---
tag: 4NCYA7GME1
kind: concept
name: Signature operator
msc: [58J20, 58A14, 57R20]
sources:
  - url: https://en.wikipedia.org/wiki/Signature_operator
same_as:
  wikidata: Q48995679
  wikipedia: Signature operator
---

## Summary

The signature operator of a closed oriented Riemannian manifold of dimension $4k$ is the operator $d + d^*$ on complex [[differential forms|0E3G0GRPZ6]].
It is restricted to the forms that are even under a natural involution built from the [[Hodge star|1Y4WX4XQWJ]].
It is an [[elliptic operator|MT9P2X85V8]], and its index equals the signature of the manifold, the signature of the [[intersection form|WK8GKRKAVD]] on middle cohomology.
Applied to it, the Atiyah–Singer index theorem gives the [[Hirzebruch signature theorem|TEG1ZX1AGJ]].
It is one of the classical operators of index theory, together with the de Rham, Dolbeault and spin Dirac operators.

## Explanations

### L1 human

The signature of a manifold of dimension $4k$ is a whole number that counts how the middle-dimensional cycles intersect, positive pairs minus negative pairs.
The signature operator is a differential operator built from the metric whose index, the number of solutions minus the number of obstructions, recovers this number.
In this way a purely topological count is expressed through analysis.
For the complex projective plane the signature is $1$, and so is the index of its signature operator.