Sphere eversion

Remarks

Such a family $H$ is a regular homotopy that turns the sphere inside out: the outward normal of $\iota$ corresponds to the inward normal of $\alpha$. The surfaces $H_t$ may pass through themselves, but they may never develop a crease or a pinch. Stephen Smale proved this as a consequence of his classification of immersions of spheres: regular homotopy classes of immersions $S^2 \to \R^3$ correspond to $\pi_2(SO(3))$, which is the trivial group. The proof was not constructive; explicit eversions were later found, for example by Bernard Morin, and by William Thurston with the method of corrugations. The analogue for circles fails: by the Whitney–Graustein theorem, immersions $S^1 \to \R^2$ are classified by the turning number, and the standard circle and its reversal have turning numbers $1$ and $-1$.

Going deeper

For a student

Sphere eversion is the standard example of the power of the Smale–Hirsch theorem and of the homotopy principle. It shows that a regular homotopy can connect immersions that look very different. The only invariants of immersions $S^2 \to \R^3$ up to regular homotopy are homotopy-theoretic, and here they vanish. Its failure for circles in the plane shows that the outcome depends on the dimensions through the relevant homotopy group.