Borsuk–Ulam Property and Sectional Category

For a Hausdorff space X, a free involution τ : X → X and a Hausdorff space Y , we discover a connection between the sectional category of the double covers q : X → X/τ and qY : F(Y , 2) → D(Y , 2) from the ordered configuration space F(Y , 2) to its unordered quotient D(Y , 2) = F(Y , 2)/ 2, and the Borsuk–Ulam property (BUP) for the triple ((X, τ); Y ). Explicitly, we demonstrate that the triple ((X, τ); Y ) satisfies the BUP if the sectional category of q is bigger than the sectional category of qY . This property connects a standard problem in Borsuk–Ulam to current research trends in sectional category. As an application of our results, we present a new lower bound for the index in terms of sectional category. We present several examples for whom the lower bound coincides with sectional category minus 1.We conjecture that the index of (M, τ) coincides with the sectional category of the quotient map q : M → M/τ minus 1 for any CW complex M.