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

TL;DR

This paper investigates the Borsuk-Ulam property for homotopy classes of fiber-preserving maps between surface bundles, linking the problem to algebraic structures like fundamental groups and parametrized braid groups, with applications to torus bundles.

Contribution

It establishes an algebraic criterion for the Borsuk-Ulam property in fiber bundles over K(π,1) spaces, connecting topological and algebraic methods for this problem.

Findings

Decides the Borsuk-Ulam property via algebraic problems involving fundamental groups.

Characterizes homotopy classes of self-maps of 2-torus bundles satisfying the property.

Provides explicit applications to surface bundles over the circle.

Abstract

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 $\mathbb{S}^1$.