Poincaré Duality Spaces
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1 Introduction
1 Definition
A Poincaré duality space of dimension consists of a space together a pair in which is a bundle of local coefficients on which is free abelian of rank one and satisfies
is an isomorphism. Here, is allowed to range over all local coefficient bundles on .
If is a finitely dominated space, then it suffices to check the above condition when is local coefficient bundle over defined by , with the fundamental groupoid of .
Note: One usually assumes that is finitely dominated.
2 Example
A closed (smooth, PL, TOP or homology) manifold of dimension is a Poincaré duality space of dimension , where is the orientation sheaf of and is the fundamental class.