Poincare duality space

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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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Definition

Let M a connected space and R a commutative ring. We say that M is a Poincare duality space of formal dimension n with respect to R if the following hold:

  • The homology of M with coefficients in R is finitely generated
  • H_n(M;R) is a free module of rank 1 over R
  • Pick a generator for H_n(M;R). Then the cap product with this generator induces a map from H^i(M) to H_{n-i}(M). This map is an isomorphism for all i.

In particular, M has nonvanishing homology groups only for 0 \le i \le n

By default, we take R = \mathbb{Z}.

Relation with other properties

Stronger properties

Weaker properties