# Lusternik-Schnirelmann category

Let $X$ be a topological space^{}. An important topological invariant^{} of $X$ called Lusternik-Schnirelmann category cat is defined as follows:

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

If $X$ is a manifold^{}, $\mathrm{cat}(X)$ coincides with the minimal number of critical points among all smooth scalars maps $X\to \mathbb{R}$.

This is equivalent^{} to saying that $X$ has a covering $\{{U}_{s}\}$ such that
it is posible to factor homotopically each ${U}_{s}\stackrel{i}{\hookrightarrow}X$ through ${U}_{s}\stackrel{a}{\to}*\stackrel{b}{\to}X$ i.e

$$i\simeq b\circ a.$$ |

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

We can ask about the minimal number of open sets ${U}_{s}$ that cover $X$ and are homotopically equivalent to ${S}^{1}$, say, the inclusion ${U}_{s}\stackrel{i}{\hookrightarrow}X$ and ${U}_{s}\stackrel{a}{\to}{S}^{1}\stackrel{b}{\to}X$ are $i\simeq b\circ a$.

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

## References

- 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 |