Partially Periodic Point Free Self-Maps on Product of Spheres and Lie Groups

Let X be a topological manifold and f : X -> X a continuous map. We say the map f is partially periodic point free up to period n if f does not have periodic points of periods smaller than n + 1. A weaker notion is Lefschetz partially periodic point free up to period n, i.e. the Lefschetz numbers of the iterates of f up to n are all zero. Similarly f is Lefschetz periodic point free if the Lefschetz numbers of all iterates of f are zero. In the present article we consider continuous self-maps on products of any number spheres of different dimensions. We give necessary and sufficient conditions for such maps to be Lefschetz periodic point free and Lefschetz partially periodic point free. We apply these results to continuous self-maps on Lie groups.