Obstruction theory for coincidences of multiple maps

Let f1,...,fk:X→N be maps from a complex X to a compact manifold N, k≥2. In previous works [1], [12], a Lefschetz type theorem was established so that the non-vanishing of a Lefschetz type coincidence class L(f1,...,fk) implies the existence of a coincidence x∈X such that f1(x)=...=fk(x). In this paper, we investigate the converse of the Lefschetz coincidence theorem for multiple maps. In particular, we study the obstruction to deforming the maps f1,...,fk to be coincidence free. We construct an example of two maps f1,f2:M→T from a sympletic 4-manifold M to the 2-torus T such that f1 and f2 cannot be homotopic to coincidence free maps but for anyf:M→T, the maps f1,f2,f are deformable to be coincidence free.