Example of two non-homeomorphic spaces with the same de Rham group Can anyone give an example of two non-homeomorphic spaces with the same de Rham cohomology? I was thinking of $[0,1]$ and $\{0\}$ but does anyone have a more spectacular example?
 A: Your example generalizes to any space $X$ and $X \times [0, 1].$ Andrew Hwang's example is essentially the same.
A: For a slightly more exciting example, try a torus with one puncture and a "pair of pants" (shown below). These are both manifolds (with boundary) which deformation retract to a figure 8.
 $\quad\quad\quad$
A deep result (de Rham's Theorem) says that the de Rham cohomology of a manifold actually agrees with the singular cohomology (computed as a mere topological space)! Of course, the singular cohomology only depends on the homotopy type. In particular, we have:
$$
\begin{aligned}
H_\text{de Rham}^n(\text{punctured torus}) 
&\cong H_\text{singular}^n(\text{punctured torus}) \\
&\cong H_\text{singular}^n(\text{figure 8}) \\
&\cong H_\text{singular}^n(\text{pair of pants}) \\
&\cong H_\text{de Rham}^n(\text{pair of pants})
\end{aligned}
$$
Lastly, we should check that a punctured torus is not homeomorphic to a pair of pants. But the punctured torus has only one boundary component (the puncture) while the pair of pants has three (the waist and two legs).

I hope this helps ^_^
