  In this paper we continue to study (`strong') Nielsen coincidence numbers
(which were introduced recently for pairs of maps between manifolds of
arbitrary dimensions) and the corresponding minimum numbers of coincidence
points and pathcomponents. We explore compatibilities with fibrations and, more
specifically, with covering maps, paying special attention to selfcoincidence
questions. As a sample application we calculate each of these numbers for all
maps from spheres to (real, complex, or quaternionic) projective spaces. Our
results turn out to be intimately related to recent work of D Goncalves and D
Randall concerning maps which can be deformed away from themselves but not by
small deformations; in particular, there are close connections to the Strong
Kervaire Invariant One Problem.
