The Nielsen coincidence theory on topological manifolds

We generalize the coincidence semi-index introduced in [D-J] to pairs of maps between topological manifolds. This permits extending the Nielsen theory to this class of maps. Introduction. In this paper we generalize the coincidence semi-index theory, introduced in [D-J] in the smooth case, to pairs of maps between topological manifolds. It will be based on the topological transversality lemma (1.1). To show that this new theory generalizes the previous one it will be necessary to reformulate [D-J] since the graphs of any two maps are never topologically transverse. This is done in Section 2: we give three equivalent versions of the semi-index in the smooth case and then we show that one of them coincides with the semi-index defined in Section 1. In Section 3 we prove a Wecken type theorem on realizing the above Nielsen number. 1. The coincidence semi-index on topological manifolds. Throughout this paper we consider pairs of maps f, g : M → N such that M,N are topological separable manifolds without boundary and the coincidence set Φ(f, g) = {x ∈M : fx = gx} is compact. The construction of the semi-index we present is based on the transversality lemma (1.1) below. Let P ⊂W and V be topological manifolds and ξ a normal microbundle of P in W , i.e. the total space of ξ is an open subset of W containing P . Recall that a map h : V → W is called topologically transverse (briefly t-transverse) to ξ if h−1P is a topological submanifold in V admitting a normal microbundle ν such that for any x ∈ h−1P a neighbourhood of x in νx is mapped by h homeomorphically onto a neighbourhood of hx in ξhx [K-S]. 1991 Mathematics Subject Classification: Primary 55M20; Secondary 57N99.