In classical fixed point and coincidence theory the notion of Nielsen
numbers has proved to be extremely fruitful. We extend it to pairs
of
maps between manifolds of arbitrary dimensions, using nonstabilized normal bordism
theory as our main tool. This leads to estimates of the minimum numbers
(and
, resp.) of path
components (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to
. Furthermore, we deduce
finiteness conditions for .
As an application we compute both minimum numbers explicitly in various concrete
geometric sample situations.
The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain
path space
into path components. Its higher dimensional topology captures further
crucial geometric coincidence data. In the setting of homotopy groups
the resulting invariants are closely related to certain Hopf–Ganea
homomorphisms which turn out to yield finiteness obstructions for
.
Keywords
coincidence manifold, normal bordism, path space, Nielsen
number, Ganea-Hopf invariant
