Find the DeRham cohomology of the spaces minus a line and a circle Find the DeRham's cohomology of the following open sets, and then looking at the product conclude that they are not diffeomorphic.
a) $M=\mathbb{R}^3\setminus (L_1\cup C)$,  $L_1=\{x=y=0\}$ and $C=\{x^2+y^2=1,z=0\}$.
b) $N=\mathbb{R}^3\setminus (L_2\cup C)$,  $L_2=\{x=3,y=0\}$ and $C=\{x^2+y^2=1,z=0\}$.
Can you give me an idea?
I tried:
Idea
I write
$$M=
\left(\mathbb{R^3}\setminus L_1\right)\cap \left(\mathbb{R^3}\setminus C\right)$$
So I had to find first the cohomology of $\left(\mathbb{R^3}\setminus L_1\right)$ and $\left(\mathbb{R^3}\setminus C\right)$. But I dont how to find them and neither how to use the different positions of the line respecto to the circle
Thank you
 A: Perhaps some visual hints will help, if you'll excuse the hand-drawn examples...

First we see $M$ deformation retracts to a torus. Intuitively, the missing line at $x=y=0$ becomes the hole through the donut, while the missing circle at $x^2+y^2 = 1, z=0$ becomes the hole inside the donut. 
Next, we consider $N$. Since the circle and the line aren't interlinked, we may consider them separately, and then glue them together at the end with the Mayer-Vietoris Sequence. 
On the left, we have $\mathbb{R}^3 \setminus S^1$, which is "well known" to deformation retract to $S^2 \vee S^1$. Instead of thinking about this as a sphere with a circle on its side, think of it as a sphere with a diameter linking its two poles. Then the missing circle becomes the hollow bit between this diameter and the surface of the sphere... Sorry I don't have a better picture for this. Perhaps the graphics here will help. This can also be found on page 46 of Hatcher.
On the right, we have $\mathbb{R}^3$ without a line. This deformation retracts to the plane without a point (by squishing down), and of course this is well known to be a circle. The missing point becomes the hole in the center of the circle.
Do you think you can use this information (as well as the Mayer-Vietoris Sequence for $N$) to discover the cohomology of these spaces, and thereby deduce they aren't diffeomorphic? 

I hope this helps ^_^
