We show that an infinite cyclic covering space M' of a PD (n) -complex M is a PD (n-1)-complex if and only if chi(M) = 0, M' is homotopy equivalent to a complex with finite [(n-1)/2]-skeleton and pi(1)(M') is finitely presentable. This is best possible in terms of minimal finiteness assumptions on the covering space. We give also a corresponding result for covering spaces M (nu) with covering group a PD (r) -group under a slightly stricter finiteness condition.25614556