Metric trees of generalized roundness one

Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable metric trees that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual combinatorial metric (SSTs). Using a simple geometric argument we show how to determine decent upper bounds on the generalized roundness of finite SSTs that depend only on the downward degree sequence of the tree in question. By considering limits it follows that if the downward degree sequence $(d_{0}, d_{1}, d_{2}...)$ of a SST $(T,\rho)$ satisfies $|\{j \, | \, d_{j} > 1 \}| = \aleph_{0}$, then $(T,\rho)$ has generalized roundness one. Included among the trees that satisfy this condition are all complete $n$-ary trees of depth $\infty$ ($n \geq 2$), all $k$-regular trees ($k \geq 3$) and inductive limits of Cantor trees. The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs.Comment: 14 pages, 2 figures, 2 table

On manifolds with nonhomogeneous factors

We present simple examples of finite-dimensional connected homogeneous spaces (they are actually topological manifolds) with nonhomogeneous and nonrigid factors. In particular, we give an elementary solution of an old problem in general topology concerning homogeneous spaces

Rokhlin Dimension for Flows

This research was supported by GIF Grant 1137/2011, SFB 878 Groups, Geometry and Actions and ERC Grant No. 267079. Part of the research was conducted at the Fields institute during the 2014 thematic program on abstract harmonic analysis, Banach and operator algebras, and at the Mittag–Leffler institute during the 2016 program on Classification of Operator Algebras: Complexity, Rigidity, and Dynamics.Peer reviewedPostprin

A fibration that does not accept two disjoint many-valued sections

AbstractThe fibration η:∏∞i=0S(2i) → ∏∞i=1RP(2i) does not accept two disjoint many-valued sections

Extension of maps to nilpotent spaces. II

AbstractLet M be a nilpotent CW-complex with finitely generated fundamental group. We give necessary and sufficient cohomological dimension theory conditions for a finite-dimensional metric compactum X so that every map A→M, where A is a closed subset of X can be extended to a map X→M.This is a generalization of a result by Dranishnikov [Mat. Sb. 182 (1991)] where such conditions were found for simply-connected CW-complexes M, and Cencelj and Dranishnikov forthcoming paper [Cannad. Bull. Math.] where such conditions were found for nilpotent CW-complexes M with finitely generated homotopy groups