Topology/Dynamical systems
A note on homotopy classes of nonsingular vector fields on S3
[Une note sur les classes d'homotopie de champs de vecteurs sans singularité sur S3]

Présenté par : Étienne Ghys

Yu, Bin1
1 Department of Mathematics, Tongji University, Shanghai 2000 92, China
Comptes Rendus. Mathématique, Tome 352 (2014) no. 4, pp. 351-355.

Les classes d'homotopie (à homéomorphisme près) de champs de vecteurs sans singularité sur la sphère S3 sont indexées, via le nombre d'homotopie, par les entiers positifs. Nous montrons que chaque classe de nombre d'homotopie non nul peut être représentée par deux champs de vecteurs de type Morse–Smale sans singularité, avec trois orbites périodiques. Ce résultat est optimal, puisqu'on sait déjà que tout champ avec deux orbites périodiques a 0 pour nombre d'homotopie.

The homotopy class (up to homeomorphism) of nonsingular vector fields on S3 are in one-to-one correspondence with N via the homotopy number. We prove that each homotopy class with a nonzero homotopy number can be represented by two nonsingular Morse–Smale vector fields with three periodic orbits. Notice that it is already known that the nonsingular Morse–Smale vector field with two periodic orbits has homotopy number 0.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.01.016
Yu, Bin 1

1 Department of Mathematics, Tongji University, Shanghai 2000 92, China 
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Yu, Bin. A note on homotopy classes of nonsingular vector fields on $ {S}^{3}$. Comptes Rendus. Mathématique, Tome 352 (2014) no. 4, pp. 351-355. doi : 10.1016/j.crma.2014.01.016. https://www.numdam.org/articles/10.1016/j.crma.2014.01.016/

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  • Yu, Bin Behavior 0 nonsingular Morse Smale flows on S3, Discrete and Continuous Dynamical Systems, Volume 36 (2015) no. 1, p. 509 | DOI:10.3934/dcds.2016.36.509

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