Abstract
Assume that f:X→Y is a proper map of a connected
n-manifold X into a Hausdorff, connected, locally
path-connected, and semilocally simply connected space Y, and y0∈Y has a neighborhood homeomorphic to Euclidean n-space.
The proper Nielsen number of f at y0 and the absolute degree
of f at y0 are defined in this setting. The proper Nielsen
number is shown to a lower bound on the number of roots at y0 among all maps properly homotopic to f, and the absolute degree
is shown to be a lower bound among maps properly homotopic to f and transverse to y0. When n>2, these bounds are shown to be
sharp. An example of a map meeting these conditions is given in
which, in contrast to what is true when Y is a manifold, Nielsen
root classes of the map have different multiplicities and
essentialities, and the root Reidemeister number is strictly
greater than the Nielsen root number, even when the latter is
nonzero.