Roots of mappings from manifolds
Tóm tắt
Assume that
is a proper map of a connected
-manifold
into a Hausdorff, connected, locally path-connected, and semilocally simply connected space
, and
has a neighborhood homeomorphic to Euclidean
-space. The proper Nielsen number of
at
and the absolute degree of
at
are defined in this setting. The proper Nielsen number is shown to a lower bound on the number of roots at
among all maps properly homotopic to
, and the absolute degree is shown to be a lower bound among maps properly homotopic to
and transverse to
. When
, 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
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.