2016/12/30 by Alexandru Chirvăsitu, Chirvasitu, Alexandru, Ludwik Dąbrowski +3 · 1 citation
Mathematics · #22C05 #54H15 #57S10 #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #General Topology (math.GN) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.1612.09567
openalex publication_date 2016/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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→ 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The proof relies on the non-existence of G-equivariant maps G*(n+1)→ G*n, which in turn is a slight strengthening of an unpublished result of M. Bestvina and R. Edwards.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.