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  "body": "\n## Summary\n\nIndex theory studies the [[Fredholm index|2V9Q4KHY1T]] of [[elliptic operators|XY8GEQSWPR]] on manifolds.\nIts central result is the Atiyah–Singer index theorem.\nIt says that the [[analytic index|44GBFZ16SG]] of an elliptic operator on a closed manifold equals its [[topological index|QDCD33CS21]].\nThe topological index is an integer built from characteristic classes.\nThe theory links analysis, topology and geometry.\nClassical results such as the [[Chern–Gauss–Bonnet theorem|GG6NYPKMHX]], the [[Hirzebruch signature theorem|TEG1ZX1AGJ]] and the [[Hirzebruch–Riemann–Roch theorem|G3X02FKQCR]] are special cases.\n\n## Explanations\n\n### L1 human\n\nAn elliptic equation $Pu = f$ on a closed manifold has finitely many independent solutions of $Pu = 0$ and finitely many independent obstructions to solving it.\nThe index is the first number minus the second.\nEach number alone can jump when the operator changes a little, but the difference stays fixed.\nIndex theory explains this stability: the index depends only on the topology of the manifold and of the operator.\nFor example, on a closed surface of genus $g$ the operator $d + d^*$ from even to odd differential forms has index $2 - 2g$, the Euler characteristic.\n",
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