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 $M\to ℝ$ 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

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

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.