# Foliations of surfaces I : an ideal boundary

• Volume: 32, Issue: 1, page 235-261
• ISSN: 0373-0956

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## Abstract

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Let $F$ be a foliation of the punctured plane $P$. Any non-compact leaf of $F$ has two ends, which we call leaf-ends. The set $ℰ$ of leaf-ends which converge to the origin has a natural cyclic order. In the case $ℰ$ is infinite, we show that the cyclicly ordered set $\beta$, obtained by identifying neighbors in $ℰ$ and filling in the holes according to the Dedeking process, is equivalent to a circle. We show that the set $P\coprod \beta$ has a natural topology, and it is homeomorphic to ${S}^{1}×\left[0,\infty \right)$ with respect to this topology.

## How to cite

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Mather, John N.. "Foliations of surfaces I : an ideal boundary." Annales de l'institut Fourier 32.1 (1982): 235-261. <http://eudml.org/doc/74527>.

@article{Mather1982,
abstract = {Let $F$ be a foliation of the punctured plane $P$. Any non-compact leaf of $F$ has two ends, which we call leaf-ends. The set $\{\cal E\}$ of leaf-ends which converge to the origin has a natural cyclic order. In the case $\{\cal E\}$ is infinite, we show that the cyclicly ordered set $\beta$, obtained by identifying neighbors in $\{\cal E\}$ and filling in the holes according to the Dedeking process, is equivalent to a circle. We show that the set $P\coprod \beta$ has a natural topology, and it is homeomorphic to $S^1 \times [0,\infty )$ with respect to this topology.},
author = {Mather, John N.},
journal = {Annales de l'institut Fourier},
keywords = {foliation of the punctured plane; leaf-ends},
language = {eng},
number = {1},
pages = {235-261},
publisher = {Association des Annales de l'Institut Fourier},
title = {Foliations of surfaces I : an ideal boundary},
url = {http://eudml.org/doc/74527},
volume = {32},
year = {1982},
}

TY - JOUR
AU - Mather, John N.
TI - Foliations of surfaces I : an ideal boundary
JO - Annales de l'institut Fourier
PY - 1982
PB - Association des Annales de l'Institut Fourier
VL - 32
IS - 1
SP - 235
EP - 261
AB - Let $F$ be a foliation of the punctured plane $P$. Any non-compact leaf of $F$ has two ends, which we call leaf-ends. The set ${\cal E}$ of leaf-ends which converge to the origin has a natural cyclic order. In the case ${\cal E}$ is infinite, we show that the cyclicly ordered set $\beta$, obtained by identifying neighbors in ${\cal E}$ and filling in the holes according to the Dedeking process, is equivalent to a circle. We show that the set $P\coprod \beta$ has a natural topology, and it is homeomorphic to $S^1 \times [0,\infty )$ with respect to this topology.
LA - eng
KW - foliation of the punctured plane; leaf-ends
UR - http://eudml.org/doc/74527
ER -

## References

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1. [1] I. BENDIXSON, Sur les courbes définies par des équations différentielles, Acta Math., 24 (1901), 1-88. JFM31.0328.03
2. [2] A. HAEFLIGER and G. REEB, Variétés (non séparées) à une dimension et structures feuilletées du plan, Enseignement Math., 3 (1957), 107-125. Zbl0079.17101MR19,671c
3. [3] B. VON KERÉKJÁRTÓ, Vorlesungen über Topologie, Berlin Springer-Verlag (1923).
4. [4] M.H.A. NEWMAN, Elements of the Topology of Plane Sets of Points, Cambridge Univ. Press (1939). Zbl0021.06704JFM65.0873.04
5. [5] I. RICHARDS, On the classification of noncompact surfaces, Trans. Amer. Math. Soc., 106 (1963), 259-269. Zbl0156.22203MR26 #746

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