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  "body": "\n## Definition\n\nLet $M$ be a closed smooth manifold, let $E$ and $F$ be complex vector bundles over $M$, and let $P \\colon C^\\infty(M; E) \\to C^\\infty(M; F)$ be an [[elliptic differential operator|MT9P2X85V8]].\nIts [[analytic index|44GBFZ16SG]] $\\operatorname{ind} P = \\dim \\ker P - \\dim \\operatorname{coker} P$ is a finite integer.\n*Index theory* is the part of global analysis that computes such indices, and their refinements, in terms of topological data.\nIts main tool is the [[topological index|QDCD33CS21]], which depends only on the homotopy class of the principal symbol of $P$.\n\n## Remarks\n\nThe Atiyah–Singer index theorem states that the analytic index and the topological index of $P$ are equal.\nRefinements replace the integer index by an element of a representation ring (equivariant index) or of a K-theory group of a parameter space (families index).\nOther refinements add correction terms on manifolds with boundary.\n\nExample: on the circle $S^1 = \\R / 2\\pi\\Z$, the operator $P = d/d\\theta$ on complex functions is elliptic.\nIts kernel is the constants and its image is the functions of mean zero, so $\\operatorname{ind} P = 1 - 1 = 0$.\n\nNon-example: the dimension of the kernel alone is not an index quantity, because it is not stable.\nFor $P_c = d/d\\theta + c$ with $c \\in \\C$, $\\dim \\ker P_c$ is $1$ when $c \\in i\\Z$ and $0$ otherwise, while $\\operatorname{ind} P_c = 0$ for all $c$.\n",
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