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 ${\rm roc}(M)$

Then

 ${\rm roc}(M)=\min\#\{\mbox{critical loops of\quad}f\quad|\quad f\colon M\to% \mathbb{R}\quad\mbox{is round function}\}$

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, $t-cat\leq roc$

Bibliography

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

Title round complexity RoundComplexity 2013-03-22 15:54:52 2013-03-22 15:54:52 juanman (12619) juanman (12619) 11 juanman (12619) Definition msc 55M30 LusternikSchnirelmannCategory TCat