Sectional Category and the parametrized Borsuk-Ulam property

Abstract

In this paper, for fibrations $p:E\to B$, $p':E'\to B$ over $B$ and a free involution $τ:E\to E$ which satisfy the equality $p\circτ=p$, we discover a connection between the sectional category of the double covers $q:E\to E/τ$ and $q^Y:F_{p'}(E',2)\to F_{p'}(E',2)/\mathbb{Z}_2$ from the $2$-ordered fibre-wise configuration space $F_{p'}(E',2)$ to its unordered quotient $ F_{p'}(E',2)/\mathbb{Z}_2$, and the parametrized Borsuk-Ulam property (PBUP) for the triple $\left((p,τ);p'\right)$. Explicitly, we demonstrate that the triple $\left((p,τ);p'\right)$ satisfies the PBUP if the sectional category of $q$ is bigger than the sectional category of $q^{p'}$. This property connects a standard problem in parametrized Borsuk-Ulam theory to current research trends in sectional category. As applictions of our results, we explore the PBUP for $E, E'$ one of the following fibrations: trivial fibration, Hopf fibration and the Fadell-Neuwirth fibration.