---
tag: B3AZ4W0NGT
kind: concept
name: Conley index
msc: [37B30, 37B35]
sources:
  - url: https://en.wikipedia.org/wiki/Conley_index_theory
same_as:
  wikidata: Q5161242
  wikipedia: Conley index theory
---

## Summary

The Conley index is a topological invariant attached to an isolated [[invariant set|WWZTW11664]] of a flow.
It is the [[homotopy type|SK0ZJGWXDP]] of a pointed space obtained from a compact neighbourhood by collapsing the part through which orbits leave.
It extends the Morse index of a nondegenerate critical point to invariant sets that may be large and complicated.
The index is unchanged under small perturbations of the flow, and a nontrivial index proves that the invariant set is nonempty.

## Explanations

### L1 human

Draw a box around a region where something interesting happens in a flow, such that nothing interesting touches the walls of the box.
Mark the part of the wall where the flow exits the box, and glue that exit part to a single point.
The shape of what results, up to continuous deformation, is the Conley index.
Take a saddle point in the plane, with one direction in and one direction out.
The box is a square, the exit is two opposite sides, and gluing them to a point gives a circle.
If the index is not the shape of a single point, there must be something invariant hidden inside the box.