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 of these examples...

It is proved, for various spaces A, such as a surface of genus 2, a figure-eight, or a sphere of dimension ≠ 1,3,7, and for any set Σ of equations, that Σ cannot be modeled by continuous operations on A unless Σ is undemanding (a form of triviality that is defined in the paper).