Some remarks on the parametrized Borsuk–Ulam theorem

Given a locally trivial fibre bundle $$E\rightarrow B$$ (with fibres and base finite complexes), an orthogonal real line bundle $$\lambda $$ over E and a real vector bundle $$\xi $$ over B, we consider a fibrewise map $$f: S(\lambda ) \rightarrow \xi $$ over B defined on the unit sphere bundle of $$\lambda $$ . 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 $$\{ v\in S(\lambda ) \vert f(v)=f(-v)\}$$ .