On the computation of the relative Nielsen number

The Reidemeister number R(f) is an upper bound for the Nielsen number N(f) of a selfmap f. For Jiang-type spaces, N(f)=0 or N(f)=R(f) according to whether L(f)=0 or L(f)≠0 where L(f) denotes the Lefschetz number of f. In this paper, we use a relative Reidemeister number to estimate the relative Nielsen number of a selfmap of a pair of spaces. We show that the relative Reidemeister number can be used to calculate the relative Nielsen number for maps of pairs of Jiang-like spaces. We also apply the relative Reidemeister number to study fixed points of fiber-preserving maps.