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 B 2 ( 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 π 1 ( X / τ ) is isomorphic to the fundamental group of such a surface. If S is different from the 2-sphere S 2 and the real projective plane R P 2 , then we show that the Borsuk–Ulam property does not hold for ( X , τ , S ) unless either π 1 ( X / τ ) ≅ π 1 ( R P 2 ) , or π 1 ( 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 π 1 ( X / τ ) → Z 2 induced by the double covering X → X / τ . The cases S = S 2 , R P 2 are treated separately.