Codimension one minimal foliations and the fundamental groups of leaves

Tomoo Yokoyama[1]; Takashi Tsuboi[2]

  • [1] The University of Tokyo Graduate School of Mathematical Sciences Komaba Meguro, Tokyo 153-8914, Japan
  • [2] The University of Tokyo Graduate School of Mathematical Sciences Komaba Meguro, Tokyo 153-8914 (Japan)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 2, page 723-731
  • ISSN: 0373-0956

Abstract

top
Let ℱ be a transversely orientable transversely real-analytic codimension one minimal foliation of a paracompact manifold M . We show that if the fundamental group of each leaf of ℱ is isomorphic to Z , then ℱ is without holonomy. We also show that if π 2 ( M ) ≅ 0 and the fundamental group of each leaf of ℱ is isomorphic to Z k ( k ∈ Z ≥ 0 ), then ℱ is without holonomy.

How to cite

top

Yokoyama, Tomoo, and Tsuboi, Takashi. "Codimension one minimal foliations and the fundamental groups of leaves." Annales de l’institut Fourier 58.2 (2008): 723-731. <http://eudml.org/doc/10330>.

@article{Yokoyama2008,
abstract = {Let $\mathcal\{F\}$ be a transversely orientable transversely real-analytic codimension one minimal foliation of a paracompact manifold $M$. We show that if the fundamental group of each leaf of $\mathcal\{F\}$ is isomorphic to $Z$, then $\mathcal\{F\}$ is without holonomy. We also show that if $\pi _2( M )\cong 0$ and the fundamental group of each leaf of $\mathcal\{F\}$ is isomorphic to $Z ^k $ ($k \in Z_\{\ge 0\}$), then $\mathcal\{F\}$ is without holonomy.},
affiliation = {The University of Tokyo Graduate School of Mathematical Sciences Komaba Meguro, Tokyo 153-8914, Japan; The University of Tokyo Graduate School of Mathematical Sciences Komaba Meguro, Tokyo 153-8914 (Japan)},
author = {Yokoyama, Tomoo, Tsuboi, Takashi},
journal = {Annales de l’institut Fourier},
keywords = {Foliations; real-analytic; holonomy; fundamental groups of leaves; foliations; fundamental group of leaves},
language = {eng},
number = {2},
pages = {723-731},
publisher = {Association des Annales de l’institut Fourier},
title = {Codimension one minimal foliations and the fundamental groups of leaves},
url = {http://eudml.org/doc/10330},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Yokoyama, Tomoo
AU - Tsuboi, Takashi
TI - Codimension one minimal foliations and the fundamental groups of leaves
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 2
SP - 723
EP - 731
AB - Let $\mathcal{F}$ be a transversely orientable transversely real-analytic codimension one minimal foliation of a paracompact manifold $M$. We show that if the fundamental group of each leaf of $\mathcal{F}$ is isomorphic to $Z$, then $\mathcal{F}$ is without holonomy. We also show that if $\pi _2( M )\cong 0$ and the fundamental group of each leaf of $\mathcal{F}$ is isomorphic to $Z ^k $ ($k \in Z_{\ge 0}$), then $\mathcal{F}$ is without holonomy.
LA - eng
KW - Foliations; real-analytic; holonomy; fundamental groups of leaves; foliations; fundamental group of leaves
UR - http://eudml.org/doc/10330
ER -

References

top
  1. John Cantwell, Lawrence Conlon, Leaf prescriptions for closed 3 -manifolds, Trans. Amer. Math. Soc. 236 (1978), 239-261 Zbl0398.57009MR645738
  2. John Cantwell, Lawrence Conlon, Endsets of exceptional leaves; a theorem of G. Duminy, Foliations: geometry and dynamics (Warsaw, 2000) (2002), 225-261, World Sci. Publ., River Edge, NJ Zbl1011.57009MR1882772
  3. D. B. A. Epstein, K. C. Millett, D. Tischler, Leaves without holonomy, J. London Math. Soc. (2) 16 (1977), 548-552 Zbl0381.57007MR464259
  4. F. T. Farrell, L. E. Jones, The surgery L -groups of poly-(finite or cyclic) groups, Invent. Math. 91 (1988), 559-586 Zbl0657.57015MR928498
  5. André Haefliger, Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes, Comment. Math. Helv. 32 (1958), 248-329 Zbl0085.17303MR100269
  6. M. Hirsch, A stable analytic foliation with only exceptional minimal sets, Dynamical Systems 468 (1975), 9-10, Springer, Berlin, Heidelberg, New York Zbl0309.53053
  7. B. Kerékjártó, Vorlesungen uber Topologie, I (1923), Springer, Berlin Zbl49.0396.07
  8. S. P. Novikov, Topology of foliations, Trans. Mosc. Math. Soc. 14 (1965), 268-304 Zbl0247.57006MR200938
  9. I. Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963), 259-269 Zbl0156.22203MR143186
  10. D. Tischler, On fibering certain foliated manifolds over S 1 , Topology 9 (1970), 153-154 Zbl0177.52103MR256413

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.