A Van Kampen type theorem for coincidences

Abstract

The Nielsen coincidence theory is well understood for a pair of maps (f,g) :M n→N n where M and N are compact manifolds of the same dimension greater than two. We consider coincidence theory of a pair (f,g) :K→N n , where the complex K is the union of two compact manifolds of the same dimension as N n . We define a number N( f, g: K 1, K 2) which is a homotopy invariant with respect to the maps. This number is certainly a lower bound for the number of coincidence points, and we prove a minimizing theorem with respect to this number. Finally, we consider the case where the target is a Jiang space and we obtain a nicer description of N( f, g: K 1, K 2) in terms of the Nielsen coincidence numbers of the maps restricted to the subspaces K 1, K 2.