Surplus Nielsen type numbers for periodic points on the complement (Q1646411)

Let \(f : (X, A) \to (X, A)\) be a self-map of a pair of compact polyhedra with \(X\) connected. In this paper the authors introduce two new Nielsen type numbers \(SNP_n\left(f;X-A\right)\) and \(SN\phi_n\left(f;X-A\right)\) which provide sharp lower bounds for the minimum number of periodic points of period exactly \(n\) of maps \(g\) that are homotopic to \(f\) as a map of pairs, and that lie in \(X-A\) (\(MP_n\left(f;X-A\right)\)) as well as for the minimum number of periodic points of all periods dividing \(n\) of maps \(g\) that are homotopic to \(f\) as a map of pairs, and that lie in \(X-A\) (\(M\phi_n\left(f;X-A\right)\)) respectively. This is a generalization of surplus fixed point theory by \textit{X. Zhao} [Topology Appl. 37, No. 3, 257--265 (1990; Zbl 0713.55001)]. The results answer an open question posed by [\textit{P. R. Heath} and \textit{X. Zhao}, ibid. 102, No. 3, 253--277 (2000; Zbl 0948.55002)].