A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it has pro-(finitely generated free) fundamental group at infinity and semi-stable ends. Conjecturally the quasi-simply filtration assumption is superfluous. Using these restrictions we provide the first examples of finitely presented groups which are not properly 3-realizable, for instance large families of Coxeter groups.
COBISS.SI-ID: 16297817
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 apply a recently obtained three-critical-point theorem of B. Ricceri to prove the existence of at least three solutions of certain two-parameter Dirichlet problems defined on the Sierpinski gasket. We also show the existence of at least three nonzero solutions of certain perturbed two-parameter Dirichlet problems on the Sierpinski gasket, using both the mountain pass theorem of Ambrosetti and Rabinowitz and that of Pucci and Serrin.
COBISS.SI-ID: 15657049
We extend the definition of Bockstein basis \sigma(G) to nilpotent groups G. A metrizable space X is called a Bockstein space if \dim_G(X) = \sup\{\dim_H(X)\vert H \in \sigma(G)\} for all Abelian groups G. The Bockstein First Theorem says that all compact spaces are Bockstein spaces. Here are the main results of the paper: Theorem 0.1. Let X be a Bockstein space. If G is nilpotent, then \dim_G(X) \le 1 if and only if \sup \{ \dim_H(X)\vert H \in \sigma(G)\} \le 1. Theorem 0.2. X is a Bockstein space if and only if \dim_{Z(l)}(X) = \dim_{\hat{Z}(l)}(X) for all subsets l of prime numbers.
COBISS.SI-ID: 15493977