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  "body": "\n## Example\n\nLet $M$ be a closed oriented Riemannian manifold of dimension $n$, and let $\\Omega^k(M)$ be the space of smooth real $k$-forms.\nLet $d^*$ be the formal adjoint of the exterior derivative $d$.\nThe operator\n$$D = d + d^* \\colon \\Omega^{\\mathrm{even}}(M) \\to \\Omega^{\\mathrm{odd}}(M)$$\nis elliptic, since $D^* D$ is the Hodge Laplacian on even forms.\nIts kernel is the space of harmonic even forms, and its cokernel is isomorphic to $\\ker D^*$, the space of harmonic odd forms.\nBy the [[Hodge theorem|A127WFVYKW]], the harmonic $k$-forms form a space of dimension $b_k$, the $k$-th [[Betti number|ST71ZV3FES]].\nSo the [[analytic index|44GBFZ16SG]] is\n$$\\operatorname{ind} D = \\sum_{k = 0}^{n} (-1)^k b_k = \\chi(M),$$\nthe [[Euler characteristic|RNDDWDAFJ5]] of $M$.\n\n## Remarks\n\nFor $n$ odd, $\\chi(M) = 0$ by Poincaré duality, and the index is $0$.\nFor $n$ even, the [[topological index|QDCD33CS21]] of $D$ is the Euler number $\\langle e(TM), [M] \\rangle$.\nSo the Atiyah–Singer index theorem gives $\\chi(M) = \\langle e(TM), [M] \\rangle$, and with the Chern–Weil form of the Euler class this is the [[Chern–Gauss–Bonnet theorem|GG6NYPKMHX]].\nFor the round sphere $S^2$: $b_0 = b_2 = 1$ and $b_1 = 0$, so the index is $2$.\n\n## Explanations\n\n### L1 human\n\nThe de Rham operator $d + d^*$ sends forms of even degree to forms of odd degree.\nIts solutions are the harmonic forms, which by Hodge theory count the holes of the manifold in each dimension.\nSo its index is the alternating count of holes, the Euler characteristic.\nThis is the simplest case where the index theorem recovers a classical theorem, here Gauss–Bonnet.\n",
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