Poincaré Duality Spaces
From Manifold Atlas
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{{Authors|Klein}}{{Stub}} | {{Authors|Klein}}{{Stub}} | ||
− | ==Introduction== | + | == Introduction == |
<wikitex>; | <wikitex>; | ||
− | + | </wikitex> | |
==Definition== | ==Definition== | ||
+ | <wikitex>; | ||
+ | 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 | ||
+ | *$\mathcal{L}$ 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 $X$ | + | but in fact it is sufficient to check the condition when $\mathcal{B}$ is the local coefficient bundle over $X$ |
− | $\Bbb Z[\pi]$, | + | associated with $\Bbb Z[\pi]$, where $\pi$ is the fundamental groupoid of $X$. |
− | + | </wikitex> | |
==Notes== | ==Notes== | ||
− | + | <wikitex>; | |
− | * If $\partial X = \emptyset$, one says that $X$ is a ''Poincaré duality space.'' (In view of this, perhaps better terminology would be to call $(X,\partial X)$ a ''Poincaré space with boundary.'') | + | * If $\partial X = \emptyset$, one says that $X$ is a ''Poincaré duality space.'' (In view of this, perhaps better terminology would be to call $(X,\partial X)$ a ''Poincaré duality space with boundary.'') |
* $\mathcal L$ is called an ''orientation sheaf'' and $[X]$ is called a ''fundamental class.'' The pair $(\mathcal L,[X])$ is unique up to unique isomorphism. | * $\mathcal L$ is called an ''orientation sheaf'' and $[X]$ is called a ''fundamental class.'' The pair $(\mathcal L,[X])$ is unique up to unique isomorphism. | ||
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* If $(X,\partial X)$ with respect to $(\mathcal{L},[X])$ a Poincar\'e pair of dimension $d$, then $\partial X$ is a Poincaré space of dimension $d-1$ with respect to $(\mathcal {L}_{|\partial X},\partial [X])$, where $\partial: H_d(X;\mathcal{L}) \to H_{d-1}(\partial X;\mathcal{L}_{|\partial X})$ is the boundary homomorphism. | * If $(X,\partial X)$ with respect to $(\mathcal{L},[X])$ a Poincar\'e pair of dimension $d$, then $\partial X$ is a Poincaré space of dimension $d-1$ with respect to $(\mathcal {L}_{|\partial X},\partial [X])$, where $\partial: H_d(X;\mathcal{L}) \to H_{d-1}(\partial X;\mathcal{L}_{|\partial X})$ is the boundary homomorphism. | ||
+ | * A finite CW complex $X$ admits the structure of a Poincaré duality space of dimension $n$ if and only if there exists a framed compact smooth manifold $M$ of dimension $m \ge n+3$ such $M$ is homotopy equivalent to $X$ and the inclusion $\partial M \subset M$ has homotopy fiber homotopy equivalent to $S^{m-n-1}$. | ||
+ | </wikitex> | ||
==Example== | ==Example== | ||
− | + | <wikitex>; | |
A compact (smooth, PL, TOP or homology) manifold $(X,\partial X)$ of dimension $d$ is a Poincaré duality pair of dimension $d$, where $\mathcal L$ is the orientation sheaf of $X$ and $[X]$ is the manifold fundamental class. | A compact (smooth, PL, TOP or homology) manifold $(X,\partial X)$ of dimension $d$ is a Poincaré duality pair of dimension $d$, where $\mathcal L$ is the orientation sheaf of $X$ and $[X]$ is the manifold fundamental class. | ||
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</wikitex> | </wikitex> | ||
Latest revision as of 17:01, 12 June 2013
The user responsible for this page is Klein. No other user may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
2 Definition
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 the local coefficient bundle over associated with , where is the fundamental groupoid of .
3 Notes
- If , one says that is a Poincaré duality space. (In view of this, perhaps better terminology would be to call a Poincaré duality 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.
- A finite CW complex admits the structure of a Poincaré duality space of dimension if and only if there exists a framed compact smooth manifold of dimension such is homotopy equivalent to and the inclusion has homotopy fiber homotopy equivalent to .
4 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.