Addition formulae for Nielsen numbers and for Nielsen type numbers of fibre preserving maps

In this paper we generalize well known product formulae for the Nielsen number of a fibre preserving map, to give addition formulae for such maps. We give necessary and sufficient conditions for when a naïve addition formula expressing the Nielsen number of the fibre map as a simple sum of Nielsen numbers on the fibres is valid. In the second part of the paper we extend to the nonorientable situation the definition and properties of a Nielsen type number of a fibre preserving map introduced by the first author.