De Rham Cohomology of Sphere

Theorem

Let $S^n$ denote the $n$-Sphere.

Then the de Rham Cohomology of $S^n$ are:

$\map {H^k} {S^n} = \begin {cases} \Z^2 & : k = 0, n = 0 \\ \Z & : k = 0 \text { or } k = n, n > 0 \\ 0 & : 0 < k < n \end {cases}$

and Higher de Rham Cohomology Vanishes.

This article, or a section of it, needs explaining. In particular: The condition for the middle case is not uniquely readable, and need to be clarified. If the cases are exhaustive, the middle one might be "otherwise". You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Proof

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