Abstract
We prove that every 3-manifold possesses aC 1, volume-preserving flow with no fixed points and no closed trajectories. The main construction is a volume-preserving version of the Schweitzer plug. We also prove that every 3-manifold possesses a volume-preserving,C ∞ flow with discrete closed trajectories and no fixed points (as well as a PL flow with the same geometry), which is needed for the first result. The proof uses a Dehn-twisted Wilson-type plug which also preserves volume.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. H. Hardy andE. M. Wright.An Introduction to the Theory of Numbers. Oxford University Press, Oxford, 1979.
D. Hart,On the smoothness of generators, Topology22 (1983), 357–363.
J. Harrison,C 2 counterexamples to the Seifert conjecture, Topology27 (1988), 249–278.
H. Hofer,Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), 515–563.
K. Kuperberg,A smooth counterexample to the Seifert conjecture, Ann. of Math.140 (1994), 723–732.
G. Kuperberg andK. Kuperberg,Generalized counterexamples to the Seifert conjecture, to appear in Ann. of Math.
J. Moser,On the volume elements of a manifold, Trans. Amer. Math. Soc.120 (1965), 286–294.
B. Dacorogna andJ. Moser,On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non-Linear7 (1990) no. 1, 1–26.
J. F. Plante,Foliations with measure-preserving holonomy, Ann. of Math.102 (1975), 327–361.
D. Rolfsen.Knots and Links, volume 7 ofMathematics Lecture Series, Publish or Perish, Inc., Wilmington, DE, 1976.
P. A. Schweitzer,Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Ann. of Math.100 (1974), 386–400.
H. Seifert,Closed integral curves in 3-space and isotopic two-dimensional deformations, Proc. Amer. Math. Soc.1 (1950), 287–302.
W. P. Thurston, private communication.
A. Weinstein,On the hypothesis of Rabinowitz's periodic orbit theorems, J. Diff. Eq.33 (1979), 353–358.
F. W. Wilson,On the minimal sets of non-singular vector fields, Ann. of Math.84 (1966), 529–536.
Author information
Authors and Affiliations
Additional information
The author was supported by an NSF Postdoctoral Fellowship, grant #DMS-9107908.
Rights and permissions
About this article
Cite this article
Kuperberg, G. A volume-preserving counterexample to the Seifert conjecture. Commentarii Mathematici Helvetici 71, 70–97 (1996). https://doi.org/10.1007/BF02566410
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02566410