Finiteness conditions and \(\text{PD}_{r}\)-group covers of \(\text{PD}_{n}\)-complexes
This paper shows that an infinite cyclic covering \(M'\) of an \(n\)-dimensional Poincaré duality space is an \((n-1)\)-dimensional Poincaré duality space, if and only if \(M'\) is homotopy equivalent to a complex with finite \([(n-1)/2]\)-skeleton and \(\pi_{1}(M')\) is finitely presentable. It is further shown that this is best possible in that it makes the minimal finiteness assumptions on the covering space. A corresponding result is shown for covering spaces whose covering group \(G\) is a PD\(_{r}\)-group (that is \(H^{q}(G;Z[G]) \cong Z\) if \(q=r\) and is 0 otherwise) under slightly stronger finiteness assumptions.
scientific article; zbMATH DE number 5159756
scientific article