The relative coincidence

We define a relative coincidence Nielsen number Nrel(f, g) for pairs of maps between manifolds, prove a Wecken type theorem for this invariant and give some formulae expressing Nrel(f, g) by the ordinary Nielsen numbers. Introduction. In [S2] pairs of spaces A ⊂ X and maps f : X → X such that f(A) ⊂ A were considered. A relative Nielsen number of such maps was defined, i.e. a lower bound of the cardinality of fixed points which is invariant with the respect to homotopies preserving A. In this paper we generalize this construction to coincidences. We consider pairs of maps f, g : M → N between n-manifolds sending a fixed k-submanifold M0 ⊂ M into a fixed k-submanifold N0 ⊂ N . We define a relative coincidence Nielsen number Nrel(f, g) for such pairs of maps, i.e. a homotopy invariant which is a lower bound for the number of coincidence points. We prove that in dimension ≥ 3 it is the best such lower bound (a Wecken type theorem). Finally, we express Nrel(f, g) by similar invariants of lifts f̃ , g̃ and we present some computations. 1. Preliminaries. We will base on the definitions of Nielsen and Reidemeister classes given in [Je1] (compare [Y]). In this section we recall them and show how the same definitions may be obtained by means of covering spaces. In fact, the identification of the sets ∇(f, g) and lift′(f, g) given below is the equivalence of the two approaches to coincidence theory: the first, “traditional”, using the fundamental group (see [B] or [Y] for fixed points), and the approach via covering spaces [Ji]. Let X and Y be path connected spaces and f, g : X → Y a pair of maps. The Nielsen relation (x ' y if there is a path ω from x to y such that fω and gω are fixed end point homotopic in Y ) splits the coincidence set Φ(f, g) = {x ∈ X : fx = gx} into Nielsen classes, and the quotient 1991 Mathematics Subject Classification: Primary 55M20; Secondary 57N99.