On Four-Dimensional Poincare Duality Cobordism Groups

摘要

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(PD)(4) (P). The main result states the existence of the exact sequence 0 - L-4(pi(1)(P))/A(4)(H-2(B pi(1)(P), L-2)) - (Omega) over tilde (PD)(4) (P) - Z(8) - 0, where (Omega) over tilde (PD)(4) (P) is the kernel of the canonical map Omega(PD)(4) (P) - H-4(P, Z) similar or equal to Z and A(4) : H-4(B pi(1), L) - L-4(pi(1)(P)) is the assembly map. It turns out that Omega(PD)(4) (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(TOP)(4) (P) - Omega(PD)(4) (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