On Four-Dimensional Poincaré Duality Cobordism Groups

This paper continues the study of four-dimensional Poincare duality cobordism theory from our previous work Cavicchioli et al. (Homol. Homotopy Appl. 18(2):267–281, 2016). Let P be an oriented finite Poincare duality complex of dimension 4. Then, we calculate the Poincare duality cobordism group \(\Omega _{4}^{{\text {PD}}}(P)\). The main result states the existence of the exact sequence \(0 \rightarrow L_4 (\pi _1 (P))/A_4 (H_2 (B\pi _1 (P), L_2)) \rightarrow {{\widetilde{\Omega }}}_{4}^{\mathrm{PD}}(P) \rightarrow \mathbb Z_8 \rightarrow 0\), where \({{\widetilde{\Omega }}}_{4}^{\mathrm{PD}}(P)\) is the kernel of the canonical map \({\Omega }_{4}^{\mathrm{PD}}(P) \rightarrow H_4 (P, \mathbb Z) \cong \mathbb Z\) and \(A_4 : H_4 (B\pi _1, \mathbb L) \rightarrow L_4 (\pi _1 (P))\) is the assembly map. It turns out that \({\Omega }_{4}^{\mathrm{PD}}(P)\) depends only on \(\pi _1 (P)\) and the assembly map \(A_4\). This does not hold in higher dimensions. Then, we discuss several examples. The cases in which the canonical map \(\Omega _{4}^{{\text {TOP}}}(P) \rightarrow \Omega _{4}^{{\text {PD}}}(P)\) is not surjective are of particular interest. Its image coincides with the kernel of the total surgery obstruction map. In fact, we establish an exact sequence Open image in new window where s is Ranicki’s total surgery obtruction map. In the above cases, there are \({\text {PD}}_4\)-complexes X which cannot be homotopy equivalent to manifolds.