Let α be a homotopy
class of maps of X, a connected compact metric ANR, into itself and let Lα
denote the Lefschetz number of α. A converse to the Lefschetz fixed point
theorem is: if Lα= 0 then α contains a fixed point free map. The converse is
true if X is a compact connected simply-connected topological n-manifold
(Fadell) or if X is a compact connected topological n-manifold, with or without
boundary, and α contains the identity map (Brown-Fadell). Let μ(α) denote the
fixed point class invariant of α, then every map in α has at least μ(α) fixed
points. The purpose of this paper is to generalize the preceding results by
proving that if X is a compact connected topological n-manifold, n ≧ 3,
with or without boundary, then there is a map in α which has exactly μ(α)
fixed points. It follows that the converse to the Lefschetz theorem will hold
whenever α contains a map all of whose fixed points are in a single fixed point
class.
