Finiteness conditions and PD_r-group covers of PD_n-complexes

J.A. Hillman and D.H. Kochloukova

Abstract

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 χ(M)=0 and M' is homotopy equivalent to a complex with finite [(n-1)/2]-skeleton and ν=π₁(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_ν with covering group a PD_r-group π₁(M)/ν under a slightly stricter finiteness condition.

Keywords: coinduced module, finiteness condition, finite domination, infinite cyclic cover, Novikov ring, Poincaré duality.

AMS Subject Classification: Primary 57P10; secondary 55R99.