# round complexity

Mimicking the Lusternik-Schnirelmann category invariant for a smooth manifold^{} $M$ we can ask about the minimal number of critical loops of smooth scalar maps $M\to \mathbb{R}$ which are round functions, that is functions^{} whose critical points^{} are aligned in a disjoint union^{} of closed curves (a link).

This number is called the round complexity of $M$ and it is symbolized as $\mathrm{roc}(M)$

Then

$$\mathrm{roc}(M)=\mathrm{min}\mathrm{\#}\{\text{critical loops of}f\mathit{\hspace{1em}}|\mathit{\hspace{1em}}f:M\to \mathbb{R}\mathit{\hspace{1em}}\text{is round function}\}$$ |

This concept is related to the invariant called t-cat.

Theorem 1: The round complexity for the 2-torus and the Klein bottle is two; all the other closed surfaces have a round complexity of three.

Theorem 2: For each closed manifold, $t\mathrm{-}c\mathit{}a\mathit{}t\mathrm{\le}r\mathit{}o\mathit{}c$

Bibliography

D. Siersma, G. Khimshiasvili, On minimal^{} round functions, Preprint 1118, Department of Mathematics, Utrecht University, 1999, pp. 18.

Title | round complexity |
---|---|

Canonical name | RoundComplexity |

Date of creation | 2013-03-22 15:54:52 |

Last modified on | 2013-03-22 15:54:52 |

Owner | juanman (12619) |

Last modified by | juanman (12619) |

Numerical id | 11 |

Author | juanman (12619) |

Entry type | Definition |

Classification | msc 55M30 |

Related topic | LusternikSchnirelmannCategory |

Related topic | TCat |