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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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

• Compact connected orientable manifold

Weaker properties

• Space with finitely generated homology