Obstruction theory for coincidences of multiple maps

Abstract Let f 1 , . . . , f k : 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 ( f 1 , . . . , f k ) implies the existence of a coincidence x ∈ X such that f 1 ( x ) = . . . = f k ( 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 f 1 , . . . , f k to be coincidence free. We construct an example of two maps f 1 , f 2 : M → T from a sympletic 4-manifold M to the 2-torus T such that f 1 and f 2 cannot be homotopic to coincidence free maps but for any f : M → T , the maps f 1 , f 2 , f are deformable to be coincidence free.