Realization of homotopy invariants by PD3-pairs

Beatrice Bleile


Up to oriented homotopy equivalence, a PD3-pair (X, ∂ X) with aspherical boundary components is uniquely determined by the Π1-system { κi: Π1(∂ Xi, *) → Π1(X, *) }i in J, the orientation character ωX in H1(X;Z / 2 Z) and the image of the fundamental class [X, ∂ X] in H3(X, ∂ X; Zω) under the classifying map. We call the triple ({ κi }i in J, ωX, [X, ∂ X]) the fundamental triple of the PD3-pair (X,∂ X).

Using Peter Hilton's homotopy theory of modules, Turaev gave a condition for realization in the absolute case of PD3-complexes X with ∂ X = ∅. Given a finitely presentable group G and ω in H1(G;Z / 2 Z), he defined a homomorphism ν: H3(G; Z^{ω}) → [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]-module A to the Z[G]-module B. Turaev showed that, given µ in H3(G; Zω), the triple (G, ω, µ) is relized by a PD3-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 PD3-pairs (X, ∂ X), where ∂ X is not necessarily empty.

Keywords: PD3-Pairs.

This paper is available as a pdf (416kB) file.
Date:Wednesday, November 19, 2003