Borsuk-Ulam property and Sectional Category

TL;DR

This paper establishes a new link between the Borsuk-Ulam property and sectional category, providing criteria for the property based on sectional categories of certain double covers, with applications to paracompact spaces.

Contribution

It introduces a novel connection between Borsuk-Ulam theory and sectional category, offering new criteria and results for understanding the property in various spaces.

Findings

The Borsuk-Ulam property holds when the sectional category of the double cover exceeds that of the configuration space.

The index of a space with involution equals the sectional category of its quotient map minus one.

New relationships between Borsuk-Ulam theory and sectional category are established.

Abstract

For a Hausdorff space $X$, a free involution $\tau:X\to X$ and a Hausdorff space $Y$, we discover a connection between the sectional category of the double covers $q:X\to X/\tau$ and $q^Y:F(Y,2)\to D(Y,2)$ from the ordered configuration space $F(Y,2)$ to its unordered quotient $D(Y,2)=F(Y,2)/\Sigma_2$, and the Borsuk-Ulam property (BUP) for the triple $\left((X,\tau);Y\right)$. Explicitly, we demonstrate that the triple $\left((X,\tau);Y\right)$ satisfies the BUP if the sectional category of $q$ is bigger than the sectional category of $q^Y$. This property connects a standard problem in Borsuk-Ulam theory to current research trends in sectional category. As an application of our results, we show that the index of $(X,\tau)$ coincides with the sectional category of the quotient map $q:X\to X/\tau$ minus 1 for any paracompact space $X$. In addition, we present some new results relating Borsuk-Ulam theory and sectional category.