Coincidences of projections and linear \(n\)-valued maps of tori (Q977464)

An \(n\)-valued map \(\phi : X\multimap Y\) is a both upper and lower semi-continuous multivalued function such that, for each \(x\in X\), \(\varphi (x)\) is an unordered subset of exactly \(n\) points of \(Y\). The authors prove that the Nielsen fixed point number \(N(\varphi)\) of an \(n\)-valued map \(\phi : X\multimap X\) of a compact connected triangulated orientable \(q\)-manifold is equal to the Nielsen coincidence number of the projections of the graph \(\Gamma (\varphi) \subset X\times X\) to the two factors. They prove that for certain \(q\times q\) integer matrices \(A\) there exist ``linear'' \(n\)-valued maps \(\Phi_{n,A,\sigma}: T^q \multimap T^q\) of \(q\)-tori that generalize the single-valued maps \(f_A: T^q \to T^q\) induced by the linear transformations \(T_A: \theta R^q \to \theta R^q\) defined by \(T_A(v)=Av\). By calculating the Nielsen coincidence number of the projections of the graph, the numbers \(N(\Phi_{n,A,\sigma})\) for a large class of \(n\)-valued maps are obtained.