Bing-Whitehead Cantor集的判别
Distinguishing Bing-Whitehead Cantor Sets
摘要：
Bing-Whitehead Cantor集是由DeGryse和Osborne在3维及更高维中引入的，以产生非标准（野生）但仍有单连通补的Cantor集的例子。与Kirkor的早期例子相比，构造技术可以推广到大于3的维度。$S^{3}$中的这些Cantor集是通过使用Bing或Whitehead链接作为定义序列的阶段来构造的。Ancel和Starbird以及Wright分别描述了在这样的构造中所需的Bing链接的数量，以便产生Cantor集。然而，它是未知的，是否改变的数量，冰和怀特黑德链接的建设将产生非等效康托集。利用几何指标的推广和三维相交模式的仔细分析，证明了Bing-Whitehead Cantor集等价嵌入$S^3$当且仅当它们的定义序列相差有限个Whitehead构造.因此，在亏格为1的环面单连通补的$S^{3}$中，存在无数个这样的非等价Cantor集。
英文摘要：
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.
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