Homology of the Group of Leaf Preserving Homeomorphisms

Rybicki, Tomasz

doi:10.1515/dema-1996-0223

Cited by 3 publications

(4 citation statements)

References 5 publications

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“…Homotopy properties of groups of leaf-preserving diffeomorphisms for nonsingular foliations are studied in many papers, see e.g. [1], [2], [3], [4], [5], [6], [7], [8] and references therein. Most of the results concern with the extension of results by M. Herman [9], W. Thurston [10], J. Mather [11], [12], D. B.…”

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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Section: Introductionmentioning

confidence: 99%

Diffeomorphisms preserving Morse–Bott functions

Khohliyk

Maksymenko

2020

Indagationes Mathematicae

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“…Homotopy properties of groups H(X, ∆) were studied e.g. in [28,30,29,31,32,14,1,19,21] and references therein. Most of them extend the results by M. Herman [16], W. Thurston [33], J. Mather [26,27] and D. B.…”

Section: Introductionmentioning

confidence: 99%

Actions of groups of foliated homeomorphisms on spaces of leaves

Maksymenko¹,

Polulyakh²

2020

Preprint

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Let ∆ be a foliation on a topological manifold X, Y be the space of leaves, and p : X → Y be the natural projection. Endow Y with the factor topology with respect to p. Then the group H(X, ∆) of foliated (i.e. mapping leaves onto leaves) homeomorphisms of X naturally acts on the space of leaves Y , which gives a homomorphism ψ : H(X, ∆) → H(Y ). We present sufficient conditions when ψ is continuous with respect to the corresponding compact open topologies.In fact similar results hold not only for foliations but for a more general class of partitions ∆ of locally compact Hausdorff spaces X.

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“…The groups of leaf preserving diffeomorphisms and homeomorphisms of foliations are studied intensively. Most of the results in the literature are concerned with regular foliations, see [2,3,28,29] and references in these papers. For singular foliations the situation is much more difficult.…”

Section: Introductionmentioning

confidence: 99%

“…Most of the results concern with regular foliations, see e.g. [B77,Ryb1,Ryb2,AF03] and references in these papers. For singular foliations the situation is much more difficult.…”

Section: Introductionmentioning

confidence: 99%

∞-jets of diffeomorphisms preserving orbits of vector fields

Maksymenko

2009

Open Mathematics

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Let F be a C ∞ vector field defined near the origin O ∈ R , F (O) = 0, and (F ) be its local flow. Denote bŷ (F ) the set of germs of orbit preserving diffeomorphisms : R → R at O, and letˆ id (F ) , ( ≥ 0), be the identity component ofˆ (F ) with respect to the weak Whitney W topology. Thenˆ id (F ) ∞ contains a subset S (F ) consisting of maps of the form F α( ) ( ), where α : R → R runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, thenŜ (F ) =ˆ id (F ) 0In this paper we present a class of examples of vector fields with degenerate singularities at O for whichŜ (F ) formally coincides withˆ id (F ) 1 , i.e. on the level of ∞-jets at O. We also establish parameter rigidity of linear vector fields and "reduced" Hamiltonian vector fields of real homogeneous polynomials in two variables. MSC:37C10

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Homology of the Group of Leaf Preserving Homeomorphisms

Cited by 3 publications

References 5 publications

Diffeomorphisms preserving Morse–Bott functions

Diffeomorphisms preserving Morse–Bott functions

Actions of groups of foliated homeomorphisms on spaces of leaves

∞-jets of diffeomorphisms preserving orbits of vector fields

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