Poincaré Duality Spaces
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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 $M$ | + | * 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 $S^{m-n-1}$. |
− | 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 $S^{m-n-1}$. | + | |
Revision as of 23:28, 24 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 the local coefficient bundle over associated with , where is 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 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 .
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.