---
tag: GX74C07T0G
kind: concept
name: de Rham cohomology
msc: [58A12, 57R19]
sources:
  - url: https://en.wikipedia.org/wiki/De_Rham_cohomology
same_as:
  wikidata: Q1179446
  wikipedia: De Rham cohomology
---

## Summary

De Rham cohomology is a cohomology theory of [[smooth manifolds|F45VQ47GPV]] built from [[differential forms|0E3G0GRPZ6]].
Its groups measure how far closed forms, those with $d\omega = 0$, are from being exact, of the form $\omega = d\eta$.
They are real vector spaces that are diffeomorphism invariants, and by the [[de Rham theorem|K7Z2PQZ4WH]] they agree with singular cohomology with real coefficients.
This links calculus on a manifold to its topology.

## Explanations

### L1 human

In calculus, a vector field with zero curl on all of space is a gradient, but on the plane with a hole removed this can fail.
The swirling field around the hole has zero curl but is not a gradient: its line integral around the hole is not zero.
De Rham cohomology counts such obstructions in every degree.
For the punctured plane, there is exactly one independent obstruction in degree $1$, which detects the single hole.