Realization of Fixed Point Sets

Let f:X->X be an selfmap of a compact connected polyhedron, and A a nonempty closed subset of X. In this paper, we shall deal with the question whether or non there is a map g: X->X homotopic to f such that the fixed point set Fixg of g equals A. We introduce a necessary condition for the existance of such a map g. It is shown that this condition is easy to check, and hebce some sufficient conditions are obtained.