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

In this paper and its sequel we give results and methods for evaluating the Nielsen type numbers NΦn(f) and NPn(f) for periodic points of self maps f of nilmanifolds and solvmanifolds (which includes the Klein bottle). Using fibre space techniques we relate these numbers to the various ordinary Nielsen numbers N(fm) of the mth iterate (for m¦n), and thus capitalize on known techniques for our calculations. In this first paper we deal with the specific situation where for each m¦n either N(fm) = R(fm) (the Reidemeister number of fm) or N(fm) = 0. In fact this is the vast majority of cases. In the case where N(fn) = R(fn) we derive the simple formula NΦm(f) = N(fm) for all m¦n.