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 usually does not estimate the
number of fixed points. It is known that N(f) = |L(f)| for all maps on nilmanifolds
(homogeneous spaces of nilpotent Lie groups) and that N(f) ≥|L(f)| for all maps on
solvmanifolds (homogeneous spaces of solvable Lie groups). Typically, though,
the strict inequality holds, so the Nielsen number cannot be completely
computed from the Lefschetz number. In the present work, we produce a large
class of solvmanifolds for which N(f) = |L(f)| for all self maps. This class
includes exponential solvmanifolds: solvmanifolds for which the corresponding
exponential map is surjective. Our methods provide Nielsen and Lefschetz number
product theorems for the Mostow fibrations of these solvmanifolds, even
though the maps on the fibers in general will belong to varying homotopy
classes.
