Reidemeister coincidence invariants of fiberwise maps

Given two fiberwise maps f 1 , f 2 between smooth fiber bundles over a base manifold B, we develop techniques for calculating their Nielsen coincidence number. In certain settings we can describe the Reidemeister set of ( f 1 , f 2 ) as the orbit set of a group action of π 1 ( B ) . The size and number of orbits captures crucial extra information. E.g. for torus bundles of arbitrary dimensions over the circle this determines the minimum coincidence numbers of the pair ( f 1 , f 2 ) completely. In particular we can decide when f 1 and f 2 can be deformed away from one another or when a fiberwise self-map can be made fixed point free by a suitable homotopy. In two concrete examples we calculate the minimum and Nielsen numbers for all pairs of fiberwise maps explicitly. Odd order orbits turn out to play a special rôle.