Poincaré Duality Spaces
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are isomorphisms. | are isomorphisms. | ||
− | is an isomorphism. Here, $\mathcal B$ is allowed to range over all local coefficient bundles on $X$ | + | is an isomorphism. Here, $\mathcal B$ is allowed to range over all local coefficient bundles on $X$, |
− | + | but in fact it is sufficient to check the condition when $\mathcal{B}$ is local coefficient bundle over $X$ defined by | |
− | + | ||
$\Bbb Z[\pi]$, with $\pi$ the fundamental groupoid of $X$. | $\Bbb Z[\pi]$, with $\pi$ the fundamental groupoid of $X$. | ||
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1 Introduction
1 Definition
A Poincaré pair of dimension consists of a pair of finitely dominated spaces such that there exists a pair in which is a bundle of local coefficients on which is free abelian of rank one and are such that
and
are isomorphisms.
is an isomorphism. Here, is allowed to range over all local coefficient bundles on , but in fact it is sufficient to check the condition when is local coefficient bundle over defined by , with the fundamental groupoid of .
2 Notes
- If , one says that is a Poincaré duality space. (In view of this, perhaps better terminology would be to call a Poincaré 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.
3 Example
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.