Homotopy periodicity and coherence Journal Articles

If f : Z → Z f\colon \,Z \rightarrow Z is a periodic homotopy equivalence ( id Z ≃ f p \operatorname {id}_{Z} \simeq f^{p} ) or a homotopy idempotent ( f ≃ f 2 f \simeq f^{2} ), the question arises whether this periodicity property can be achieved by a homotopy “compatible with” f f . These coherence questions are answered.