Fibre techniques in nielsen periodic point theory on solmanifolds III: Calculations

This third paper of the series gives the necessarily lengthy illustrations of the main results of the first two, delayed until now for reasons of space. The illustrations in question concern calculations of the Nielsen type periodic point numbers NPn And NΦn(f) for self maps f of solvmanifolds. We indicate that for low dimensional solvmanifolds, we can often give formulae (as opposed to algorithms) for these numbers, which of course include formulae for the ordinary Nielsen numbers N(fn). We give a complete analysis of all maps on two very different example generalizations of the Klein bottle K2. Both examples admit non-weakly Jiang maps, which is where the more complex calculations occur. Our methods employ matrix theory and modular arguments with periodic matrices. Among other things, our results include the promised completion of the more difficult calculations on K2 itself, as well as a generalization of a surprising result first observed in Nielsen periodic point theory on K2. More precisely we show for each m 〉 1 that there is a solvmanifold Sof dimension m, and self map f on S for which, NPn(f) = N(fn) for an infinite number of n. We also discuss the generality of our considerations, indicating that the type of map and solvmanifold given here and the methods used, provide the data needed to make calculations for all maps on all solvmanifolds. Finally we indicate that data that allow the exhibited modular patterns, though widely available, do not hold universally. Mathematics Subject Classification (2000): 55M20. Key words: Nielsen numbers; periodic points; nilmanifolds; solvmanifolds. Quaestiones Mathematicae 25 (2002), 177-20