Poincaré duality pairs of dimension three

Abstract

We extend Hendriks' classification theorem and Turaev's realisation and splitting theorems for PD³-complexes to the relative case of PD³-pairs. The results for PD³-complexes are recovered by restricting the results to the case of PD³-pairs with empty boundary. Up to oriented homotopy equivalence, PD³-pairs are classified by their fundamental triple consisting of the fundamental group system, the orientation character and the image of the fundamental class under the classifying map. Using the derived module category we provide necessary and sufficient conditions for a given triple to be realised by a PD³-pair. The results on classification and realisation yield splitting or decomposition theorems for PD³-pairs, that is, conditions under which a given PD³-pair decomposes as interior or boundary connected sum of two PD³-pairs.