Poincaré Duality Spaces
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Here, $\mathcal B$ is allowed to range over all local coefficient bundles on $X$, | 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 the local coefficient bundle over $X$ defined by | but in fact it is sufficient to check the condition when $\mathcal{B}$ is the local coefficient bundle over $X$ defined by | ||
− | $\ | + | taking the fiberwise free abelian group on $\tilde X \to X$, the universal cover of $X$. |
==Notes== | ==Notes== |
Revision as of 18:20, 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 the local coefficient bundle over defined by taking the fiberwise free abelian group on , the universal cover 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.