Lusternik-Schnirelmann category

Let X be a topological spaceMathworldPlanetmath. An important topological invariantPlanetmathPlanetmath of X called Lusternik-Schnirelmann category cat is defined as follows:

cat(X)=min{#(C):where C are the coverings of X by contractible open sets}.

If X is a manifoldMathworldPlanetmath, cat(X) coincides with the minimal number of critical points among all smooth scalars maps X.

This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to saying that X has a covering {Us} such that it is posible to factor homotopically each UsiX through Usa*bX i.e


This allows us to define another category, e.g.:

We can ask about the minimal number of open sets Us that cover X and are homotopically equivalent to S1, say, the inclusion UsiX and UsaS1bX are iba.

It is becoming standard to speak of the t-cat of X. This is related to the round complexity of the space.


  • 1 R.H. Fox, On the Lusternik-Schnirelmann category, Annals of Math. 42 (1941), 333-370.
  • 2 F. Takens, The minimal number of critical points of a function on compact manifolds and the Lusternik-Schnirelmann category, Invent. math. 6,(1968), 197-244.
Title Lusternik-Schnirelmann category
Canonical name LusternikSchnirelmannCategory
Date of creation 2013-03-22 15:53:30
Last modified on 2013-03-22 15:53:30
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 13
Author juanman (12619)
Entry type Definition
Classification msc 55M30
Related topic Topology
Related topic RoundComplexity