Poincare duality space

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 $$R = \mathbb{Z}$$.

Stronger properties

 * Compact connected orientable manifold

Weaker properties

 * Space with finitely generated homology