---
tag: 675MAXBQN6
kind: concept
name: Index theory
msc: [58J20, 19K56, 47A53]
sources:
  - url: https://en.wikipedia.org/wiki/Atiyah–Singer_index_theorem
---

## Summary

Index theory studies the [[Fredholm index|2V9Q4KHY1T]] of [[elliptic operators|XY8GEQSWPR]] on manifolds.
Its central result is the Atiyah–Singer index theorem.
It says that the [[analytic index|44GBFZ16SG]] of an elliptic operator on a closed manifold equals its [[topological index|QDCD33CS21]].
The topological index is an integer built from characteristic classes.
The theory links analysis, topology and geometry.
Classical results such as the [[Chern–Gauss–Bonnet theorem|GG6NYPKMHX]], the [[Hirzebruch signature theorem|TEG1ZX1AGJ]] and the [[Hirzebruch–Riemann–Roch theorem|G3X02FKQCR]] are special cases.

## Explanations

### L1 human

An elliptic equation $Pu = f$ on a closed manifold has finitely many independent solutions of $Pu = 0$ and finitely many independent obstructions to solving it.
The index is the first number minus the second.
Each number alone can jump when the operator changes a little, but the difference stays fixed.
Index theory explains this stability: the index depends only on the topology of the manifold and of the operator.
For 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.