Poincaré Duality Spaces
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− | A Poincaré duality space of dimension $d$ consists of a space $X$ together a pair $(\mathcal{L},[X])$ in which $\mathcal{L}$ is a bundle of local coefficients on $X$ which is free abelian of rank one and $[X] \in H_d(X;\mathcal {L})$ | + | A Poincaré duality space of dimension $d$ consists of a space $X$ together a pair $(\mathcal{L},[X])$ in which $\mathcal{L}$ is a bundle of local coefficients on $X$ which is free abelian of rank one and $[X] \in H_d(X;\mathcal {L})$ satisfies |
$$ \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X;\mathcal{B} \otimes \mathcal{L})$$ | $$ \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X;\mathcal{B} \otimes \mathcal{L})$$ | ||
− | 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 $X$. |
+ | |||
+ | If $X$ is a finitely dominated space, then it suffices to check the above | ||
+ | condition when $\mathcal{B}$ is local coefficient bundle over $X$ defined by | ||
+ | $\Bbb Z[\pi]$, with $\pi$ the fundamental groupoid of $X$. | ||
+ | |||
+ | Note: One usually assumes that $X$ is finitely dominated. | ||
+ | |||
==Example== | ==Example== |
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1 Introduction
1 Definition
A Poincaré duality space of dimension consists of a space together a pair in which is a bundle of local coefficients on which is free abelian of rank one and satisfies
is an isomorphism. Here, is allowed to range over all local coefficient bundles on .
If is a finitely dominated space, then it suffices to check the above condition when is local coefficient bundle over defined by , with the fundamental groupoid of .
Note: One usually assumes that is finitely dominated.
2 Example
A closed (smooth, PL, TOP or homology) manifold of dimension is a Poincaré duality space of dimension , where is the orientation sheaf of and is the fundamental class.