I am having problems calculating the cohomology with compact support of the open Möbius strip (without the bounding edge).
I am using the Mayer Vietoris sequence: U and V are two open subsets diffeomorphic to $\mathbb{R}^2$ and $U\cap V$ is diffeomorphic to two copies of $\mathbb{R}^2$.
$H^0_C(M)=0$ and that's ok, but then I get the exact sequence
$ 0 \rightarrow H^1_C(M) \rightarrow H^2_C(U\cap V) \rightarrow H^2_C(U)\oplus H^2_C(V) \rightarrow H^2_C(M)\rightarrow 0$
where both $H^2_C(U\cap V)$ and $H^2_C(U)\oplus H^2_C(V)$ have dimension 2.
I would say that the function $\delta:H^2_C(U\cap V) \rightarrow H^2_C(U)\oplus H^2_C(V)$ in the exact sequence sends $(\phi,\psi)$ to $(-j_U(\phi + \psi),j_V(\phi + \psi))$ where $\phi$ and $\psi$ are generators of $H^2_C(U\cap V)$.
So I would say that $Im\ \delta$ is one dimensional and spanned by $(-j_U(\phi + \psi),j_V(\phi + \psi))$ and $\dim H^1_C(M)=\dim H^2_C(M)=1$. But, I have read that all compact cohomology classes of the Möbius strip are zero. So I must be wrong somewhere.