Realization of Fixed Point Sets

Let f: X→X be a selfmap of a compact connected polyhedron, and A a nonemptyclosed subset of X. In this paper, we shall deal with the question whether or not there is a mapg: X→X homotopic to f such that the fixed point set Fixg of g equals A. We introduce anecessary condition for the existence of such a map g. It is shown that this condition is easy tocheck, and hence some sufficient conditions are obtained.