On minimal fixed point numbers of relative maps

Let f :(X,A)→(X,A) be a self map of a pair of compact polyhedra. We define two new Nielsen type numbers m(f;X,A) and m(f;X−A), which are lower bounds for the number of fixed points on X and on Cl(X−A), the closure of X−A in X, respectively. These relative homotopy theoretic lower bounds can be realized without the by now familiar by-passing condition. f our intention is that m(f;X,A) and m(f;X−A) be companion numbers to the surplus number SN(f;X−A) in the sense that, as much as possible, these three numbers should be realizable simultaneously. This is analogous to the realizability of the numbers N(f;X,A), Ñ(f;X,A) and N(f;X−A) which, in the presence of by-passing (together with the usual Wecken type conditions), can be realized simultaneously. Thus we answer open questions posed by Schirmer.