Some remarks on the parametrized Borsuk-Ulam theorem

Abstract

Given a locally trivial fibre bundle $E\to B$ (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 $f:S(\lambda )\to \xi$ 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 of points $v$ in $S(\lambda )$ such that $f(v)=f(-v)$.

15 pages

References in corpus (2)

• Projective product spaces
• Parametrized Borsuk-Ulam problem for projective space bundles