The Borsuk-Ulam property for homotopy classes on bundles, parametrized braids groups and applications for surfaces bundles

Let \(M\) and \(N\) be fiber bundles over the same base \(B\), where \(M\) is endowed with a free involution \(\tau\) over \(B\). A homotopy class \(\delta \in [M,N]_{B}\) (over \(B\)) is said to have the Borsuk-Ulam property with respect to \(\tau\) if for every fiber-preserving map \(f\colon M \to N\) over \(B\) which represents \(\delta\) there exists a point \(x \in M\) such that \(f(\tau(x)) = f(x)\). In the cases that \(B\) is a \(K(\pi ,1)\)-space and the fibers of the projections \(M \to B\) and \(N \to B\) are \(K(\pi,1)\) closed surfaces \(S_M\) and \(S_N\), respectively, we show that the problem of decide if a homotopy class of a fiber-preserving map \(f\colon M \to N\) over \(B\) has the Borsuk-Ulam property is equivalent of an algebraic problem involving the fundamental groups of \(M\), the orbit space of \(M\) by \(\tau\) and a type of generalized braid groups of \(N\) that we call parametrized braid groups. As an application, we determine the homotopy classes of self fiber-preserving maps of some 2-torus bundles over \(\mathbb{S}^1\) that satisfy the Borsuk-Ulam property with respect to certain involutions \(\tau\) over \(\mathbb{S}^1\).