# Foliations of surfaces I : an ideal boundary

Annales de l'institut Fourier (1982)

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

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topMather, 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

top- [1] I. BENDIXSON, Sur les courbes définies par des équations différentielles, Acta Math., 24 (1901), 1-88. JFM31.0328.03
- [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] B. VON KERÉKJÁRTÓ, Vorlesungen über Topologie, Berlin Springer-Verlag (1923).
- [4] M.H.A. NEWMAN, Elements of the Topology of Plane Sets of Points, Cambridge Univ. Press (1939). Zbl0021.06704JFM65.0873.04
- [5] I. RICHARDS, On the classification of noncompact surfaces, Trans. Amer. Math. Soc., 106 (1963), 259-269. Zbl0156.22203MR26 #746

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