The Nielsen root number
N(f;c) of a map f:M → N at a point c ∈ N is a homotopy invariant lower bound
for the number of roots at c, that is, for the cardinality of f−1(c). There is a formula
for calculating N(f;c) if M and N are closed oriented manifolds of the same
dimension. We extend the calculation of N(f; c) to manifolds that are not
orientable, and also to manifolds that have non-empty boundaries and are not
compact, provided that the map f is boundary-preserving and proper. Because
of its connection with degree theory, we introduce the transverse Nielsen
root number for maps transverse to c, obtain computational results for it in
the same setting, and prove that the two Nielsen root numbers are sharp
lower bounds in dimensions other than 2. We apply these extended root
theory results to the degree theory for maps of not necessarily orientable
manifolds introduced by Hopf in 1930. Thus we re-establish, in a new and
modern treatment, the relationship of Hopf’s Absolutgrad and the geometric
degree with homotopy invariants of Nielsen root theory, a relationship that is
present in Hopf’s work but not in subsequent re-examinations of Hopf’s degree
theory.
