Example of two non-homeomorphic spaces with the same de Rham group

Question

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?

Answer 1

Your example generalizes to any space $X$ and $X \times [0, 1].$ Andrew Hwang's example is essentially the same.

Answer 2

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.

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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).

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I hope this helps ^_^