Poincare duality space
This article defines a property of topological spaces that depends only on the homology of the topological space, viz it is completely determined by the homology groups. In particular, it is a homotopy-invariant property of topological spaces
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Let a connected space and a commutative ring. We say that is a Poincare duality space of formal dimension with respect to if the following hold:
- The homology of with coefficients in is finitely generated
- is a free module of rank over
- Pick a generator for . Then the cap product with this generator induces a map from to . This map is an isomorphism for all .
In particular, has nonvanishing homology groups only for
By default, we take .