The Borsuk-Ulam theorem for maps into a surface

Let (X, τ, S) be a triple, where S is a compact, connected surface without boundary, and τ is a free cellular involution on a CW -complex X. The triple (X, τ, S) is said to satisfy the Borsuk-Ulam property if for every continuous map f : X −→ S, there exists a point x ∈ X satisfying f(τ(x)) = f(x). In this paper, we formulate this property in terms of a relation in the 2-string braid group B2(S) of S. If X is a compact, connected surface without boundary, we use this criterion to classify all triples (X, τ, S) for which the Borsuk-Ulam property holds. We also consider various cases where X is not necessarily a surface without boundary, but has the property that π1(X/τ) is isomorphic to the fundamental group of such a surface. If S is different from the 2-sphere S and the real projective plane RP , then we show that the Borsuk-Ulam property does not hold for (X, τ, S) unless either π1(X/τ) ∼= π1(RP ), or π1(X/τ) is isomorphic to the fundamental group of a compact, connected non-orientable surface of genus 2 or 3 and S is non-orientable. In the latter case, the veracity of the Borsuk-Ulam property depends further on the choice of involution τ ; we give a necessary and sufficient condition for it to hold 1 ha l-0 04 63 15 6, v er si on 1 11 M ar 2 01 0 in terms of the surjective homomorphism π1(X/τ) −→ Z2 induced by the double covering X −→ X/τ . The cases S = S,RP 2 are treated separately.