---
tag: W6SSJN7TVW
kind: concept
name: Cerf theory
msc: [57R70, 58K05, 57R65]
sources:
  - url: https://en.wikipedia.org/wiki/Cerf_theory
same_as:
  wikidata: Q5064229
  wikipedia: Cerf theory
---

## Summary

Cerf theory studies one-parameter families of smooth real-valued functions on a manifold and the ways in which they fail to be [[Morse functions|KPQ96640ZJ]].
In a generic family, the function is a Morse function with distinct critical values at all but finitely many times.
At those times either a pair of critical points is born or dies, or two critical values cross.
Jean Cerf developed it to study diffeomorphisms of manifolds, in particular to prove pseudo-isotopy results.
Through the correspondence between Morse functions and [[handle decompositions|9V11N8VVTP]], it describes how any two handle decompositions of a manifold are related.

## Explanations

### L1 human

Morse theory looks at one function on a manifold; Cerf theory looks at a film in which the function changes with time.
Most frames of the film show a Morse function, with clean peaks, pits and saddles.
At a few moments something happens: a peak and a saddle melt together and disappear, or appear out of nothing, or two of them swap heights.
For example, $f_t(x) = x^3 - t x$ on the line has no critical points for $t < 0$ and two for $t > 0$, which are born at $t = 0$.