\(PD_4\)-complexes: constructions, cobordisms and signatures (Q505363)

This paper computes the Poincaré duality cobordism group \(\Omega^{PD}_{4}(P)\) associated to a Poincaré duality complex \(P\) of formal dimension \(4\). The authors show that there is a morphism \(\Omega^{PD}_{4}(P) \rightarrow L^{0}(\Lambda) \oplus {\mathbb Z}\); where \(\Lambda = {\mathbb Z}[\pi_{1}(P)]\) and \(L^{0}(\Lambda)\) is the Witt group of non degenerate hermitian forms on finitely generated stably free \(\lambda\)-modules. Further if \(\pi_{1}(P)\) does not contain elements of order \(2\) the morphism is an isomorphism. The proof depends on Wall's results on the homotopy type of Poincaré duality complexes of dimension \(4\).