Let M and N be topological spaces such that M admits a free involution τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}. A homotopy class β∈[M,N]\documentclass[12pt]{minimal}
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\begin{document}$$\beta \in [ M , N ] $$\end{document} is said to have the Borsuk–Ulam property with respect toτ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} if for every representative map f:M→N\documentclass[12pt]{minimal}
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\begin{document}$$f:\,M \rightarrow N$$\end{document} of β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document}, there exists a point x∈M\documentclass[12pt]{minimal}
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\begin{document}$$x \in M$$\end{document} such that f(τ(x))=f(x)\documentclass[12pt]{minimal}
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\begin{document}$$f ( \tau ( x) ) = f(x)$$\end{document}. 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 τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}. If M=N\documentclass[12pt]{minimal}
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\begin{document}$$M=N$$\end{document} is either the 2-torus T2\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {T}^2$$\end{document} or the Klein bottle K2\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {K}^2$$\end{document}, we then solve the problem of deciding which homotopy classes of [M, M] have the Borsuk–Ulam property. First, if τ:T2→T2\documentclass[12pt]{minimal}
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\begin{document}$$\tau :\,\mathbb {T}^2\rightarrow \mathbb {T}^2$$\end{document} is a free involution that preserves orientation, we show that no homotopy class of [T2,T2]\documentclass[12pt]{minimal}
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\begin{document}$$[ \mathbb {T}^2, \mathbb {T}^2]$$\end{document} has the Borsuk–Ulam property with respect to τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}. Second, we prove that up to a certain equivalence relation, there is only one class of free involutions τ:T2→T2\documentclass[12pt]{minimal}
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\begin{document}$$\tau :\,\mathbb {T}^2\rightarrow \mathbb {T}^2$$\end{document} that reverse orientation, and for such involutions, we classify the homotopy classes in [T2,T2]\documentclass[12pt]{minimal}
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\begin{document}$$[\mathbb {T}^2, \mathbb {T}^2]$$\end{document} that have the Borsuk–Ulam property with respect to τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} in terms of the induced homomorphism on the fundamental group. Finally, we show that if τ:K2→K2\documentclass[12pt]{minimal}
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\begin{document}$$\tau :\,\mathbb {K}^2\rightarrow \mathbb {K}^2$$\end{document} is a free involution, then a homotopy class of [K2,K2]\documentclass[12pt]{minimal}
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\begin{document}$$[\mathbb {K}^2, \mathbb {K}^2]$$\end{document} has the Borsuk–Ulam property with respect to τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} if and only if the given homotopy class lifts to the torus.
