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

Abstract

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 $\delta \in {[M,N]}_B$ (over B) is said to have the Borsuk–Ulam property with respect to τ if for every fiber-preserving map $f:M\to N$ over B which represents δ there exists a point $x\in M$ such that $f\left(\tau \right(x\left)\right)=f\left(x\right)$. 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 $f:M\to N$ 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 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 $S^1$ that satisfy the Borsuk-Ulam property, with respect to all involutions τ over $S^1$, for the torus bundles over $S^1$ with $M=N=MA$ and $A=\left[\begin{array}{cc}1& n\\ 0& 1\end{array}\right]$.