Some remarks on the parametrized Borsuk-Ulam theorem

TL;DR

This paper explores lower bounds on the cohomological dimension of points where a fibrewise map on a sphere bundle equates to its antipodal point, extending the parametrized Borsuk-Ulam theorem in fiber bundle contexts.

Contribution

It provides new lower bounds for the cohomological dimension of symmetric point sets in fiber bundles, generalizing previous results in the parametrized Borsuk-Ulam theorem.

Findings

Derived bounds on cohomological dimensions of symmetric point sets.

Extended the parametrized Borsuk-Ulam theorem to more general fiber bundle settings.

Connected the theorem to cohomological methods in topology.

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 $f(v) = f(-v)$.