Nielsen coincidence theory of fibre-preserving maps and Dold’s fixed point index Topological Methods in Nonlinear Analysis

Dedicated to Albrecht Dold on the occasion of his 80th birthday Abstract. Let M → B, N → B be fibrations and f1, f2: M → N be a pair of fibre-preserving maps. Using normal bordism techniques we define an invariant which is an obstruction to deforming the pair f1, f2 over B to a coincidence free pair of maps. In the special case where the two fibrations are the same and one of the maps is the identity, a weak version of our ω-invariant turns out to equal Dold’s fixed point index of fibre-preserving maps. The concepts of Reidemeister classes and Nielsen coincidence classes over B are developed. As an illustration we compute e.g. the minimal number of coincidence components for all homotopy classes of maps between S1−bundles over S1 as well as their Nielsen and Reidemeister numbers