Singular levels and topological invariants of Morse--Bott foliations on non orientable surfaces

Martínez-Alfaro, José; Meza-Sarmiento, Ingrid S.; Oliveira, Regilene

doi:10.12775/tmna.2017.051

Cited by 7 publications

(6 citation statements)

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“…(iii) Finally, for K 2 there are two graphs, P 2 and C 2 , in which both vertices are of the same type, so [12], Theorem 5.7 again gives two classes.…”

Section: A Generalization: Topological Morse-bott Functionsmentioning

confidence: 97%

“…(ii) In the case of RP 2 , the Reeb graph is again P 2 , but its vertices correspond to different critical contours: one is a center, the other is a singular circle with nonorientable normal bundle. In terms of [11], Theorem 22, we have two different labeled digraphs, so [12], Theorem 5.7 gives two classes of topological conjugacy.…”

Section: A Generalization: Topological Morse-bott Functionsmentioning

confidence: 99%

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Morse-Bott functions with two critical values on a surface

Gelbukh¹

2021

Czech.Math.J.

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“…(iii) Finally, for K 2 there are two graphs, P 2 and C 2 , in which both vertices are of the same type, so [12], Theorem 5.7 again gives two classes.…”

Section: A Generalization: Topological Morse-bott Functionsmentioning

confidence: 97%

Section: A Generalization: Topological Morse-bott Functionsmentioning

confidence: 99%

Morse-Bott functions with two critical values on a surface

Gelbukh¹

2021

Czech.Math.J.

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“…About realization of Reeb graphs, there have been a lot of studies, e.g. by Sharko [29], Martínez-Alfaro, Meza-Sarmiento and Oliveira [18,19,20], Masumoto and Saeki [21], Gelbukh [2,3,4,5], Kaluba, Marzantowicz and Silva [12], Michalak [22,23], Michalak and Marzantowicz [17], Kitazawa [13,14,15], etc. Our theorems generalize some of them.…”

Section: Realization IImentioning

confidence: 99%

Reeb spaces of smooth functions on manifolds

Saeki

2020

Preprint

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The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite connected graph without loops that induces an epimorphism between the fundamental groups is identified with the natural quotient map to the Reeb space of a certain smooth function with finitely many critical values, up to homotopy.

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“…These foliations, in particular, foliations by level sets of Morse-Bott functions, play an important role in Hamiltonian dynamics and Poisson geometry, see e.g. [18], [19], [20], [21], [22].…”

Section: Introductionmentioning

confidence: 99%

Diffeomorphisms preserving Morse–Bott functions

Khohliyk

Maksymenko

2020

Indagationes Mathematicae

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Let f : M → R be a Morse-Bott function on a closed manifold M , so the set Σ f of its critical points is a closed submanifold whose connected components may have distinct dimensions. Denote by S(f ) = {h ∈ D(M ) | f • h = f } the group of diffeomorphisms of M preserving f and let D(Σ f ) be the group of diffeomorphisms of Σ f . We prove that the "restriction to Σ f " map ρ : S(f ) → D(Σ f ), ρ(h) = h| Σ f , is a locally trivial fibration over its image ρ(S(f )).2010 Mathematics Subject Classification. 57R30, 57R45, 53B15,

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Singular levels and topological invariants of Morse--Bott foliations on non orientable surfaces

Cited by 7 publications

References 0 publications

Morse-Bott functions with two critical values on a surface

Morse-Bott functions with two critical values on a surface

Reeb spaces of smooth functions on manifolds

Diffeomorphisms preserving Morse–Bott functions

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