Realization of homotopy invariants by PD³-pairs

Beatrice Bleile

Abstract

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