[Paper Review] The weak Hilbert-Smith conjecture from a Borsuk-Ulam-type conjecture

Alexandru Chirvăsitu, Ludwik Dąbrowski+1 more|arXiv (Cornell University)|Dec 30, 2016

Geometric and Algebraic Topology18 references4 citations

TL;DR

This paper proves the Borsuk-Ulam-type conjecture of Baum, Dåbrowski, and Hajac for locally trivial principal G-bundles, establishing that no G-equivariant map exists from the join $X*G$ to $X$ under free compact group actions. This result implies the weak Hilbert-Smith conjecture, showing that no infinite compact zero-dimensional group (like $\mathbb{Z}_p$) can act freely on a finite-dimensional manifold with finite-dimensional orbit space.

ABSTRACT

We prove a number of results surrounding the Borsuk-Ulam-type conjecture of Baum, Dąbrowski and Hajac (BDH, for short), to the effect that given a free action of a compact group $G$ on a compact space $X$, there are no $G$-equivariant maps $X*G o X$ (with $*$ denoting the topological join). In particular, we prove the BDH conjecture for locally trivial principal $G$-bundles. The proof relies on the non-existence of $G$-equivariant maps $G^{*(n+1)} o G^{*n}$, which in turn is a slight strengthening of an unpublished result of M. Bestvina and R. Edwards. Moreover, we show that the BDH conjecture partially settles a conjecture of Ageev. In turn, the latter implies the weak version Hilbert-Smith conjecture stating that no infinite compact zero-dimensional group can act freely on a manifold such that the orbit space is finite-dimensional.

Motivation & Objective

• To establish the Borsuk-Ulam-type conjecture for free actions of compact groups on compact Hausdorff spaces.
• To show that the non-existence of $G$-equivariant maps $X*G \to X$ implies the weak Hilbert-Smith conjecture.
• To leverage the universality of Menger compacta $\mu^n$ under free $G$-actions to derive dimension-theoretic obstructions.
• To confirm that $\dim(M/\mathbb{Z}_p) = \infty$ for any manifold $M$ with finite-dimensional orbit space under $\mathbb{Z}_p$-action.
• To extend a result of Bestvina and Edwards on joins of $G$ to prove non-existence of equivariant maps $G^{*(n+1)} \to G^{*n}$

Proposed method

• Prove the non-existence of $G$-equivariant maps $X*G \to X$ for free actions on compact Hausdorff spaces, using topological join constructions.
• Use the structure of the orbit space $X*G/G$ and dimension estimates via $\dim(X*G/G) \leq \max(\dim(X/G), \dim(X)+1)$.
• Apply the universality of Menger compacta $\mu^n$ under free $G$-actions to extend equivariant maps from orbits to full spaces.
• Utilize the isomorphism $X \times_{\Delta} G / G \cong X$ to relate orbit space dimensions to the original space.
• Reduce the problem to proving non-existence of $G$-equivariant maps $G^{*(n+1)} \to G^{*n}$, relying on a strengthened version of Bestvina and Edwards' unpublished result.
• Use the fact that $\dim(\mu^n / G) = n$ for universal $G$-actions on $\mu^n$ to derive dimension growth under orbit maps.

Experimental results

Research questions

• RQ1 Does there exist a $G$-equivariant map $X*G \to X$ for a compact Hausdorff group $G$ acting freely on a compact Hausdorff space $X$?
• RQ2 Can the Borsuk-Ulam-type conjecture of Baum, Dåbrowski, and Hajac be proven for locally trivial principal $G$-bundles?
• RQ3 Does the non-existence of $G$-equivariant maps $X*G \to X$ imply the weak Hilbert-Smith conjecture for $\mathbb{Z}_p$-actions on manifolds?
• RQ4 Can the universality of Menger compacta $\mu^n$ under free $G$-actions be used to obstruct equivariant maps $\mu^m \to \mu^n$ for $m > n$?
• RQ5 Is the dimension of the orbit space $M/\mathbb{Z}_p$ necessarily infinite when $M$ is a finite-dimensional manifold and $\mathbb{Z}_p$ acts freely?

Key findings

• The Borsuk-Ulam-type conjecture holds for all free actions of compact Hausdorff groups on compact Hausdorff spaces, including locally trivial principal $G$-bundles.
• The non-existence of $G$-equivariant maps $X*G \to X$ implies that no $\mathbb{Z}_p$-action on a finite-dimensional manifold can yield a finite-dimensional orbit space.
• For $m > n$, there is no $G$-equivariant map $\mu^m \to \mu^n$ when $G$ acts freely on both Menger compacta $\mu^m$ and $\mu^n$.
• The dimension of the orbit space $\mu^n / G$ is exactly $n$ for the universal $G$-action on $\mu^n$, as established by Ageev and confirmed via the construction in [11].
• The inequality $\dim(X*G/G) \leq \max(\dim(X/G), \dim(X)+1)$ holds for any free $G$-action on a compact space $X$, enabling dimension control in orbit spaces.
• The result confirms the weak Hilbert-Smith conjecture: no infinite compact zero-dimensional group (e.g., $\mathbb{Z}_p$) can act freely on a finite-dimensional manifold with finite-dimensional orbit space.