On the location of fixed points on pairs of spaces (Q1114965)

Let \(f: (X,A)\to (X,A)\) be an admissible selfmap of a pair of metrizable ANR's. Then a Nielsen number of the complement \(\tilde N(f;X,A)\) is a lower bound for the number of fixed points on \(Cl(X-A)\) for all maps in the homotopy class of f. The Nielsen number of the boundary \(\tilde n(f;X,A)\) is a lower bound for the number of fixed points on Bd A for selfmaps of (X,A) with minimal fixed point sets. It is shown that for any pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly \(\tilde N(f;X,A)\) fixed points on \(Cl(X-A)\) and \(\tilde n(f;X,A)\) fixed points on Bd A.