Smale–Hirsch theorem

Remarks

On path components the theorem says: two immersions are regularly homotopic, that is homotopic through immersions, if and only if their differentials are homotopic through fibrewise injective bundle maps. It also says that $M$ immerses in $N$ as soon as there is a fibrewise injective bundle map $TM \to TN$. Smale first proved the case of spheres in Euclidean space. For $m < n$, regular homotopy classes of immersions $S^m \to \R^n$ correspond to $\pi_m(V_{n, m})$. Here $V_{n, m}$ is the Stiefel manifold of orthonormal $m$-frames in $\R^n$. Hirsch extended the result to general $M$ and $N$. Example: $\pi_2(V_{3, 2}) = \pi_2(SO(3)) = 0$, so all immersions $S^2 \to \R^3$ are regularly homotopic, which gives sphere eversion. The hypothesis cannot be dropped: for $m = n = 1$ and $M = S^1$, $N = \R$, the set $\mathrm{Mono}(TS^1, T\R)$ is not empty, but there is no immersion $S^1 \to \R$.

Going deeper

For a student

The theorem is the first major instance of the homotopy principle. Its proof builds the regular homotopy cell by cell over a handle decomposition. It uses a covering homotopy property of the restriction map to a smaller submanifold. The extra room needed comes from the dimension condition or from the openness of $M$. Combined with computations of homotopy groups of Stiefel manifolds, it classifies immersions of spheres, and with obstruction theory it gives existence results for immersions of general manifolds.