The weak Hilbert-Smith conjecture from a Borsuk-Ulam-type conjecture

Abstract

We prove a number of results surrounding the Borsuk-Ulam-type conjecture of Baum, D\k{a}browski 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 $X*G\to X$ (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on the non-existence of $G$-equivariant maps $G^{*(n+1)}\to G^{*n}$, which in turn is a slight strengthening of an unpublished result of M. Bestvina and R. Edwards.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.