  Minimum numbers of fixed points or of coincidence components (realized by
maps in given homotopy classes) are the principal objects of study in
topological fixed point and coincidence theory. In this paper we investigate
fiberwise analoga and represent a general approach e.g. to the question when
two maps can be deformed until they are coincidence free. Our method involves
normal bordism theory, a certain pathspace EB and a natural generalization of
Nielsen numbers. As an illustration we determine the minimum numbers for all
maps between torus bundles of arbitrary (possibly different) dimensions over
spheres and, in particular, over the unit circle. Our results are based on a
careful analysis of the geometry of generic coincidence manifolds. They allow
also a simple algebraic description in terms of the Reidemeister invariant (a
certain selfmap of an abelian group) and its orbit behavior (e.g. the number of
odd order orbits which capture certain nonorientability phenomena). We carry
out several explicit sample computations, e.g. for fixed points in
(S1)2-bundles. In particular, we obtain existence criteria for fixed point free
fiberwise maps.
