Coincidences of projections and linear n-valued maps of tori

Abstract

We prove that the Nielsen fixed point number N ( φ ) of an n-valued map φ : X ⊸ X of a compact connected triangulated orientable q-manifold without boundary is equal to the Nielsen coincidence number of the projections of the graph of φ, a subset of X × X , to the two factors. For certain q × q integer matrices A, there exist “linear” n-valued maps Φ n , A , σ : T q ⊸ T q of q-tori that generalize the single-valued maps f A : T q → T q induced by the linear transformations T A : R q → R q defined by T A ( v ) = A v . By calculating the Nielsen coincidence number of the projections of its graph, we calculate N ( Φ n , A , σ ) for a large class of linear n-valued maps.