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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 $\mathbb{S}^1$.

In this work we analysed the validity of a type of Borsuk-Ulam theorem for multimaps between surfaces. We developed an algebraic technique involving braid groups to study this problem for $n$-valued maps. As a first application we described when the Borsuk-Ulam theorem holds for splits and non-splits multimaps $\phi \colon X \multimap Y$ in the following two cases: $(i)$ $X$ is the $2$-sphere eqquiped with the antipodal involution and $Y$ is either a closed surface or the Euclidean plane; $(ii)$ $X$ is a closed surface different of the $2$-sphere eqquiped with a free involution $\tau$ and $Y$ is the Euclidean plane. The results are exhaustive and in the case $(ii)$ are described in terms of an algebraic condition involving the first integral homology group of the orbit space $X / \tau$.

Let $M$ and $N$ be topological spaces, let $G$ be a group, and let $\tau \colon\thinspace G \times M \to M$ be a proper free action of $G$. In this paper, we define a Borsuk-Ulam-type property for homotopy classes of maps from $M$ to $N$ with respect to the pair $(G,\tau)$ that generalises the classical antipodal Borsuk-Ulam theorem of maps from the $n$-sphere $\mathbb{S}^n$ to $\mathbb{R}^n$. In the cases where $M$ is a finite pathwise-connected CW-complex, $G$ is a finite, non-trivial Abelian group, $\tau$ is a proper free cellular action, and $N$ is either $\mathbb{R}^2$ or a compact surface without boundary different of $\mathbb{S}^2$ and $\mathbb{RP}^2$, we give an algebraic criterion involving braid groups to decide whether a free homotopy class $\beta \in [M,N]$ has the Borsuk-Ulam property. As an application of this criterion, we consider the case where $M$ is a compact surface without boundary equipped with a free action $\tau$ of the finite cyclic group $\mathbb{Z}_n$. In terms of the orientability of the orbit space $M_\tau$ of $M$ by the action $\tau$, the value of $n$ modulo $4$ and a certain algebraic condition involving the first homology group of $M_\tau$, we are able to determine if the single homotopy class of maps from $M$ to $\mathbb{R}^2$ possesses the Borsuk-Ulam property with respect to $(\mathbb{Z}_n,\tau)$. Finally, we give some examples of surfaces on which the symmetric group acts, and for these cases, we obtain some partial results regarding the Borsuk-Ulam property for maps whose target is $\mathbb{R}^2$., Comment: 18 pages, 2 figures

Let $M$ be a topological space that admits a free involution $\tau$, and let $N$ be a topological space. A homotopy class $\beta \in [ M,N ]$ is said to have the Borsuk-Ulam property with respect to $\tau$ if for every representative map $f: M \to N$ of $\beta$, there exists a point $x \in M$ such that $f(\tau(x))= f(x)$. In this paper, we determine the homotopy class of maps from the $2$-torus $T^2$ to the Klein bottle $K^2$ that possess the Borsuk-Ulam property with respect to any free involution of $T^2$ for which the orbit space is $K^2$. Our results are given in terms of a certain family of homomorphisms involving the fundamental groups of $T^2$ and $K^2$. This completes the analysis of the Borsuk-Ulam problem for the case $M=T^2$ and $N=K^2$, and for any free involution $\tau$ of $T^2$.

Let $M$ be a topological space that admits a free involution $\tau$, and let $N$ be a topological space. A homotopy class $\beta \in [ M,N ]$ is said to have {\it the Borsuk-Ulam property with respect to $\tau$} if for every representative map $f: M \to N$ of $\beta$, there exists a point $x \in M$ such that $f(\tau(x))= f(x)$. In this paper, we determine the homotopy classes of maps from the $2$-torus $T^2$ to the Klein bottle $K^2$ that possess the Borsuk-Ulam property with respect to a free involution $\tau_1$ of $T^2$ for which the orbit space is $T^2$. Our results are given in terms of a certain family of homomorphisms involving the fundamental groups of $T^2$ and $K^2$.

Let M and N be topological spaces such that M admits a free involution $\\tau$. A homotopy class $\beta$ $\in$ [M, N ] is said to have the Borsuk-Ulam property with respect to $\\tau$ if for every representative map f : M $\rightarrow$ N of $\beta$, there exists a point x $\in$ M such that f ($\\tau$ (x)) = f (x). In the case where M is a compact, connected manifold without boundary and N is a compact, connected surface without boundary different from the 2-sphere and the real projective plane, we formulate this property in terms of the pure and full 2-string braid groups of N , and of the fundamental groups of M and the orbit space of M with respect to the action of $\\tau$. If M = N is either the 2-torus T^2 or the Klein bottle K^2 , we then solve the problem of deciding which homotopy classes of [M, M ] have the Borsuk-Ulam property. First, if $\\tau$ : T^2 $\rightarrow$ T^2 is a free involution that preserves orientation, we show that no homotopy class of [T^2 , T^2 ] has the Borsuk-Ulam property with respect to $\\tau$. Secondly, we prove that up to a certain equivalence relation, there is only one class of free involutions $\\tau$ : T^2 $\rightarrow$ T^2 that reverse orientation, and for such involutions, we classify the homotopy classes in [T^2 , T^2 ] that have the Borsuk-Ulam property with respect to $\\tau$ in terms of the induced homomorphism on the fundamental group. Finally, we show that if $\\tau$ : K^2 $\rightarrow$ K^2 is a free involution, then a homotopy class of [K^2 , K^2 ] has the Borsuk-Ulam property with respect to $\\tau$ if and only if the given homotopy class lifts to the torus.