On Four-Dimensional Poincaré Duality Cobordism Groups

Abstract

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 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.