DISTINGUISHING BING WHITEHEAD CANTOR SETS

Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were nonstandard (wild) but still had a simply connected complement In contrast to an earlier example of Kirkor the construction techniques could be generalized to dimensions greater 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 nonequivalent Cantor sets Using a generalization of the geometric index and a careful analysis of three dimensional into section pattern we prove that Bing Whitehead Cantor sets are equivalently embedded in S-3 if and of Whitehead constructions As a consequence there are uncountably many nonequivalent such Cantor sets in S-3 constructed with genus one tori and with a simply connected complement.