The Poincaré Lemma and the de Rham Theorem

Will J. Merry
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Will J. Merry: ETH Zürich, Departement Mathematik

Chapter Chapter 27 in Lectures on Differential Geometry I, 2026, pp 273-284 from Springer

Abstract: Abstract 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.

Date: 2026
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DOI: 10.1007/978-3-032-03733-6_27

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