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  "body": "\n## Summary\n\nThe topological index of an elliptic operator on a closed manifold is an integer built only from topological data.\nThese data are the K-theory class of the principal symbol and the characteristic classes of the manifold.\nAtiyah and Singer defined it by pushing the symbol class forward along an embedding of the manifold into a Euclidean space.\nTheir index theorem states that it equals the [[analytic index|44GBFZ16SG]].\nIn cohomology it is an integral of the Chern character of the symbol times the [[Todd class|2YDB5089FR]].\n\n## Explanations\n\n### L1 human\n\nThe analytic index is computed by solving equations, which is hard.\nThe topological index is computed from the symbol alone, the top-order part of the operator, read as a family of matrices over the cotangent bundle.\nA small change of the operator does not change the topology of its symbol, so the topological index cannot jump.\nFor the operator $d + d^*$ from even to odd forms, the topological index is the Euler characteristic of the manifold.\n",
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