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Where is the cohomology with compact support useful? It seems that, a part from proving Poincaré duality, we also use it to compute the top dimensional cohomology group of closed manifolds: isn't singular cohomology sufficient here?

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Sufficient for what? –  Mariano Suárez-Alvarez May 15 '11 at 14:49
    
In any case it is easy to underestimate the importance and usefulness of having a cohomology theory with a nice duality property such as Poincaré duality. –  Mariano Suárez-Alvarez May 15 '11 at 14:52
    
@Mariano I meant sufficient to compute top dimensional cohomology group or module of a closed manifold –  El Moro May 15 '11 at 15:02

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up vote 5 down vote accepted

For a closed manifold, cohomology and cohomology with compact supports coincide, but for open manifolds they do not. The top dimensional cohomology with compact support is always one-dimensional for a connected orientable manifold, regardless of whether or not the manifold is closed. Furthermore, on any (connected orientable) manifold, closed or not, Poincare duality is true when expressed as a duality between cohomology and cohomology with compact support in the complementary dimension. Thus cohomology with compact support is a natural tool when working with non-closed manifolds.

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