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 S2 and the real projective plane RP2, then we show that the Borsuk–Ulam property does not hold for (X,τ,S) unless either π1(X/τ)≅π1(RP2), 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 in terms of the surjective homomorphism π1(X/τ)→Z2 induced by the double covering X→X/τ. The cases S=S2,RP2 are treated separately.