The Borsuk-Ulam theorem for surfaces

Abstract

Given a pair (S, T) whereS is a closed surface and T is a free Z ₂ action on S, we classify which pairs have the property that the Borsuk-Ulam theorem holds with respect to R ². In particular if S is orientable and its Euler characteristic is congruent to 2 mod 4, this is always the case independent of the action. We say that the Borsuk-Ulam theorem holds for (S, T) with respect to R ² if for any map f : S → R ² there is a point x ∈ S such that f(x) = f(T(x)).