In this final lecture 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.

In the bonus section we give a (mostly complete and entirely non-examinable) proof of the de Rham Theorem, which states that de Rham cohomology is naturally isomorphic to singular cohomology with real coefficients. We conclude with a very brief discussion of Poincaré Duality.

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