no. 2
Some examples of nonsingular Morse-Smale vector fields on S 3
Annales de l'Institut Fourier, Tome 27 (1977) no. 2, pp. 145-159.

On examine la possibilité de déterminer la classe d’homotopie d’un champ de vecteurs en considérant des invariants algébriques relatifs à sa propriété qualitative. Les invariants algébriques associés avec certains exemples de champs de vecteurs non singuliers de Morse-Smale sur la 3-sphère sont étudiés ici. Pour ces exemples, les invariants algébriques usuels associés aux solutions périodiques ne peuvent pas être utilisés pour prédire la classe d’homotopie du champ de vecteurs.

One wonders or not whether it is possible to determine the homotopy class of a vector field by examining some algebraic invariants associated with its qualitative behavior. In this paper, we investigate the algebraic invariants which are usually associated with the periodic solutions of non-singular Morse-Smale vector fields on the 3-sphere. We exhibit some examples for which there appears to be no correlation between the algebraic invariants of the periodic solutions and the homotopy classes of the vector fields.

@article{AIF_1977__27_2_145_0,
     author = {Wilson Jr, F. Wesley},
     title = {Some examples of nonsingular {Morse-Smale} vector fields on $S^3$},
     journal = {Annales de l'Institut Fourier},
     pages = {145--159},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {27},
     number = {2},
     year = {1977},
     doi = {10.5802/aif.654},
     zbl = {0357.57002},
     mrnumber = {58 #13072},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.654/}
}
TY  - JOUR
AU  - Wilson Jr, F. Wesley
TI  - Some examples of nonsingular Morse-Smale vector fields on $S^3$
JO  - Annales de l'Institut Fourier
PY  - 1977
SP  - 145
EP  - 159
VL  - 27
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.654/
DO  - 10.5802/aif.654
LA  - en
ID  - AIF_1977__27_2_145_0
ER  - 
%0 Journal Article
%A Wilson Jr, F. Wesley
%T Some examples of nonsingular Morse-Smale vector fields on $S^3$
%J Annales de l'Institut Fourier
%D 1977
%P 145-159
%V 27
%N 2
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.654/
%R 10.5802/aif.654
%G en
%F AIF_1977__27_2_145_0
Wilson Jr, F. Wesley. Some examples of nonsingular Morse-Smale vector fields on $S^3$. Annales de l'Institut Fourier, Tome 27 (1977) no. 2, pp. 145-159. doi : 10.5802/aif.654. https://aif.centre-mersenne.org/articles/10.5802/aif.654/

[1] D. Asimov, Round Handles and Nonsingular Morse-Smale Flows, to appear.

[2] A. Davis, Singular Foliations, Doctoral Dissertation, Univ. of Colorado, 1971.

[3] A. Davis and F. W. Wilson, Tangent vector fields to foliations I: Reeb foliations, Journal Diff. Equations, 11 (1972), 491-498. | MR | Zbl

[4] F. B. Fuller, Note on trajectories on a solid torus, Ann. Math., 56 (1952), 438-439. | MR | Zbl

[5] H. Hopf, Uber die abbildungen von Spharen auf Spharen neidrigerer dimension, Fund. Math., 25 (1935), 427-440. | JFM | Zbl

[6] J. Palis and S. Smale, Structural stability theorems, global analysis, A.M.S. Proc. Symp. Pure Math., 14 (1970), 223-231. | MR | Zbl

[7] M. Peixoto, Structural stability on 2-dimensional manifolds, Topology, 1 (1962), 101-120. | MR | Zbl

[8] P. Percell and F. W. Wilson, Plugging Flows, to appear. | Zbl

[9] C. Pugh, R. Walker and F. W. Wilson, On Morse-Smale approximations: a counter example, Jour. Diff. Equations, to appear. | Zbl

[10] B. L. Reinhart, Line elements on the torus, Am. J. Math., 81 (1959), 617-631. | MR | Zbl

[11] S. Smale, Differential dynamical systems, Bull. A.M.S., 73 (1967), 747-817. | Zbl

[12] F. W. Wilson, Some examples of vector fields on the 3-sphere, Ann. Four. Inst., Grenoble, 20 (1970), 1-20. | Numdam | MR | Zbl

[13] F. W. Wilson, On the minimal sets of nonsingular vector fields, Ann. Math., 84 (1966), 529-536. | MR | Zbl

Cité par Sources :