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 τ over B. A homotopy class δ∈[M,N]B (over B) is said to have the Borsuk–Ulam property with respect to τ if for every fiber-preserving map f:M→N over B which represents δ there exists a point x∈M such that f(τ(x))=f(x). In the cases that B is a K(π,1)-space and the fibers of the projections M→B and N→B are K(π,1) closed surfaces SM and SN, respectively, we show that the problem of decide if a homotopy class of a fiber-preserving map f:M→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 τ and a type of generalized braid groups of N that we call parametrized braid groups. As an application, we determine the homotopy classes of fiber-preserving self maps over S1 that satisfy the Borsuk-Ulam property, with respect to all involutions τ over S1, for the torus bundles over S1 with M=N=MA and A=[1n01].