De Rham Cohomology Computed
One may often find the general de Rham cohomologies of a manifold using the above fact about the zero cohomology and a Mayer–Vietoris sequence. Another useful fact is that the de Rham cohomology is a homotopy invariant. While the computation is not given, the following are the computed de Rham cohomologies for some common topological objects:
The n-sphere:
For the n-sphere, and also when taken together with a product of open intervals, we have the following. Let n > 0, m ≥ 0, and I an open real interval. Then
The n-torus:
Similarly, allowing n > 0 here, we obtain
Punctured Euclidean space:
Punctured Euclidean space is simply Euclidean space with the origin removed. For n > 0, we have:
The Möbius strip, M:
This follows from the fact that the Möbius strip can be deformation retracted to the 1-sphere:
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