---
tag: 7R8W704VGW
kind: statement
role: theorem
name: Smale–Hirsch theorem
msc: [57R42, 58A20]
uses: [F45VQ47GPV, 6RW68JGBHC, WY6K3P4YCG, 6P9PXQDSEN, P976WYEFMB]
about: [6RW68JGBHC, Y2S9GWG1J1]
sources:
  - url: https://en.wikipedia.org/wiki/Immersion_(mathematics)
---

## Statement

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]] of dimensions $m$ and $n$.
Assume that $m < n$, or that $m = n$ and no connected component of $M$ is compact.
Let $\mathrm{Imm}(M, N)$ be the space of [[immersions|6RW68JGBHC]] $M \to N$ with the $C^1$ topology.
Let $\mathrm{Mono}(TM, TN)$ be the space of continuous bundle maps $TM \to TN$ between [[tangent bundles|WY6K3P4YCG]] that are injective on each fibre, with the compact-open topology.
Then the map $\mathrm{Imm}(M, N) \to \mathrm{Mono}(TM, TN)$, $f \mapsto df$, which sends an immersion to its [[differential|6P9PXQDSEN]], is a weak homotopy equivalence.
That is, it induces a bijection on path components and isomorphisms on all [[homotopy groups|P976WYEFMB]].

## 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 $S^1 \to \R$.

## Explanations

### L1 human

Can one manifold be drawn smoothly inside another, possibly with self-crossings?
To decide this, one only needs to match up tangent directions continuously, without integrating them into an actual map.
The same is true for deciding when two such drawings can be deformed into each other.
This turns a hard geometric question into a question of algebraic topology.

### L2 human

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.