Weckenʹs theorem for periodic points

Boju Jiang introduced a homotopy invariant NFn(f) which is a lower bound for the cardinality of periodic points, up to n-periodic, of a selfmap of a compact polyhedron. We prove that any selfmap of a compact PL-manifold f : M→M (dim M⩾4) is homotopic to a map g satisfying #Fix(gn)=NFn(f) i.e. NFn(f) is the best such homotopy invariant. This answers positively the conjecture of B.Halpern ( Wecken theorem for periodic points: see Theorem 4.14 in Chapter III (Contemp. Math. Vol. 14, Amer. Math. Soc., Providence, RI, 1983).