Abstract
In this article parametric versions of Wilson’s plug and Kuperberg’s plug are discussed. We show that there is a weak homotopy equivalence induced by the inclusion between the space of non-singular vector fields tangent to a foliation and its subspace comprised of those without closed orbits, as long as the leaves of the foliation have dimension at least 3. We contrast this with the case of foliations by surfaces in 3-manifolds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Ghys, Construction de champs de vecteurs sans orbite périodique, Astérisque 227 (1995), Exp. No. 785, 5, 283–307.
G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Part A, Aspects of Mathematics, Vol. 1, Friedrich Vieweg & Sohn, Braunschweig, 1986.
S. Hurder and A. Rechtman, The dynamics of generic Kuperberg flows, Astérisque 377 (2016).
H. Imanishi, Structure of codimension one foliations which are almost without holonomy, Journal of Mathematics of Kyoto University 16 (1976), 93–99.
H. Imanishi and K. Yagi, On Reeb components, Journal of Mathematics of Kyoto University 16 (1976), 313–324.
K. Kuperberg, A smooth counterexample to the Seifert conjecture, Annals of Mathematics 140 (1994), 723–732.
S. Matsumoto, Kuperberg’s C8 counterexample to the Seifert conjecture, Sugaku 47 (1995), 38–45.
T. Nishimori, Compact leaves with abelian holonomy, Tohoku Mathematical Journal 27 (1975), 259–272.
A.delPino and F.Presas, The foliated Weinstein conjecture, arXiv: 1509.05268.
P. A. Schweitzer, Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Annals of Mathematics 100 (1974), 386–400.
P. A. Schweitzer, Riemannian manifolds not quasi-isometric to leaves in codimension one foliations, Université de Grenoble. Annales de l’Institut Fourier 61 (2011), 1599–1631.
H. Seifert, Closed integral curves in 3-space and isotopic two-dimensional deformations, Proceedings of the American Mathematical Society 1 (1950), 287–302.
F. W. Wilson, On the minimal sets of non-singular vector fields, Annals of Mathematics 84 (1966), 529–536.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Peralta-Salas, D., del Pino, Á. & Presas, F. Foliated vector fields without periodic orbits. Isr. J. Math. 214, 443–462 (2016). https://doi.org/10.1007/s11856-016-1336-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-016-1336-3