On one hand, a closed disk $D^2$ is contractible so that it has the same cohomology as a point, so $H^2(D^2)=0$.
On the other hand, $D^2$ is a compact orientable manifold (isn't it?). According to Top deRham cohomology group of a compact orientable manifold is 1-dimensional, $H^2(D^2)\cong\mathbb R$ and I got a conflict.
Where am I wrong? I guess "$H^n(M)\cong\mathbb R$ where $M$ is a compact orientable smooth manifold of dimension n" is not applicable to a manifold with boundary?
