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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]], and for $k \\ge 0$ let $\\Omega^k(M)$ be the real vector space of smooth [[differential forms|0E3G0GRPZ6]] of degree $k$ on $M$.\nLet $d \\colon \\Omega^k(M) \\to \\Omega^{k+1}(M)$ be the [[exterior derivative|0ZHSTAG8ZY]]; it satisfies $d \\circ d = 0$.\nA form $\\omega$ is *closed* if $d\\omega = 0$, and *exact* if $\\omega = d\\eta$ for some $\\eta \\in \\Omega^{k-1}(M)$; for $k = 0$ only the zero form counts as exact.\nThe $k$-th *de Rham cohomology group* of $M$ is the quotient vector space\n$$H^k_{\\mathrm{dR}}(M) = \\frac{\\ker\\big(d \\colon \\Omega^k(M) \\to \\Omega^{k+1}(M)\\big)}{\\im\\big(d \\colon \\Omega^{k-1}(M) \\to \\Omega^k(M)\\big)}.$$\n\n## Remarks\n\nSince $d \\circ d = 0$, every exact form is closed, so the quotient makes sense.\n$H^0_{\\mathrm{dR}}(M)$ is the space of locally constant functions, so when $M$ has finitely many connected components its dimension is their number.\nExample: on $\\R^n$ every closed $k$-form with $k \\ge 1$ is exact, by the Poincaré lemma, so $H^k_{\\mathrm{dR}}(\\R^n) = 0$ for $k \\ge 1$ and $H^0_{\\mathrm{dR}}(\\R^n) = \\R$.\nNon-example of exactness: on $M = \\R^2 \\setminus \\set{0}$ the form $\\omega = (x \\, dy - y \\, dx)/(x^2 + y^2)$ is closed but not exact.\nIts integral over the unit circle is $2\\pi$, while the integral of an exact form $df$ over a closed loop is $0$; so $H^1_{\\mathrm{dR}}(\\R^2 \\setminus \\set{0}) \\neq 0$, and in fact it is $\\R$.\nA smooth map $f \\colon M \\to N$ induces linear maps $f^* \\colon H^k_{\\mathrm{dR}}(N) \\to H^k_{\\mathrm{dR}}(M)$ by pullback of forms, and the wedge product of forms makes $\\bigoplus_k H^k_{\\mathrm{dR}}(M)$ a graded ring.\n",
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