---
tag: QARH15D2PN
kind: concept
name: Lusternik–Schnirelmann theory of critical points
msc: ["58E05", 55M30]
specializes: [9ZQC7ARWTT]
sources:
  - url: https://en.wikipedia.org/wiki/Lusternik–Schnirelmann_category
---

## Summary

Lusternik–Schnirelmann theory gives lower bounds for the number of [[critical points|0YTCMFB4HC]] of a function from the topology of its domain, without any nondegeneracy assumption.
Its main result is that a $C^1$ function on a closed manifold has at least $\operatorname{cat}(M) + 1$ critical points, where $\operatorname{cat}(M)$ is the normalized [[Lusternik–Schnirelmann category|W5BHHXCYXA]].
The critical values are obtained by minimax over closed sets of increasing category.
It is a branch of [[critical point theory|9ZQC7ARWTT]], complementary to Morse theory, and it extends to infinite dimensions under the [[Palais–Smale condition|4724KRNHAT]].

## Explanations

### L1 human

Morse theory counts critical points of a function with only clean critical points, and gives bounds from the homology of the space.
Lusternik and Schnirelmann asked what can be said for any smooth function, however degenerate.
Their answer is a number measuring how many contractible pieces are needed to cover the space.
For example, a torus needs three such pieces, so every smooth function on a torus has at least three critical points.
On a sphere the bound is two: a maximum and a minimum.