Foliations with non-compact leaves on surfaces

Maksymenko, Sergiy; Polulyakh, Eugene

doi:10.15673/tmgc.v8i3-4.1603

Cited by 9 publications

(4 citation statements)

References 7 publications

(9 reference statements)

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“…Moreover, see [16,Lemma 4.1], h 0 (x, y) = α(x, y), β(y) , where ). Let q : Z 0 → Z be a reduced striped atlas such that the family ∆ spec of all special leaves of the canonical foliation ∆ is locally finite.…”

Section: 1mentioning

confidence: 99%

Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves

Maksymenko,

Polulyakh

2022

Preprint

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Let Z be a non-compact two-dimensional manifold obtained from a family of open strips R × (0, 1) with boundary intervals by gluing those strips along some pairs of their boundary intervals. Every such strip has a natural foliation into parallel lines R × t, t ∈ (0, 1), and boundary intervals which gives a foliation ∆ on all of Z. Denote by H(Z, ∆) the group of all homeomorphisms of Z that maps leaves of ∆ onto leaves and by H(Z/∆) the group of homeomorphisms of the space of leaves endowed with the corresponding compact open topologies. Recently, the authors identified the homeotopy group π 0 H(Z, ∆) with a group of automorphisms of a certain graph G with additional structure which encodes the combinatorics of gluing Z from strips. That graph is in a certain sense dual to the space of leaves Z/∆.On the other hand, for every h ∈ H(Z, ∆) the induced permutation k of leaves of ∆ is in fact a homeomorphism of Z/∆ and the correspondence h → k is a homomorphism ψ : H(∆) → H(Z/∆). The aim of the present paper is to show that ψ induces a homomorphism of the corresponding homeotopy groups ψ 0 : π 0 H(Z, ∆) → π 0 H(Z/∆) which turns out to be either injective or having a kernel Z 2 . This gives a dual description of π 0 H(Z, ∆) in terms of the space of leaves.

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Section: 1mentioning

confidence: 99%

Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves

Maksymenko,

Polulyakh

2022

Preprint

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“…The latter spaces are often non Hausdorff manifolds, see e.g. [15,25,26]. For two sets M, N we will denote by Map(M, N ) the set of all maps M Ñ N .…”

Section: Lemma 32 For Three Topological Spaces L M N the Followinmentioning

confidence: 99%

Smooth approximations and their applications to homotopy types

Khokhliuk

Maksymenko

2020

PIGC

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Let $M, N$ the be smooth manifolds, $\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with the corresponding weak Whitney topology, and $\mathcal{B} \subset \mathcal{C}^{r}(M,N)$ an open subset.It is proved that for $0<r<s\leq\infty$ the inclusion $\mathcal{B} \cap \mathcal{C}^{s}(M,N) \subset \mathcal{B}$ is a weak homotopy equivalence.It is also established a parametrized variant of such a result.In particular, it is shown that for a compact manifold $M$, the inclusion of the space of $\mathcal{C}^{s}$ isotopies $\eta:[0,1]\times M \to M$ fixed near $\{0,1\}\times M$ into the space of loops $\Omega(\mathcal{D}^{r}(M), \mathrm{id}_{M})$ of the group of $\mathcal{C}^{r}$ diffeomorphisms of $M$ at $\mathrm{id}_{M}$ is a weak homotopy equivalence.

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“…Remark 4.1. In [22,24,23,20,25] we called those points "special ", but in the present paper we decided to change their name to "branch" as in [13,12] since it better reflects the structure of a space near such points. Also in [18] there were considered families of pairwise T 2 -non-disjoint points called sets of compatible appartion points.…”

Section: Branch Points Of T 1 Spacesmentioning

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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Foliations with non-compact leaves on surfaces

Cited by 9 publications

References 7 publications

Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves

Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves

Smooth approximations and their applications to homotopy types

Actions of groups of foliated homeomorphisms on spaces of leaves

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