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 r over B . A homotopy class delta E [M, N I B (over B ) is said to have the Borsuk-Ulam property with respect to r if for every fiber-preserving map f : M-* N over B which represents delta there exists a point x E M such that f (r(x)) = f (x). In the cases that B is a K(7r, 1)-space and the fibers of the projections M-* B and N-* B are K(7r, 1) closed surfaces SM and S N , 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 r 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 S 1 that satisfy the Borsuk-Ulam property, with respect to all involutions r over S 1 , for the torus 1 n ] bundles over S 1 with M = N = MA and A = . 0 1 (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.