The Borsuk-Ulam property for maps from the product of two surfaces into a surface

Abstract

Let $X$, $Y$, $S$ be closed connected surfaces and $\tau \times \beta$ a diagonal involution on $X \times Y$ where $\tau$ and $\beta$ are free involutions on $X$ and $Y$, respectively. In this work we study when the triple $(X \times Y, \tau \times \beta, S)$ satisfies the Borsuk-Ulam property. The problem is formulated in terms of an algebraic diagram, involving the 2-string braid group $B_{2}(S)$.