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  "body": "\n## Summary\n\nDe Rham cohomology is a cohomology theory of [[smooth manifolds|F45VQ47GPV]] built from [[differential forms|0E3G0GRPZ6]].\nIts groups measure how far closed forms, those with $d\\omega = 0$, are from being exact, of the form $\\omega = d\\eta$.\nThey are real vector spaces that are diffeomorphism invariants, and by the [[de Rham theorem|K7Z2PQZ4WH]] they agree with singular cohomology with real coefficients.\nThis links calculus on a manifold to its topology.\n\n## Explanations\n\n### L1 human\n\nIn 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.\nThe swirling field around the hole has zero curl but is not a gradient: its line integral around the hole is not zero.\nDe Rham cohomology counts such obstructions in every degree.\nFor the punctured plane, there is exactly one independent obstruction in degree $1$, which detects the single hole.\n",
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