  Let (X, t, S) be a triple, where S is a compact, connected surface without
boundary, and t is a free cellular involution on a CW-complex X. The triple (X,
t, S) is said to satisfy the Borsuk-Ulam property if for every continuous map
f:X-->S, there exists a point x belonging to X satisfying f(t(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, t, 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/t) 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, t, S) unless either \pi_1(X/t) is isomorphic to
\pi_1(RP^2), or \pi_1(X/t) is isomorphic to the fundamental group of a compact,
connected non-orientable surface of genus 2 or 3 and S is orientable. In the
latter case, the veracity of the Borsuk-Ulam property depends further on the
choice of involution t; we give a necessary and sufficient condition for it to
hold in terms of the surjective homomorphism \pi_1(X/t)-->Z_2 induced by the
double covering X-->X/t. The cases S=S^2,RP^2 are treated separately.
