The Bestvina-Edwards theorem and the Hilbert-Smith conjecture
type: Item
instance of: scholarly article
title: The Bestvina-Edwards theorem and the Hilbert-Smith conjecture
DOI: 10.1215/21562261-2022-0015
author: Alexandru Chirvasitu
author: M. Tobolski
author: Ludwik Dąbrowski
published in: Kyoto Journal of Mathematics
publication date: 2022-10-27T00:00:00Z
review text: The generalized compact-Hausdorff-space Borsuk-Ulam conjecture proposed by \textit{P. F. Baum} et al. [Banach Cent. Publ. 106, 9--18 (2015; Zbl 1343.46064)] states the following: \textbf{Conjecture.} Let \(X\) be a compact Hausdorff space equipped with a continuous free action of a nontrivial compact Hausdorff group \(G\). Then for the diagonal action of \(G\) on the topological join \(X \ast G\), there does not exist a \(G\)-equivariant continuous map \(f : X \ast G \to X\). Note that the BDH conjecture is known to be true for actions of compact Hausdorff groups with torsion elements on compact Hausdorff spaces. In the paper under review, the authors prove a number of interesting results surrounding the BDH conjecture. More precisely, in Section~1, the authors prove the generalized Bestvina-Edwards theorem (Theorem 1.1): \textbf{Theorem.} Let \(G\) be a nontrivial compact Hausdorff group, and for positive integers let \(n\) denote \(E_n G := G^{\ast (n+1)}\). Then there are no \(G\)-equivariant maps \(E_{n+1} G \to E_N G\). This theorem is used in Section 2 to prove the BDH-conjecture under the additional assumption that the canonical quotient map \(X \to X/G\) is a locally trivial principal bundle (Theorem 2.1). Section 3 focuses on the weaker version of the Hilbert-Smith conjecture: \textbf{The weak Hilbert-Smith conjecture.} Let \(G\) be a locally compact group acting freely and properly on a connected finite-dimensional topological manifold \(M\) such that the orbit space \(M/G\) is finite- dimensional. Then \(G\) is a Lie group. The authors show in Theorem 3.10 that the weak Hilbert-Smith conjecture can be reduced to the case of \(p\)-adic integers which reads as: \textbf{The weak Hilbert-Smith conjecture for $p$-adic actions.} Let \(p\) be a prime number, and let \(M\) be a connected finite-dimensional topological manifold. Then the groups \(A_p\) of \(p\)-adic integers cannot act freely on \(M\) such that \(M/A_p\) is finite-dimensional. Finally, in Section …
reviewed by: Stefan Wagner
Mathematics Subject Classification ID: 57S10
Mathematics Subject Classification ID: 46L85
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54H15
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 22C05
zbMATH DE Number: 7608273
zbMATH Keywords: Ageev conjecture
zbMATH Keywords: Borsuk-Ulam theorem
zbMATH Keywords: dimension
zbMATH Keywords: free action
zbMATH Keywords: Hilbert-Smith conjecture
zbMATH Keywords: Menger compactum
zbMATH Keywords: \(p\)-adic integers
Wikidata QID: Q113751492
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/21562261-2022-0015
OpenAlex ID: W4285612517
cites work: CLASSIFYING SPACES FOR FREE ACTIONS, AND THE HILBERT-SMITH CONJECTURE
cites work: A characterization of the free actions of zero-dimensional compact groups on $ k$-dimensional Menger compacta
cites work: Dimensionstheorie. Ein Beitrag zur Geometrie der abgeschlossenen Mengen
cites work: Invariant pseudometrics on Palais proper \(G\)-spaces
cites work: Dimension of proper \(G\)-spaces
cites work: Noncommutative Borsuk-Ulam-type conjectures
cites work: Characterizing 𝑘-dimensional universal Menger compacta
cites work: Une caractérisation des rétractes absolus de voisinage
cites work: Compact group actions on topological and noncommutative joins
cites work: Q4421932
cites work: ON FREE ACTIONS OF ZERO-DIMENSIONAL COMPACT GROUPS
cites work: On some problems related to the Hilbert-Smith conjecture
cites work: Q4185470
cites work: Groups without small subgroups
cites work: Mathematical problems
cites work: The structure of compact groups. A primer for the student -- A handbook for the expert
cites work: Q5522732
cites work: Q5837247
cites work: On flat bundles
cites work: Q4954867
cites work: The Hilbert-Smith conjecture for Hölder actions
cites work: The Hilbert-Smith conjecture for quasiconformal actions
cites work: Q3999514
cites work: Using the Borsuk-Ulam theorem. Lectures on topological methods in combinatorics and geometry. Written in cooperation with Anders Björner and Günter M. Ziegler
cites work: Small subgroups of finite-dimensional groups
cites work: Q4086457
cites work: Local Cross Sections in Locally Compact Groups
cites work: A THEOREM ON PERIODIC TRANSFORMATIONS OF SPACES
cites work: On the existence of slices for actions of non-compact Lie groups
cites work: The Hilbert–Smith conjecture for three-manifolds
cites work: Free actions on C*-algebra suspensions and joins by finite cyclic groups
cites work: A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps
cites work: Q5846106
cites work: Q4950141
cites work: Hilbert's fifth problem and related topics
cites work: Coincidence points of maps of $ \Bbb Z_p^n$-spaces
cites work: Ext(Q,Z) is the additive group of real numbers
cites work: A generalization of a theorem of Gleason
cites work: \(p\)-adic transformation groups
cites work: A construction of classifying spaces for \(p\)-adic group actions
description: scientific article; zbMATH DE number 7608273