Computing the de Rham cohomology for the torus. I'm computing the de Rham cohomology for the $2$-dimensional torus using the Mayer-Vietoris sequence with the following cover given on Wikipedia.
What I currently have is that for $H^0$ we have the sequence $$0 \longrightarrow H^0(\mathbb{T}^2)  \xrightarrow{i_0^*} H^0(U ) \oplus H^0 (V)  \xrightarrow{j_0^*} H^0(U \cap V)  \xrightarrow{d_0^*}  \dots$$
which continues as $$\dots   \xrightarrow{d_0^*}   H^1(\mathbb{T}^2) \xrightarrow{i_1^*} H^1(U ) \oplus H^1 (V)  \xrightarrow{j_1^*} H^1(U \cap V)  \xrightarrow{d_1^*}  \dots$$ up to $$\dots \xrightarrow{d_1^*} H^2(\mathbb{T}^2) \longrightarrow 0. $$
Now the first bit concering $H^0$ reduced to $$0 \longrightarrow \mathbb{R} \xrightarrow{i_0^*} \mathbb{R} \oplus \mathbb{R}  \xrightarrow{j_0^*} \mathbb{R} \oplus \mathbb{R}  \xrightarrow{d_0^*}  \dots$$ due to connectedness and the fact that $U$ and $V$ are homotopy equivalent with $S^1$'s.
The part concering $H^1$ reduced to $$\dots   \xrightarrow{d_0^*}   H^1(\mathbb{T}^2) \xrightarrow{i_1^*} \mathbb{R} \oplus \mathbb{R}  \xrightarrow{j_1^*} \mathbb{R} \oplus \mathbb{R}\xrightarrow{d_1^*}  \dots$$ again due to the homotopy equivalences.
Now we only need to figure out $H^1(\mathbb{T}^2)$ and $H^2(\mathbb{T}^2)$. For the latter we have $$H^2(\mathbb{T}^2) = \operatorname{im}(d_1^*) = \frac{\mathbb{R} \oplus \mathbb{R}}{\ker(d_1^*)} = \frac{\mathbb{R} \oplus \mathbb{R}}{\operatorname{im}(j_1^*)}.$$ My question how do I figure out what $\operatorname{im}(j_1^*)$ should be? I think I'll get the same conclusion for $H^1(\mathbb{T}^2)$ where I would need to figure out what $\operatorname{im}(j_0^*)$ is. How can these be found?

When you have an exact sequence, the alternate sum of the dimensions of the cohomology groups is zero. Here this gives
$$1-2+2-dim\ H^1+2-2+dim\ H^2 = 0,$$
$$dim\ H^1 = dim\ H^2 + 1.$$
Since the torus is a connected orientable manifold, its top dimensional cohomology group is one dimensional, generated by the volume form. So $dim \ H^2 = 1$ and $dim\ H^1 = 2$.
