A note on homotopy classes of nonsingular vector fields on \(\mathcal S^3\) (Q2449604)

The author is interested in finding simple representatives of homotopy classes of nonsingular vector fields on 3-manifolds. The question is addressed for the case where the manifold is the 3-sphere. Homotopy classes up to homeomorphism of nonsingular vector fields on the 3-sphere are classified by the set of non-negative integers, via the homotopy number. The author proves that each homotopy class with nonzero homotopy number can be represented by two nonsingular vector fields of Morse-Smale type with three periodic orbits. The author notes that this result is optimal since it is known that a nonsingular Morse-Smale vector field with two periodic orbits has homotopy number 0.