The Essence of de Rham Cohomology

TL;DR

This paper explains de Rham cohomology, its computational tools, and proves its equivalence to singular cohomology, highlighting its importance in understanding the topology of manifolds.

Contribution

It provides an intuitive introduction to de Rham cohomology and demonstrates its equivalence to singular cohomology with detailed proofs and computational methods.

Findings

De Rham cohomology is equivalent to singular cohomology.

Presented computational tools: Mayer-Vietoris, homotopy invariance, Poincaré duality, Künneth formula.

Proved de Rham's theorem establishing the equivalence.

Abstract

The study of differential forms that are closed but not exact reveals important information about the global topology of a manifold, encoded in the de Rham cohomology groups $H^k(M)$, named after Georges de Rham (1903-1990). This expository paper provides an explanation and exploration of de Rham cohomology and its equivalence to singular cohomology. We present an intuitive introduction to de Rham cohomology and discuss four associated computational tools: the Mayer-Vietoris theorem, homotopy invariance, Poincar\'e duality, and the K\"unneth formula. We conclude with a statement and proof of de Rham's theorem, which asserts that de Rham cohomology is equivalent to singular cohomology.