The Borsuk-Ulam property for homotopy classes on bundles, parametrized braids groups and applications for surfaces bundles (Q6659419)

The paper under review explored the Borsuk-Ulam property for homotopy classes on fiber bundles, parametrized braids groups and give applications to surfaces bundles. Let \(M\) and \(N\) be fiber bundles over the same base \(B\), where \(M\) is equipped with a free involution \(\tau\) over \(B\). Then a homotopy class \(\delta \in [M,N]_B\) is said to have the Borsuk-Ulam property with respect to the involution \(\tau\) if for every fiber-preserving continuous map \(f : M \to N\) which represents the class \(\delta\), there exists a point \(x \in M\) such that \(f(\tau(x)) = f(x)\). Suppose that \(B\) is a \(K(\pi,1)\)-space and the fibers of the bundle projections \(M\to B\) and \(N\to B\) are \(K(\pi, 1)\) closed surfaces. Then, the authors show that the problem to decide if a homotopy class of a fiber-preserving map \(f : M \to N\) has the Borsuk-Ulam property is equivalent of an algebraic problem involving the fundamental groups of \(M\), the orbit space \(M/\tau\) and a type of generalized braid groups of \(N\), that the authors refer as parametrized braid groups. As an application, the authors determine the homotopy classes of fiber-preserving self maps over the circle that satisfy the Borsuk-Ulam property with respect to all involutions.