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  "body": "\n## Summary\n\nThe signature operator of a closed oriented Riemannian manifold of dimension $4k$ is the operator $d + d^*$ on complex [[differential forms|0E3G0GRPZ6]].\nIt is restricted to the forms that are even under a natural involution built from the [[Hodge star|1Y4WX4XQWJ]].\nIt is an [[elliptic operator|MT9P2X85V8]], and its index equals the signature of the manifold, the signature of the [[intersection form|WK8GKRKAVD]] on middle cohomology.\nApplied to it, the Atiyah–Singer index theorem gives the [[Hirzebruch signature theorem|TEG1ZX1AGJ]].\nIt is one of the classical operators of index theory, together with the de Rham, Dolbeault and spin Dirac operators.\n\n## Explanations\n\n### L1 human\n\nThe signature of a manifold of dimension $4k$ is a whole number that counts how the middle-dimensional cycles intersect, positive pairs minus negative pairs.\nThe signature operator is a differential operator built from the metric whose index, the number of solutions minus the number of obstructions, recovers this number.\nIn this way a purely topological count is expressed through analysis.\nFor the complex projective plane the signature is $1$, and so is the index of its signature operator.\n",
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