Poincaré Duality Spaces
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− | A ''Poincaré pair'' of dimension $d$ consists of a finitely dominated pair $(X,\partial X)$ | + | A ''Poincaré pair'' of dimension $d$ consists of a finitely dominated CW pair $(X,\partial X)$ for which there exists $$(\mathcal{L},[X])$$ in which |
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+ | *$\mathcal{L}$ is a bundle of local coefficients on $X$ which is free abelian of rank one, and | ||
+ | |||
+ | * $[X] \in H_d(X,\partial X;\mathcal {L})$ is a class such that | ||
$$ \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X,\partial X;\mathcal{B} \otimes \mathcal{L})$$ | $$ \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X,\partial X;\mathcal{B} \otimes \mathcal{L})$$ | ||
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are isomorphisms. | are isomorphisms. | ||
− | + | 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 | 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 $X$. |
Revision as of 18:17, 23 March 2011
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1 Introduction
1 Definition
A Poincaré pair of dimension consists of a finitely dominated CW pair for which there exists in which
- is a bundle of local coefficients on which is free abelian of rank one, and
- is a class such that
and
are isomorphisms.
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.