The Lefschetz fixed point
theorem states that whenever the Lefschetz number Λ(f) of a map f : X → X is
nonzero, then f must have a fixed point. The theorem is known to hold when X is an
ANR and f is a compact continuous map. The theorem has been studied for
compact, upper semi-continuous, acyclic multi-valued maps and is known to hold in
this setting for topologically complete ANR’s.
A more general class of multi-valued maps is considered in this paper: the class of
compact upper semi-continuous maps which can be written as a composition of
acyclic maps. Using this class of maps, a theorem is proved which generates spaces
for which the Lefschetz theorem holds. In particular, the Lefschetz theorem holds for
all (metric) ANR’s.
