Borsuk-Ulam property and Sectional Category

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 application of our results, we present a new lower bound for the index in terms of sectional category. We present several examples and for those the lower bound coincides with sectional category minus 1. We conjecture that the index of $(M,\tau)$ coincides with the sectional category of the quotient map $q:M\to M/\tau$ minus 1 for any CW complex $M$.Comment: 16 pages. Comments are welcome. Minor change