We prove a number of results surrounding the Borsuk–Ulam-type conjecture of Baum, Dabrowski, and Hajac (BDH, for short), which states that given a free action of a compact group G on a compact space X, there are no G-equivariant maps (with ∗ denoting the topological join). Mainly, we prove the BDH conjecture for locally trivial principal G-bundles. The proof relies on the nonexistence of G-equivariant maps , which in turn is a strengthening of an unpublished result of Bestvina and Edwards. Moreover, we show that the BDH conjecture partially settles a conjecture of Ageev which implies the weak version of the Hilbert–Smith conjecture stating that no infinite compact zero-dimensional group can act freely on a manifold so that the orbit space is finite-dimensional.
Kyoto J. Math.
62(3):
523-545
(September 2022).
DOI: 10.1215/21562261-2022-0015
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