---
tag: 1991WETZF9
kind: concept
name: Stable map
msc: [58K25, 57R45]
sources:
  - url: https://en.wikipedia.org/wiki/Singularity_theory
---

## Summary

A smooth map between [[smooth manifolds|F45VQ47GPV]] is stable if every map close enough to it is equivalent to it.
Here equivalent means equal after a change of coordinates in the source and in the target.
Small perturbations do not change its singularities in kind, only move them a little.
Stable maps are the maps whose singularities survive in nature and in experiments, which is why [[singularity theory|42V5XGZS4F]] aims to classify them.
Mather proved that stability is equivalent to a linear condition, [[infinitesimal stability|C3TFCVE2YK]], for proper maps.

## Explanations

### L1 human

Imagine a map drawn with a slightly shaky hand.
If the shaky copy can always be turned back into the original by smoothly bending the source and the target, the map is stable.
The height function on a circle is stable: tilt or wobble the circle a little and it still has one highest and one lowest point, of the same kind.
A constant function on a circle is not stable: a tiny wobble gives a function with peaks and valleys, which no change of coordinates can make constant again.