On four-dimensional Poincaré duality cobordism groups
The Poincaré duality spaces in this paper are finite CW-complexes and the paper considers the bordism groups \(\Omega^{PD}(P)\) where \(P\) is a Poincaré duality space. The main result is to show that there is an exact sequence \[ 0 \;\rightarrow \;L_{4}(\pi_{1}(P)) / A_{4}(H_{2}(B\pi_{1}(P), L_{2})) \;\rightarrow \;\tilde{\Omega}^{PD}_{4}(P) \;\rightarrow \;{\mathbb Z}_{8} \;\rightarrow \;0 \] where \(A_{4}\) is the assembly map and \(\tilde{\Omega}^{PD}_{4}(P)\) is the kernel of the canonical map \(\Omega^{PD}_{4}(P) \;\rightarrow \;H_{4}(P, {\mathbb Z}) \cong {\mathbb Z}\). The arguments make use of spectral sequences and specific information on homotopy groups of the Thom space \(\mathbf{MSG}\). In addition the authors establish the exactness of the following sequence \[ \Omega_{4}^{TOP} (P) \;\longrightarrow \;\Omega_{4}^{PD}(P) \;\overset{s}{\longrightarrow} \;{\mathcal S}_{4} \;\longrightarrow \;0 \] \noindent where \(s\) is Ranicki's total surgery obstruction map.
scientific article; zbMATH DE number 6879901
scientific article