Some remarks about Poincaré duality pairs

Bieri-Eckmann [6] introduced the concept of relative cohomology for a group pair (G, S), where G is a group and S is a family of subgroups of G and, by using that theory, they introduced the concept of Poincaré duality pairs (G, S) and provided a topological interpretation for such pairs through Eilenberg-MacLane pairs K(G, S, 1). A Poincaré duality pair is a pair (G, S) that satisfies two isomorphisms, one between absolute cohomology and relative homology and the second between relative cohomology and absolute homology. In this paper, we present a proof that those two isomorphisms are equivalent. We also present some calculations on duality pairs by using the cohomological invariant defined in [1] and studied in [2-4]. © 2012 Pushpa PublishingHouse.