I have to find the De Rham's cohomology groups of $X$, where:
a)$X=S^1\times S^1$.
b) $X=\mathbb{R}^3\setminus \mathbb{R}$.
c) $X=\mathbb{R}^3\setminus S^1$.
(d) $X=\mathbb{R}^3\setminus (L_1\cup C)$, $L_1=\{x=y=0\}$ and $C=\{x^2+y^2=1,z=0\}$.
(e) $X=\mathbb{R}^3\setminus (L_2\cup C)$, $L_2=\{x=3,y=0\}$ and $C=\{x^2+y^2=1,z=0\}$.
I've seen for (a) using Mayer-Vietoris sequence, but I would like to do it by definition. I have no idea.
For (b) I will identify $\mathbb{R}=\{(x,0,0)\}$, and then take $U=\{(x,y,z)/yz>0\}$ and $V=\{(x,y,z)/yz<0\}$, then
$$U\cup V=\mathbb{R}^3\setminus \mathbb{R},\quad U\cap V=\emptyset$$ and then use Mayer-Vietoris.
For (c) $S^1=C\cap \mathbb{R}^2$, where $C=\{(x,y,z)/x^2+y^2=1\}$ the cilynder and $\mathbb{R}^2=\{(x,y,0)\}$. Then
$$\mathbb{R}^3\setminus S^1=\mathbb{R}^3\setminus \left(C\cap \mathbb{R}^2\right) =\left(\mathbb{R}^3\setminus C\right)\cup\left(\mathbb{R}^3\setminus \mathbb{R}^2\right)=U\cup V$$ and then use Mayer-Vietoris.
Are my ideas for (b) and (c) right? Can they be solved by definition?
For the other ones, i can use properties ou M-V sequence. Here is what I tried (d) $X=\left(\mathbb{R^3}\setminus L_1\right)\cap \left(\mathbb{R^3}\setminus C\right)=M\cap N$, but $M$ and $N$ are like (b) and (c).
(e) Thinking...
I will appreciate your help.
Thank you.