Fibre techniques in Nielsen periodic point theory on nil and solvmanifolds II

In the first paper we showed for all maps f on nilmanifolds, and for weakly Jiang maps on solvmanifolds, that the Nielsen periodic numbers NPn(f) and NΦn(f) can be computed from the (highly computable) collection of ordinary Nielsen numbers {N(fm): m|n} . For non-weakly Jiang maps this is no longer true. In this second paper we prove and briefly illustrate naı̈ve type addition formulae for periodic points of fibre preserving maps. Our formulae are particularly useful for maps of solvmanifolds that are not weakly Jiang. The resulting calculations utilize certain matrix data from f , and a factorization of the space as the total space of a suitable fibration. We illustrate these considerations with examples of non weakly Jiang maps on the Klein bottle where, for an infinite number of n , we have the surprising equality NPn(f)=N(fn) . We will also give an example which shows that this pattern is not unique to the Klein bottle.