  In classical fixed point and coincidence theory the notion of Nielsen numbers
has proved to be extremely fruitful. Here we extend it to pairs (f_1, f_2) of
maps between manifolds of arbitrary dimensions. This leads to estimates of the
minimum numbers MCC(f_1, f_2) (and MC(f_1, f_2), resp.) of pathcomponents (and
of points, resp.) in the coincidence sets of those pairs of maps which are
homotopic to (f_1, f_2). Furthermore we deduce finiteness conditions for
MC(f_1, f_2). As an application we compute both minimum numbers explicitly in
four concrete geometric sample situations. The Nielsen decomposition of a
coincidence set is induced by the decomposition of a certain path space E(f_1,
f_2) into pathcomponents. Its higher dimensional topology captures further
crucial geometric coincidence data. An analoguous approach can be used to
define also Nielsen numbers of certain link maps.
