Distinguishing Bing-Whitehead Cantor Sets

TL;DR

This paper classifies Bing-Whitehead Cantor sets in three-dimensional spheres, showing that their equivalence depends on the finite differences in their defining sequences, leading to uncountably many distinct examples.

Contribution

It provides a complete classification of Bing-Whitehead Cantor sets based on their defining sequences, revealing the existence of uncountably many non-equivalent sets.

Findings

Bing-Whitehead Cantor sets are equivalent iff their sequences differ finitely.

Uncountably many non-equivalent Bing-Whitehead Cantor sets exist in $S^3$.

Classification uses geometric index and intersection pattern analysis.

Abstract

Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were non standard (wild), but still had simply connected complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions bigger than three. These Cantor sets in $S^{3}$ are constructed by using Bing or Whitehead links as stages in defining sequences. Ancel and Starbird, and separately Wright characterized the number of Bing links needed in such constructions so as to produce Cantor sets. However it was unknown whether varying the number of Bing and Whitehead links in the construction would produce non equivalent Cantor sets. Using a generalization of geometric index, and a careful analysis of three dimensional intersection patterns, we prove that Bing-Whitehead Cantor sets are equivalently embedded in $S^3$ if and only if their defining sequences differ by some finite number of Whitehead constructions. As a consequence, there are uncountably many non equivalent such Cantor sets in $S^{3}$ constructed with genus one tori and with simply connected complement.