The relative Reidemeister numbers of fiber map pairs

Abstract

The relative Reidemeister number, denoted by $\text{\rm R}(f;X,A)$, is an upper bound for the relative Nielsen number, denoted by $\text{\rm N}(f;X,A)$). If $(f,f_{A})$ is a pair of fibre-preserving maps of a pair of Hurewicz fibrations, then under certain conditions, the relative Reidemeister number can be calculated in terms of those on the base and on the fiber. In this paper, we give addition formulas for $\text{\rm R}(f;X,A)$ and for the relative Reidemeister number on the complement $\text{\rm R}(f;X-A)$. As an application, we give estimation of the asymptotic Nielsen type number $text{\rm NI}^{\infty}(f)$.