Let f : X → Y be a map of
closed orientable manifolds of the same dimension, and let a ∈ Y . The topological
degree of f is an algebraic count of the number of solutions to f(x) = a,
but not an actual count. The Nielsen number N(f,a) of roots is an actual
lower bound for the number of solutions. We investigate conditions under
which N(f,a) = |degree f|. Our question is analogous to the question in
fixed point theory: when is the Lefschetz number equal to the fixed point
Nielsen number? We find equality when X = Y is an aspherical manifold
whose fundamental group satisfies the ascending chain condition on normal
subgroups, or if X and Y are aspherical manifolds with virtually polycyclic
fundamental groups. This includes infrasolvmanifolds. Similar results are
obtained for nonorientable manifolds by considering their orientable double
covers.
