A well-known result by H. Hopf
states that every selfmap f of a polyhedron |K| can be deformed into a selfmap f′
which has only a finite number of fixed points and is arbitrarily close to the given
one. In addition one can locate all fixed points of f′ in maximal simplexes. A map
which has a finite fixed point set is here called a fix-finite map, and a homotopy
F : |K|× I →|K| is called a fix-finite homotopy if the map ft= F(⋅,t) is fix-finite
for every t ∈ I. We extend Hopf’s result to homotopies, and show that two homotopic
selfmaps f0 and f1 of a polyhedron |K| which are fix-finite and have all their fixed
points located in maximal simplexes can be related by a homotopy which is
fix-finite and arbitrarily close to the given one. All fixed points of F can
again be located in as high-dimensional simplexes as possible. Some simple
properties are derived from the fact that the fix-finite homotopy is constructed
in such a way that its fixed point set is a one-dimensional polyhedron in
|K|× I.
