finiteness of the dimensions of cohomologies of open subsets of a compact manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T07:26:36Z http://mathoverflow.net/feeds/question/99474 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99474/finiteness-of-the-dimensions-of-cohomologies-of-open-subsets-of-a-compact-manifol finiteness of the dimensions of cohomologies of open subsets of a compact manifold Alberto Jermaine 2012-06-13T16:43:52Z 2012-06-13T16:43:52Z <p>Let $M$ be a compact differentiable manifold which can be covered by two open subsets $U$ and $V$. Then $H_{\text{dR}}^n(M)$ is finite-dimensional for all $n$. But how about $U$, $V$ and $U\cap V$? Are their de Rham cohomologies necessarily finite-dimensional? The open sets $U$ and $V$ can be chosen to be very strange but I have not found a counterexample. Also, the Mayer-Vietoris sequence and the rank-nullity theorem seem not to provide any information about this.</p>