Realization of Homotopy Invariants by Pd–pairs

Up to oriented homotopy equivalence, a PD–pair (X, ∂X) with aspherical boundary components is uniquely determined by the Π1–system {κi : Π1(∂Xi, ∗) → Π1(X, ∗)}i∈J , the orientation character ωX ∈ H(X;Z/2Z) and the image of the fundamental class [X, ∂X] ∈ H3(X, ∂X;Z) under the classifying map [3]. We call the triple ({κi}i∈J , ωX , [X, ∂X]) the fundamental triple of the PD–pair (X, ∂X). Using Peter Hilton’s homotopy theory of modules, Turaev [12] gave a condition for realization in the absolute case of PD–complexes X with ∂X = ∅. Given a finitely presentable group G and ω ∈ H(G;Z/2Z), he defined a homomorphism ν : H3(G;Z) −→ [F, I] where F is some Z[G]–module, I = ker aug and [A,B] denotes the group of homotopy classes of Z[G]–morphisms from the Z[G]–moduleA to the Z[G]–moduleB. Turaev showed that, given μ ∈ H3(G;Z), the triple (G,ω, μ) is relized by a PD–complex X if and only if ν(μ) is a class of homotopy equivalences of Z[G]–modules. Using Turaev’s construction of the homomorphism ν, we generalize the condition for realization to the case of PD–pairs (X, ∂X), where ∂X is not necessarily empty.