round complexity

Mimicking the Lusternik-Schnirelmann category invariant for a smooth manifoldMathworldPlanetmath M we can ask about the minimal number of critical loops of smooth scalar maps M which are round functions, that is functionsMathworldPlanetmath whose critical pointsDlmfPlanetmath are aligned in a disjoint unionMathworldPlanetmath of closed curves (a link).

This number is called the round complexity of M and it is symbolized as roc(M)


roc(M)=min#{critical loops of f|f:Mis 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-catroc


D. Siersma, G. Khimshiasvili, On minimalPlanetmathPlanetmath 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