine and J. Alexander; see [27,Stat. 4] for detailed references.…”

“…(For N = 2, this is impossible by [48].) We believe that both Theorem 2.5 and Corollary 2.6 hold true for I k ; this probably can be obtained using ideas from [27].…”

“…In contrast to this, by results of R.H. Bing, it is impossible to place an uncountable collection of pairwise disjoint wild closed surfaces in R 3 (a short sketch of Bing's idea can be found in [14,Thm. 3.6.1]; see also [27] for detailed references). For N 4 there is a similar impossibility theorem for wild (N − 1)-spheres in R N [15, Thm.…”

“…We would like to mention that besides widely known Antoine's necklace, there is an essentially different construction of a wild Cantor set in R 3 given by P.S. Urysohn in 1922-23; for references, see [27]. At the moment, known examples of wild Cantor sets include the ones with such strong properties as simply-connectedness of the complement, slipperiness, stickiness (see Section 3.2).…”

An answer to a question of J.W. Cannon and S.G. Wayment

“…Namely, fix a Krushkal Cantor set K ⊂ Q \ ∂Q. We require F to be choosed so that F (P ) ⊃ K. This can be achieved by the feelers technique which comes back to L. Antoine, see [23,St. 4] for references.…”