We prove that a self-map f: M → M of a compact PL-manifold of dimension ≥ 3 is homotopic to a map with no periodic points of period n iff the Nielsen numbers for k dividing n all vanish. This generalizes the result from [Je] to dimension 3.
Nielsen theory, originally developed as a homotopy-theoretic approach to fixed point theory, has been translated and extended to various other problems, such as the study of periodic points, coincidence points and roots. Recently, the techniques of Nielsen theory have been applied to the study of intersections of maps. A Nielsen-type number, the Nielsen intersection number NI(f,g), was introduced, and shown to have many of the properties analogous to those of the Nielsen fixed point number. In particular,...