Poincaré Duality Spaces
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==Introduction== | ==Introduction== | ||
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+ | ==Definition== | ||
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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 $[X] \in H_d(X;\mathcal {L})$ sastifies | 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 $[X] \in H_d(X;\mathcal {L})$ sastifies | ||
− | $$ \cap [X] : H^*(X; | + | $$ \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X;\mathcal{B} \otimes \mathcal{L})$$ |
is an isomorphism. Here, $M$ is allowed to range over all local coefficient bundles on $X$. | is an isomorphism. Here, $M$ is allowed to range over all local coefficient bundles on $X$. | ||
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+ | ==Example== | ||
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+ | A closed (smooth, PL, TOP or homology) manifold $X$ of dimension $d$ is a Poincaré duality space of dimension $d$, where $\mathcal L$ is the orientation sheaf of $M$ | ||
+ | and $[X]$ is the fundamental class. | ||
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</wikitex> | </wikitex> | ||
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==References== | ==References== | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Theory]] | [[Category:Theory]] |
Revision as of 22:22, 21 March 2011
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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 sastifies
is an isomorphism. Here, is allowed to range over all local coefficient bundles on .
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.