Remarks on minimal round functions

Abstract

Abstract. We describe the structure of minimal round functions on compact closed surfaces and three-dimensional manifolds. The minimal possible number of critical loops is determined and typical non-equisingular round function germs are interpreted in the spirit of isolated line singularities. We also discuss a version of Lusternik-Schnirelmann theory suitable for round functions. Introduction.A differentiable function on a differentiable manifold is called a round function if its critical set is a union of embedded one-dimensional submanifolds. In general it is not assumed that the critical submanifolds are non-degenerate critical submanifolds in the sense of R. Bott [3] so this is a straightforward extension of the notion of round Morse function introduced by W. Thurston [30].Our main concern in this paper are round functions on a given compact closed (i.e., without boundary) smooth manifold with the minimal possible number of critical submanifolds (critical loops). By analogy with the terminology of [29] they will be called minimal round functions as in [21]. We are especially interested in describing possible changes of the topology of their Lebesgue sets (preimages of semi-infinite intervals) and typical local models of their singular behaviour near critical loops. Our setting and approach are much in the spirit of F. Takens' paper [29] which contains a comprehensive

Holomorphic Dynamics in Loop Spaces

“…In particular, a typical geometric picture of a loopy analytic disk is given by a solid torus foliated by closed curves, which exhibits an obvious analogy with Seifert fibrations and round handle decompositions [21] (cf. [15]). …”

Fundamental groups of manifolds with S 1-category 2

“…A smooth function M −→ S 1 whose critical set is a finite link in M is called a round function and Khimshiashvili and Siersma [8] show that round functions exist on all (orientable) 3-manifolds. Furthermore they show that crit S 1 (M ) = 2 if and only if M is a lens space.…”

“…If A = S 1 , then crit S 1 (M ) has been studied in ( [8]). A smooth function M −→ S 1 whose critical set is a finite link in M is called a round function and Khimshiashvili and Siersma [8] show that round functions exist on all (orientable) 3-manifolds.…”

Um invariante para sistemas com integral primeira Morse-Bott

“…16) a deformação leva cada ponto, a velocidade constante, ao longo da linha fluxo do campo vetorial, −gradg, a g −1 (−ǫ) ∪ B1 . Vide Figura 2.10.…”