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?


  • $\begingroup$ Sufficient for what? $\endgroup$ – Mariano Suárez-Álvarez May 15 '11 at 14:49
  • $\begingroup$ 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. $\endgroup$ – Mariano Suárez-Álvarez May 15 '11 at 14:52
  • $\begingroup$ @Mariano I meant sufficient to compute top dimensional cohomology group or module of a closed manifold $\endgroup$ – El Moro May 15 '11 at 15:02

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.