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 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 nonequivalent such Cantor sets in ▫$S^3$▫ constructed with genus one tori and with a simply connected complement.
COBISS.SI-ID: 15682137
We prove that if an ultrafilter ▫$\mathcal{L}$▫ is not coherent to a ▫$Q$▫-point, then each analytic non-▫$\sigma$▫-bounded topological group ▫$G$▫ admits an increasing chain ▫$\langle G_\alpha \colon \alpha ( \mathfrak{b} (\mathcal{L}) \rangle$▫ of its proper subgroups such that: (i) ▫$\bigcup_\alpha G_\alpha = G$▫; and (ii) For every ▫$\sigma$▫-bounded subgroup ▫$H$▫ of ▫$G$▫ there exists ▫$\alpha$▫ such that ▫$H \subset G_\alpha$▫. In case of the group ▫${\rm Sym}(\omega)$▫ of all permutations of ▫$\omega$▫ with the topology inherited from ▫$\omega^\omega$▫ this improves upon earlier results of S. Thomas.
COBISS.SI-ID: 15872601
We show that the dimension of the sublinear Higson corona of a metric space ▫$X$▫ is the smallest non-negative integer ▫$m$▫ with the following property: Any norm-preserving asymptotically Lipschitz function from a closed subset ▫$A$▫ of ▫$X$▫ to the Euclidean space of dimension ▫$m+1$▫ extends to a norm-preserving asymptotically Lipschitz function from ▫$X$▫ to the Euclidean space of dimension ▫$m+1$▫. As an application we obtain another proof of the following result of Dranishnikov and Smith: Let ▫$X$▫ be a cocompact proper metric space, which is ▫$M$▫-connected for some $M$, and has the asymptotic Assouad-Nagata dimension finite. Then this dimension equals the dimension of the sublinear Higson corona of ▫$X$▫.
COBISS.SI-ID: 16135001