# Hirsch-Smale theory

## 1 Introduction

Hirsch-Smale theory is the name now given to the study of regular homotopy classes of immersions and more generally the space of immersions via their derivative maps. It is one of the spectacular success stories of geometric topology and in particular the h-principle.

## 2 Results

Definition 2.1. For a submanifold $A\subset{\mathbb R}^q$${{Stub}} == Introduction == ; Hirsch-Smale theory is the name now given to the study of regular homotopy classes of immersions and more generally the space of immersions via their derivative maps. It is one of the spectacular success stories of geometric topology and in particular the [[h-principle]]. == Results == ; {{beginthm|Definition}} For a submanifold A\subset{\mathbb R}^q and a manifold N, a pair \left(f,f^\prime\right) is called an {\mathbb R}^q-immersion if - f:A\rightarrow N is an immersion, - f^\prime: T{\mathbb R}^q\mid_{A}\rightarrow TN is a linear map, and - there exists an open neighborhood U of A in {\mathbb R}^q and an immersion g:U\rightarrow N such that g\mid_{A}=f and Dg\mid_{A}=f^\prime. {{endthm}} {{beginthm|Definition}}Let \left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n be an {\mathbb R}^q-immersion. The obstruction to extending \tau\left(f,f^\prime\right), denoted by \tau\left(f,f^\prime\right)\in \pi_{k}\left(V_{n,q}\right) with V_{n,q} the Stiefel manifold of q-frames in {\mathbb R}^n, is the homotopy class of x\rightarrow f^\prime\left(e_1\left(x\right),\ldots,e_q\left(x\right)\right).{{endthm}} {{beginthm|Theorem |}} Let \left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n be a smooth {\mathbb R}^q-immersion. If k and a manifold $N$$N$, a pair $\left(f,f^\prime\right)$$\left(f,f^\prime\right)$ is called an ${\mathbb R}^q$${\mathbb R}^q$-immersion if

- $f:A\rightarrow N$$f:A\rightarrow N$ is an immersion,

- $f^\prime: T{\mathbb R}^q\mid_{A}\rightarrow TN$$f^\prime: T{\mathbb R}^q\mid_{A}\rightarrow TN$ is a linear map, and

- there exists an open neighborhood $U$$U$ of $A$$A$ in ${\mathbb R}^q$${\mathbb R}^q$ and an immersion $g:U\rightarrow N$$g:U\rightarrow N$ such that $g\mid_{A}=f$$g\mid_{A}=f$ and $Dg\mid_{A}=f^\prime$$Dg\mid_{A}=f^\prime$.

Definition 2.2.Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$$\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be an ${\mathbb R}^q$${\mathbb R}^q$-immersion. The obstruction to extending $\tau\left(f,f^\prime\right)$$\tau\left(f,f^\prime\right)$, denoted by $\tau\left(f,f^\prime\right)\in \pi_{k}\left(V_{n,q}\right)$$\tau\left(f,f^\prime\right)\in \pi_{k}\left(V_{n,q}\right)$ with $V_{n,q}$$V_{n,q}$ the Stiefel manifold of $q$$q$-frames in ${\mathbb R}^n$${\mathbb R}^n$, is the homotopy class of $\displaystyle x\rightarrow f^\prime\left(e_1\left(x\right),\ldots,e_q\left(x\right)\right).$

Theorem 2.3. Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$$\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be a smooth ${\mathbb R}^q$${\mathbb R}^q$-immersion.

If $k$k and $\tau\left(f^\prime\right)=0$$\tau\left(f^\prime\right)=0$, then $\left(f,f^\prime\right)$$\left(f,f^\prime\right)$ can be extended to an ${\mathbb R}^q$${\mathbb R}^q$-immersion $f:D^{k+1}\rightarrow {\mathbb R}^n$$f:D^{k+1}\rightarrow {\mathbb R}^n$.

[Hirsch1959], Theorem 3.9.