A well-known lower bound for
the number of fixed points of a self-map f : X → X is the Nielsen number N(f).
Unfortunately, the Nielsen number is difficult to calculate. The Lefschetz number
L(f), on the other hand, is readily computable, but does not give a lower bound for
the number of fixed points. In this paper, we investigate conditions on the space X
which guarantee either N(f) = |L(f)| or N(f) ≥|L(f)|. By considering the
Nielsen and Lefschetz coincidence numbers, we show that N(f) ≥|L(f)| for
all self-maps on compact infrasolvmanifolds (aspherical manifolds whose
fundamental group has a normal solvable subgroup of finite index). Moreover,
for infranilmanifolds, there is a Lefschetz number formula which computes
N(f).
