Can a space have a de Rham cohomology other than $\mathbb{R}^n$? I have yet to see an example. (i.e. can a space have k$^{th}$ cohomology group = n torus or something?)
2026-04-09 16:32:59.1775752379
de Rham cohomology types
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de Rham cohomology groups are actually real vector spaces, so I don't think a space can have an $n$-torus as its cohomology.
If you're asking whether all spaces have the same de Rham cohomology as $\mathbb{R}^n$, the answer is no. For example $H^1(S^1)=\mathbb{R}$ while $H^1(\mathbb{R}^n)=0$.