Sectional Category and the parametrized Borsuk-Ulam property

Abstract

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

20 pages. Comments are welcome