The Borsuk-Ulam theorem for maps into a surface

Let (X, tau, S) be a triple, where S is a compact, connected surface without boundary, and tau is a free cellular involution on a CW-complex X. The triple (X, tau, S) is said to satisfy the Borsuk-Ulam property if for every continuous map f : X -> S. there exists a point x is an element of X satisfying f(tau(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, tau, 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 pi(1)(X/tau) 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 RP(2), then we show that the Borsuk-Ulam property does not hold for (X, tau, S) unless either pi(1)(X/tau) congruent to pi(1)(RP(2)), pi(1)(X/tau) 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 tau: we give a necessary and sufficient condition for it to hold in terms of the surjective homomorphism pi(1)(X/tau) -> Z(2) induced by the double covering X -> X/tau. The cases S =S(2), RP(2) are treated separately. (C) 2010 Elsevier B.V. All rights reserved.