Corpus ID: 119142544

# Foliations with all non-closed leaves on non-compact surfaces

@article{Maksymenko2016FoliationsWA,
title={Foliations with all non-closed leaves on non-compact surfaces},
author={Sergiy Ivanovych Maksymenko and Eugene Polulyakh},
journal={arXiv: Geometric Topology},
year={2016}
}
• Published 31 May 2016
• Mathematics
• arXiv: Geometric Topology
Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $\Delta$ be a foliation on $X$ such that each leaf $\omega\in\Delta$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction… Expand
7 Citations

#### Figures from this paper

Homeotopy groups of one-dimensional foliations on surfaces
• Mathematics, Physics
• 2017
Let $Z$ be a non-compact two-dimensional manifold obtained from a family of open strips $\mathbb{R}\times(0,1)$ with boundary intervals by gluing those strips along their boundary intervals.EveryExpand
One-dimensional foliations on topological manifolds
• Mathematics
• 2017
Let $X$ be an $(n+1)$-dimensional manifold, $\Delta$ be a one-dimensional foliation on $X$, and $p: X \to X / \Delta$ be a quotient map. We will say that a leaf $\omega$ of $\Delta$ is specialExpand
Homeotopy groups of one-dimensional foliations on surfaces
• Mathematics
• 2017
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 their boundary intervals. Every such strip has aExpand
Homeotopy Groups for Nonsingular Foliations of the Plane
We consider a special class of nonsingular oriented foliations F on noncompact surfaces Σ whose spaces of leaves have a structure similar to the structure of rooted trees of finite diameter. LetExpand
Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold
• Mathematics
• 2017
This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface $M$ which are also isolated critical points of theirExpand
Fundamental groupoids and homotopy types of non-compact surfaces
• Mathematics
• 2021
The paper contains an application of van Kampen theorem for groupoids to computation of homotopy types of certain class of non-compact foliated surfaces obtained by at most countably many strips R ×Expand
Smooth approximations and their applications to homotopy types
• Mathematics
• 2020
Let $M, N$ the be smooth manifolds, $\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with weak $C^{r}$ Whitney topology, and $\mathcal{B} \subset \mathcal{C}^{r}(M,N)$ an open subset. It isExpand

#### References

SHOWING 1-10 OF 19 REFERENCES
The Topology of the Level Curves of Harmonic Functions with Critical Points
Introduction. In a previous paper,2 of which this is a continuation, topological properties of curve families which filled the Euclidean plane 7r, or a simply connected domain in r, wereExpand
Peixoto graphs of Morse-Smale foliations on surfaces
• Mathematics
• 1997
Abstract We consider non-orientable foliations, given on orientable compact 2-manifolds. Within a special class of such foliations, a Morse-Smale class, a complete topological classification isExpand
Topological classification of flows on closed two-dimensional manifolds
• Mathematics
• 1986
CONTENTS Introduction Chapter I. Development of the Poincar?-Bendixson theory for flows on closed two-dimensional manifolds ??1. Possible types of trajectories. Singular and non-singular trajectoriesExpand
The combinatorics of gradient-like flows and foliations on closed surfaces: I. Topological classification
Abstract In terms of rotation graphs we construct the complete topological invariants for gradient-like flows on closed surfaces. In terms of the so-called current graphs we also construct a completeExpand
The Topology of Regular Curve Families with Multiple Saddle Points
Introduction. It is known that the level curves of any function f (x, y) which is harmonic in a simply connected domain form a curve family which is regular (locally homeomorphic to parallel lines)Expand
ON SOME INVARIANTS OF DYNAMICAL SYSTEMS ON TWO-DIMENSIONAL MANIFOLDS (NECESSARY AND SUFFICIENT CONDITIONS FOR THE TOPOLOGICAL EQUIVALENCE OF TRANSITIVE DYNAMICAL SYSTEMS)
• Mathematics
• 1973
In this paper, topological invariants of dynamical systems given on a two-dimensional manifold M2 of genus p>1 are selected which allow one to distinguish topologically inequivalent systems whichExpand
Stabilizers and orbits of smooth functions
Abstract Let f : R m → R be a smooth function such that f ( 0 ) = 0 . We give a condition J(id) on f when for arbitrary preserving orientation diffeomorphism ϕ : R → R such that ϕ ( 0 ) = 0 theExpand
Topology of closed one-forms
The Novikov numbers The Novikov inequalities The universal complex Construction of the universal complex Bott-type inequalities Inequalities with von Neumann Betti numbers Equivariant theoryExpand
Structural stability on two-dimensional manifolds☆
THE ABOVE paper appeared in Volume 1, pp. 101-120 of Topology, and the aim of it was to prove that the set of all structurally stable differential equations is open and dense in the space, with theExpand
Equivalence of closed 1-forms on surfaces with edge
• Mathematics
• 2009
We investigate closed 1-forms with isolated zeros on surfaces with edge. A criterion for the topological equivalence of closed 1-forms is proved.