Nielsen fixed point theory for split n-valued maps on the Klein bottle

Abstract

In this work we begin the study of n-valued maps on the Klein bottle, denoted by K, where we focus on the split ones. We provide an explicit description of $P_n\left(K\right)$ (the n-th pure braid group of K) as an iterated semi-direct product of the form $F_n{⋊}_{{\theta }_{n-1}}(\cdots ⋊(F_2{⋊}_{{\theta }_1}{\pi }_1\left(K\right)\left)\right)$, where the $F_i^{\prime }s$ are free groups on i letters. Given a split n-valued map $\Phi =\{f_1,\dots ,f_n\}$ its pointed homotopy class is determined by a pair of braids in $P_n\left(K\right)$. We also provide a formula for $N\left(\Phi \right)$, the Nielsen number of Φ, which is completely determined by two braids, which in turn also determine the homotopy classes of the functions $f_i^{\prime }s$. If $\Phi =\{f_1,f_2\}$ is a 2-valued map with $N\left(\Phi \right)=0$, we show that there exists at least one 2-valued map ${\Phi }^{\prime }=\{f_1^{\prime },f_2^{\prime }\}$, such that ${\Phi }^{\prime }$ is fixed point free and for $i=1,2$ it holds that $\left[f_i\right]=\left[f_i^{\prime }\right]$, where $\left[\right]$ denotes a pointed homotopy class of maps. Finally, we display an infinite family of pointed homotopy classes in $[K,F_2(K\left)\right]$, such that $N\left(\varphi \right)=0$, for any ϕ in the family. Furthermore, the map from $[K,F_2(K\left)\right]$ to $[K,K\times K]$, induced by the inclusion $F_2\left(K\right)\to K\times K$, takes this family to one single element in $[K,K\times K]$. We do not know if these 2-valued maps of the family can be deformed to fixed point free maps.