Poincaré Duality Spaces
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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 $ | + | 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 $X$ |
and $[X]$ is the fundamental class. | and $[X]$ is the fundamental class. | ||
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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.