Homotopically periodic maps of model aspherical manifolds

For a closed orientable surface S, any map f : S --> S whose n-th power is homotopic to the identity, is homotopic to a homeomorphism g of S of order n. This famous theorem of Nielsen is known to fail in general for aspherica l manifolds. In this paper, for model aspherical manifolds M associated to a finitely extendable set of data, we, however, present a weaker version of Nielsen's result. We show that any homotopically periodic self-map f of M is homotopic to a ber preserving homeomorphism g of M of finite order ( but the order of g is not necessarily equal to the homotopy period of f).