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  "body": "\n## Definition\n\nLet $M$ be a closed smooth manifold, with cotangent bundle $\\pi \\colon T^*M \\to M$.\nLet $P \\colon C^\\infty(M; E) \\to C^\\infty(M; F)$ be an [[elliptic operator|MT9P2X85V8]] between complex [[vector bundles|P4DZM975BV]].\nIts [[principal symbol|S8GC34D5ZG]] $\\sigma(P)$ is a bundle map $\\pi^* E \\to \\pi^* F$ that is invertible outside the zero section.\nSo it defines a class $[\\sigma(P)]$ in the compactly supported K-theory $K(T^*M)$.\nChoose a smooth embedding $i \\colon M \\to \\R^N$.\nIt induces a Gysin map $i_! \\colon K(T^*M) \\to K(T^*\\R^N) = K(\\R^{2N})$, and Bott periodicity identifies $K(\\R^{2N})$ with $\\Z$.\nThe *topological index* of $P$ is the integer\n$$\\operatorname{t-ind} P = i_! [\\sigma(P)] \\in \\Z.$$\nIt does not depend on the choice of the embedding.\n\n## Remarks\n\nIn cohomology, the topological index is the integral over $T^*M$ of $\\operatorname{ch}[\\sigma(P)] \\cdot \\pi^* \\operatorname{Td}(TM \\otimes \\C)$, up to a sign fixed by orientation conventions.\nHere $\\operatorname{ch}$ is the Chern character and $\\operatorname{Td}$ is the [[Todd class|2YDB5089FR]].\nLet $M$ be a closed oriented Riemannian manifold of even dimension and $P = d + d^*$ from even to odd forms.\nThen the topological index is the Euler number $\\langle e(TM), [M] \\rangle$, which is the Euler characteristic of $M$.\n\nExample: let $P \\colon C^\\infty(M; E) \\to C^\\infty(M; F)$ be the operator of order $0$ given by a bundle isomorphism $\\phi \\colon E \\to F$.\nIts symbol is $\\pi^*\\phi$, which is invertible on all of $T^*M$, so $[\\sigma(P)] = 0$ in $K(T^*M)$ and $\\operatorname{t-ind} P = 0$.\nThis agrees with the analytic index, since $P$ is bijective.\n\nNon-example: for a non-elliptic operator, the principal symbol is not invertible at some nonzero covector.\nThen $\\sigma(P)$ does not define a class in $K(T^*M)$, and there is no topological index.\n",
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