  Let M and N be topological spaces such that M admits a free involution
$\\tau$. A homotopy class $\beta$ $\in$ [M, N ] is said to have the Borsuk-Ulam
property with respect to $\\tau$ if for every representative map f : M
$\rightarrow$ N of $\beta$, there exists a point x $\in$ M such that f ($\\tau$
(x)) = f (x). In the case where M is a compact, connected manifold without
boundary and N is a compact, connected surface without boundary different from
the 2-sphere and the real projective plane, we formulate this property in terms
of the pure and full 2-string braid groups of N , and of the fundamental groups
of M and the orbit space of M with respect to the action of $\\tau$. If M = N
is either the 2-torus T^2 or the Klein bottle K^2 , we then solve the problem
of deciding which homotopy classes of [M, M ] have the Borsuk-Ulam property.
First, if $\\tau$ : T^2 $\rightarrow$ T^2 is a free involution that preserves
orientation, we show that no homotopy class of [T^2 , T^2 ] has the Borsuk-Ulam
property with respect to $\\tau$. Secondly, we prove that up to a certain
equivalence relation, there is only one class of free involutions $\\tau$ : T^2
$\rightarrow$ T^2 that reverse orientation, and for such involutions, we
classify the homotopy classes in [T^2 , T^2 ] that have the Borsuk-Ulam
property with respect to $\\tau$ in terms of the induced homomorphism on the
fundamental group. Finally, we show that if $\\tau$ : K^2 $\rightarrow$ K^2 is
a free involution, then a homotopy class of [K^2 , K^2 ] has the Borsuk-Ulam
property with respect to $\\tau$ if and only if the given homotopy class lifts
to the torus.
