Topological classification of circle-valued simple Morse-Bott functions

Batista, Erica Boizan; Costa, João Carlos Ferreira; Meza-Sarmiento, Ingrid S.

doi:10.5427/jsing.2018.17q

Cited by 8 publications

(7 citation statements)

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“…In general, if f : M → R is an arbitrary smooth function with isolated critical points, then a certain part of its "combinatorial symmetries" is reflected by the so-called Kronrod-Reeb graph ∆ f , see, e.g., [2,3,18,19,26,33,35,38] and Section 2.3. Such a graph is obtained by shrinking each connected component of each level set f −1 (c), c ∈ R, of f into a point.…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of Cellular Divisions of 2-Sphere Induced by Functions with Isolated Critical Points

Kravchenko¹,

Maksymenko²

2020

Z. mat. fiz. anal. geom.

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Let f : S 2 → R be a Morse function on the 2-sphere and K be a connected component of some level set of f containing at least one saddle critical point. Then K is a 1-dimensional CW-complex cellularly embedded into S 2 , so the complement S 2 \ K is a union of open 2-disks D 1 , . . . , D k . Let S K (f ) be the group of isotopic to the identity diffeomorphisms of S 2 leaving invariant K and also each level set f −1 (c), c ∈ R. Then each h ∈ S K (f ) induces a certain permutation σ h of those disks. Denote by G = {σ h | h ∈ S K (f )} the group of all such permutations. We prove that G is isomorphic to a finite subgroup of SO(3).

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Section: Introductionmentioning

confidence: 99%

Automorphisms of Cellular Divisions of 2-Sphere Induced by Functions with Isolated Critical Points

Kravchenko¹,

Maksymenko²

2020

Z. mat. fiz. anal. geom.

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“…This problem has been first considered by Sharko ([12]). Related to this pioneering work, several works have been done: the work of J. Martinez-Alfaro, I. S. Meza-Sarmiento and R. Oliveira ( [6]), the work of Masumoto and Saeki ([7]) and recent works such as [1], [8] and [9]. In these studies, explicit smooth functions giving the given graphs as the Reeb graphs have been constructed and most of the functions are ones on closed surfaces or Morse functions on closed manifolds such that inverse images of regular values are disjoint unions of circles or standard spheres.…”

Section: Introductionmentioning

confidence: 99%

On Reeb graphs induced from smooth functions on closed or open manifolds

Kitazawa¹

2019

Preprint

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For a smooth function on a smooth manifold of a suitable class, the space of all the connected components of inverse images is the graph and called the Reeb graph. Reeb graphs are fundamental tools in the algebraic and differential topological theory of Morse functions and more general functions not so hard to handle: the global singularity theory.In this paper, we attack the following natural problem: can we construct a smooth function with good geometric properties inducing a given graph as the Reeb graph. This problem has been essentially launched by Sharko in 2000s and various answers have been given by Masumoto, Michalak, Saeki etc.. Recently the author has set a new explicit problem and given an answer. In the studies before the result of the author, considered functions are smooth functions on closed surfaces or Morse functions such that inverse images of regular values are disjoint unions of standard spheres: well-known most fundamental Morse functions with just 2 singular points, characterizing spheres topologically, are examples of such functions. On the other hand, the author has succeed in construction of a smooth function on a 3-dimensional closed, connected and orientable manifold inducing a given graph as the Reeb graph such that inverse images of regular values are desired. Based on the studies, especially on the result and method of the author, with several new ideas, we will consider smooth functions on surfaces which may be non-closed and give an answer.

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“…In general, if f : M → R is an arbitrary smooth function with isolated critical points, then a certain part of its "combinatorial symmetries" is reflected by a so-called Kronrod-Reeb graph ∆ f , see e.g. [2,3,18,19,26,33,35,38] and §2.4. Such a graph is obtained by shrinking each connected component of each level set…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of cellular divisions of $2$-sphere induced by functions with isolated critical points

Kravchenko¹,

Maksymenko²

2019

Preprint

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Let f : S 2 → R be a Morse function on the 2-sphere and K be a connected component of some level set of f containing at least one saddle critical point. Then K is a 1-dimensional CW-complex cellularly embedded into S 2 , so the complement S 2 \ K is a union of open 2-disks D 1 , . . . , D k . Let S K (f ) be the group of isotopic to the identity diffeomorphisms of S 2 leaving invariant K and also each level set f −1 (c), c ∈ R. Then each h ∈ S K (f ) induces a certain permutation σ h of those disks. Denote by G = {σ h | h ∈ S K (f )} be the group of all such permutations. We prove that G is isomorphic to a finite subgroup of SO(3).

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Topological classification of circle-valued simple Morse-Bott functions

Cited by 8 publications

References 0 publications

Automorphisms of Cellular Divisions of 2-Sphere Induced by Functions with Isolated Critical Points

Automorphisms of Cellular Divisions of 2-Sphere Induced by Functions with Isolated Critical Points

On Reeb graphs induced from smooth functions on closed or open manifolds

Automorphisms of cellular divisions of $2$-sphere induced by functions with isolated critical points

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