On four-dimensional Poincarè duality cobordism groups

This paper continues the study of 4-dimensional Poincarè duality cobordism theory from our previous work "PD4-complexes: constructions, cobordisms and signatures", Homology, Homotopy and Applications 18(2) (2016), 267-281. Let P be an oriented finite Poincarè duality complex of dimension 4. Then we calculate the Poincarè duality cobordism group of P ΩPD4(P) . The main result is the existence of the exact sequence of Theorem 1.1. It turns out that ΩPD4(P) depends only on the fundamental group and the assembly map A4. This does not hold in higher dimensions. Then we discuss several examples. The cases in which the canonical map ΩTOP4(P) → ΩPD4(P) is not surjective are of particular interest. Its image coincides with the kernel of the total surgery obstruction map. In these cases there are PD4-complexes which cannot be homotopy equivalent to manifolds.