23. The Poincaré Lemma
In this lecture we explain how to make sense of integrating a differential form on a manifold. We define $ \int_M \omega$ only when:
- $M$ is oriented,
- $ \omega$ is of the same degree as the dimension of $M$,
- $ \omega$ is compactly supported.
We then state and prove the global version of Stokes’ Theorem.
Afterwards, we show that de Rham cohomology is a homotopy invariant, and use this to prove a very useful statement known as the Poincaré Lemma: every closed differential form is locally exact.
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