Realization of a graph as the Reeb graph of a Morse function on a manifold

Michalak, Lukasz

doi:10.12775/tmna.2018.029

Cited by 22 publications

(35 citation statements)

References 9 publications

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“…The Reeb graph R(f ) of a Morse function f : M → R, as an invariant of the pair (M, f ), is a tool of global analysis attracting more attention recently due to its applications to computer graphics as well as its importance to purely mathematical problems (for more details see [2,5,7,8,18,19,23]). It is constructed by contracting the connected components of levels sets of the function f .…”

Section: Introductionmentioning

confidence: 99%

“…The corank of a finitely generated group G is the maximal rank r for which there exists an epimorphism G → F r . As it was defined in [19] for closed manifolds, the Reeb number R(W ) of W denotes the maximal number of cycles in Reeb graphs of Morse functions f : W → R which are constant on the connected components of ∂W . In other words, R(W ) is the maximal rank of the Reeb epimorphism of such a Morse function on W .…”

Section: Introductionmentioning

confidence: 99%

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Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

Marzantowicz¹,

Michalak²

2020

Preprint

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In this work we present a construction which gives a correspondence between epimorphisms ϕ : π1(W ) → Fr, from the fundamental group of a compact manifold W onto the free group of rank r, and systems of framed non-separating hypersurfaces in W . In consequence, any such ϕ, which corresponds to a system of hypersurfaces without boundary, can be represented by the Reeb epimorphism of a Morse function f : W → R, i.e. by the epimorphism induced by the quotient map W → R(f ) onto the Reeb graph of f . We study properties and natural relations between these three objects. In particular, from this point of view we discuss the problem of classification up to (strong-)equivalence of epimorphisms onto free groups and we provide a purely geometrical-topological proof of the solution of this problem for surface groups which was given earlier by Grigorchuk, Kurchanov and Zieshang by using other methods.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

Marzantowicz¹,

Michalak²

2020

Preprint

Self Cite

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“…It is worth mentioning that graph theory is often used to study Morse functions, for example, Reeb graphs are important for exploring Morse functions over manifolds, see, for instance, [4,7,12,13] and references therein. Reeb graphs are used also in computer graphics, see [4].…”

Section: Theoremmentioning

confidence: 99%

“…In [12], conditions are indicated under which Reeb graph is a tree. In [7,13], the authors provide the upper bound for the number of cycles in Reeb graphs and prove that the upper bound is sharp. (Compare with the main theorems in this article.)…”

Section: Theoremmentioning

confidence: 99%

On the Maximum Number of Period Annuli for Second Order Conservative Equations

Gritsans

Yermachenko

2021

Mathematical Modelling and Analysis

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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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Realization of a graph as the Reeb graph of a Morse function on a manifold

Cited by 22 publications

References 9 publications

Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

On the Maximum Number of Period Annuli for Second Order Conservative Equations

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

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