Calculate the de Rham cohomology of the Möbius band Using Mayer-Vietoris to calculate the de Rham cohomology of the Möbius band $M$, what is the choice of separation?
i.e. $M=U\cup V$, which $U$ and $V$ well chosen for calculation?
 A: You can take these two open sets (in blue and red) :

The long exact Mayer-Vietoris sequence is $$0\rightarrow H^0(M)\rightarrow H^0(U)\oplus H^0(V)\rightarrow H^0(U\cap V)\rightarrow H^1(M)\rightarrow H^1(U)\oplus H^1(V)\rightarrow$$
But the Möbius strip is connected, so $H^0(M)=\mathbb R$.
The open sets are connected, so $H^0(U)=H^0(V)=\mathbb R$.
The intersection has two connected components so $H^0(U\cap V)=\mathbb R^2$.
The opens sets are contractible so $H^1(U)=H^1(V)=0$.
So you have the exact sequence
$$0\rightarrow\mathbb R\rightarrow \mathbb R^2\rightarrow\mathbb R^2\rightarrow H^1(M)\rightarrow0$$
So you have $1-2+2-\dim H^1(M)=0\Rightarrow \dim H^1(M)=1$.
Next, $$H^1(U)\oplus H^1(V)=0\rightarrow H^1(U\cap V)=0\rightarrow H^2(M)\rightarrow H^2(U)\oplus H^2(V)=0$$ gives you $H^2(M)=0$.
Conclusion: $H^k(M)=\left\{\begin{array}{ll}\mathbb R & \text{ if }k=0,1 \\ 0 & \text{ else}\end{array}\right.$
Remark: without the MV sequence, you could notice that the median circle $\mathbb S^1$ is a deformation retract of the Möbius strip and use https://math.stackexchange.com/a/162378/33615.
