Computation of the Nielsen fixed point number for 2-valued non-split maps on the Klein bottle

In this paper we study 2-valued non-split maps, focusing on the Klein bottle. We establish a connection between a 2-valued non-split map $\phi:X\multimap Y$ and a pair of classes of maps $([f],[f\circ \delta])\in [\tilde X,Y]\times[\tilde X, Y]$, where $\delta$ is a free involution on $\tilde X$, $X=\tilde X/\delta$ and the class of the lift factor $[f]$ does not satisfy the Borsuk-Ulam Property in respect to $\delta$. We also exhibit a method to compute the Nielsen fixed point number of a 2-valued non-split map on a closed connected manifold in terms of the Nielsen coincidence number between a lift factor and a covering space map, generalizing the formula from only orientable manifolds to also non-orientable manifolds. Finally we display a formula for the Nielsen fixed point number of 2-valued non-split maps on the Klein bottle in terms of two braids of the Klein bottle.