Hirsch-Smale theory

Definition 2.1. For a submanifold ${{Stub}} == Introduction == <wikitex>; 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]]. </wikitex> == Results == <wikitex>; {{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 $k+1<n$ and $\tau\left(f^\prime\right)=0$, then $\left(f,f^\prime\right)$ can be extended to an ${\mathbb R}^q$-immersion $f:D^{k+1}\rightarrow {\mathbb R}^n$. {{endthm}} {{cite|Hirsch1959}}, Theorem 3.9. This Theorem does not hold for $n=k+1$. If $n=k+1=2$, then conditions for the extendibility of $\left(f,f^\prime\right)$ are given in {{cite|Blank1967}}, more details are worked out in {{cite|Frisch2010}}. ==References== {{#RefList:}} [[Category:Theory]]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 $Dg\mid_{A}=f^\prime$.

Definition 2.2.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

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

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