Non-affine $n$-valued maps on tori

In this paper we construct $n$-valued maps on $k$-dimensional tori, where $n,k\ge 2$, that are not homotopic to affine $n$-valued maps. This is in high contrast with the single valued case, where any such map is homotopic to an affine (even linear) map. We do this by investigating necessary and sufficient algebraic conditions on certain induced morphisms.