Poincaré Duality Spaces
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- is a bundle of local coefficients on which is free abelian of rank one, and
- is a class such that
Here, is allowed to range over all local coefficient bundles on , but in fact it is sufficient to check the condition when is the local coefficient bundle over associated with , where is the fundamental groupoid of .
- If , one says that is a Poincaré duality space. (In view of this, perhaps better terminology would be to call a Poincaré duality space with boundary.)
- is called an orientation sheaf and is called a fundamental class. The pair is unique up to unique isomorphism.
- If with respect to a Poincar\'e pair of dimension , then is a Poincaré space of dimension with respect to , where is the boundary homomorphism.
- A finite CW complex admits the structure of a Poincaré duality space of dimension if and only if there exists a framed compact smooth manifold of dimension such is homotopy equivalent to and the inclusion has homotopy fiber homotopy equivalent to .
A compact (smooth, PL, TOP or homology) manifold of dimension is a Poincaré duality pair of dimension , where is the orientation sheaf of and is the manifold fundamental class.