Sectional Category and the parametrized Borsuk-Ulam property

In this paper, for fibrations p:E→ B, p':E'→ B over B and a free involution τ:E→ E which satisfy the equality p∘τ=p, we discover a connection between the sectional category of the double covers q:E→ E/τ and qY:Fp'(E',2)→ Fp'(E',2)/Z₂ from the $2$-ordered fibre-wise configuration space Fp'(E',2) to its unordered quotient Fp'(E',2)/Z₂, and the parametrized Borsuk-Ulam property (PBUP) for the triple ((p,τ);p'). Explicitly, we demonstrate that the triple ((p,τ);p') satisfies the PBUP if the sectional category of q is bigger than the sectional category of qp'. 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.