Given two maps between smooth manifolds, the obstruction to removing their
coincidences (via homotopies) is measured by the minimum numbers. In order to
determine them we introduce and study an infinite hierarchy of Nielsen numbers
,
.
They approximate the minimum numbers from below with decreasing
accuracy, but they are (in principle) more easily computable as
grows. If the domain and the target manifold have the same dimension (eg in the
fixed point setting) all these Nielsen numbers agree with the classical definition.
However, in general they can be quite distinct.
While our approach is very geometric, the computations use the techniques of
homotopy theory and, in particular, all versions of Hopf invariants (à la Ganea,
Hilton or James). As an illustration we determine all Nielsen numbers and minimum
numbers for pairs of maps from spheres to spherical space forms. Maps into even
dimensional real projective spaces turn out to produce particularly interesting
coincidence phenomena.
Dedicated to Karl-Otto Stöhr on the
occasion of his 70th birthday.
Keywords
coincidence, Nielsen number, minimum number, Hopf
invariants, spherical space forms
