The topology of systems of hyperspaces determined by dimension functions

Given a non-degenerate Peano continuum $X$, a dimension function $D:2^X_*\to[0,\infty]$ defined on the family $2^X_*$ of compact subsets of $X$, and a subset $\Gamma\subset[0,\infty)$, we recognize the topological structure of the system (2^X,\D_{\le\gamma}(X))_{\alpha\in\Gamma}, where $2^X$ is the hyperspace of non-empty compact subsets of $X$ and $D_{\le\gamma}(X)$ is the subspace of $2^X$, consisting of non-empty compact subsets $K\subset X$ with $D(K)\le\gamma$.Comment: 12 page

End sums of irreducible open 3-manifolds

An end sum is a non-compact analogue of a connected sum. Suppose we are given two connected, oriented $n$-manifolds $M_1$ and $M_2$. Recall that to form their connected sum one chooses an $n$-ball in each $M_i$, removes its interior, and then glues together the two $S^{n-1}$ boundary components thus created by an orientation reversing homeomorphism. Now suppose that $M_1$ and $M_2$ are also open, i.e. non-compact with empty boundary. To form an end sum of $M_1$ and $M_2$ one chooses a halfspace $H_i$ (a manifold \homeo\ to ${\bold R}^{n-1} \times [0, \infty)$) embedded in $M_i$, removes its interior, and then glues together the two resulting ${\bold R}^{n-1}$ boundary components by an orientation reversing homeomorphism. In order for this space $M$ to be an $n$-manifold one requires that each $H_i$ be {\bf end-proper} in $M_i$ in the sense that its intersection with each compact subset of $M_i$ is compact. Note that one can regard $H_i$ as a regular neighborhood of an end-proper ray (a 1-manifold \homeo\ to $[0,\infty)$) \ga_i in $M_i$

Normal Smoothings for Smooth Cube Manifolds

We prove that smooth cube manifolds have normal smooth structures.Comment: The paper "Pinched Smooth Hyperbolization" [arXiv:1110.6374] has been divided in parts. "Normal Smoothings for Smooth Cube Manifolds" is one of the part

Isometry groups among topological groups

It is shown that a topological group G is topologically isomorphic to the isometry group of a (complete) metric space iff G coincides with its G-delta-closure in the Rajkov completion of G (resp. if G is Rajkov-complete). It is also shown that for every Polish (resp. compact Polish; locally compact Polish) group G there is a complete (resp. proper) metric d on X inducing the topology of X such that G is isomorphic to Iso(X,d) where X = l_2 (resp. X = Q; X = Q\{point} where Q is the Hilbert cube). It is demonstrated that there are a separable Banach space E and a nonzero vector e in E such that G is isomorphic to the group of all (linear) isometries of E which leave the point e fixed. Similar results are proved for an arbitrary complete topological group.Comment: 30 page

Complete bipartite graphs whose topological symmetry groups are polyhedral

We determine for which $n$, the complete bipartite graph $K_{n,n}$ has an embedding in $S^3$ whose topological symmetry group is isomorphic to one of the polyhedral groups: $A_4$, $A_5$, or $S_4$.Comment: 25 pages, 6 figures, latest version has minor edits in preparation for submissio