Some remarks on the parametrized Borsuk–Ulam theorem

Abstract

Given a locally trivial fibre bundle 𝐸→𝐵 (with fibres and base finite complexes), an orthogonal real line bundle 𝜆 over E and a real vector bundle 𝜉 over B, we consider a fibrewise map 𝑓:𝑆(𝜆)→𝜉 over B defined on the unit sphere bundle of 𝜆. Following the fundamental work of Jaworowski and Dold on the parametrized Borsuk–Ulam theorem, we investigate lower bounds on the cohomological dimension of the set {𝑣∈𝑆(𝜆)|𝑓(𝑣)=𝑓(−𝑣)}.