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In this final chapter of Volume I we return to the de Rham cohomology of a smooth manifold. We show that de Rham cohomology is a homotopy invariant, and use this to prove the Poincaré Lemma, which states that any closed form is locally exact. In the bonus section we prove the de Rham Theorem, which can be thought of as a massive generalisation of the homotopy invariance property.

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The Poincaré Lemma and the de Rham Theorem

  • 章节

  • 发表日期：

    2026年

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  • Will J. Merry

摘要

In this final chapter of Volume I we return to the de Rham cohomology of a smooth manifold. We show that de Rham cohomology is a homotopy invariant, and use this to prove the Poincaré Lemma, which states that any closed form is locally exact. In the bonus section we prove the de Rham Theorem, which can be thought of as a massive generalisation of the homotopy invariance property.