Splittings of Poincaré duality groups (Q1091475)

If \(G\) is an \(n\)-dimensional Poincaré duality group and \(S\) is an \(n-1\)-dimensional Poincaré duality subgroup then it is quite common for \(G\) to split over \(S\), either as free product with amalgamation, or as an HNN-extension. In this paper we introduce an obstruction \(\text{sing}_G(S)\), in the form of a certain cohomology class. A splitting of \(G\) over a subgroup commensurable with \(S\) exists if and only if this obstruction vanishes. The algebraic theory has close parallels in the theory of aspherical manifolds.