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Let $S^3$ denote the set of points with modulus one in euclidean 4-space $R^4$ ; and let ${\Gamma }_0^1(S^3)$ denote the space of nonsingular vector fields on $S^3$ with the $C^1$ topology. Under what conditions are two elements from ${\Gamma }_0^1(S^3)$ homotopic ? There are several examples of nonsingular vector fields on $S^3$. However, they are all homotopic to the tangent fields of the fibrations of $S^3$ due to H. Hopf (there are two such classes). We construct some new examples of vector fields which can be classified geometrically. Each...

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...