The Bestvina-Edwards theorem and the Hilbert-Smith conjecture
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 4 it is shown that the positive answer to the metrizable case (with \(G\) and \(X\) metrizable) of the BDH conjecture implies the stronger form of the weak Hilbert-Smith conjecture for \(p\)-adic actions with any ANR (for absolute neighbourhood retract) instead of a manifold (Corollary 4.3).
scientific article; zbMATH DE number 7608273
scientific article