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}}. ==Applications== {{beginthm|Theorem |}} Let $M$ be a smooth manifold of dimension $k<n$. Then the following assertions are equivalent: (i) $M$ can be immersed into ${\mathbb R}^n$, (ii) there exists a $GL\left(k,{\mathbb R}\right)$-equivariant map $T_k\left(M\right)\rightarrow V_{n,k}$, where $T_k\left(M\right)\rightarrow M$ is the $k$-frame bundle and $V_{n,k}$ is the Stiefel manifold, (iii) the bundle associated to $T_k\left(M\right)$ with fiber $V_{n,k}$ has a cross section. {{endthm}} {{cite|Hirsch1959}}, Theorem 6.1. The equivalence between (i) and (ii) is proved by induction over the dimension of subsimplices in a triangulation of $M$ using Theorem 3.9 (which can be adapted from $\left(D^k,S^{k-1}\right)$ to $\left(\Delta^k,\partial \Delta^k\right)$) for the inductive step. The equivalence between (ii) and (iii) is a general fact from the theory of fiber bundles. {{beginthm|Corollary |}} Parallelizable $k$-manifolds can be immersed into ${\mathbb R}^{k+1}$.{{endthm}} {{beginthm|Corollary |}} Exotic $-spheres can be immersed into ${\mathbb R}^8$.{{endthm}} ==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 $f:D^{k+1}\rightarrow {\mathbb R}^n$.

Theorem 4.1. Let $M$ be a smooth manifold of dimension $k<n$. Then the following assertions are equivalent:

(i) $M$ can be immersed into ${\mathbb R}^n$,

(ii) there exists a $GL\left(k,{\mathbb R}\right)$-equivariant map $T_k\left(M\right)\rightarrow V_{n,k}$, where $T_k\left(M\right)\rightarrow M$ is the $k$-frame bundle and $V_{n,k}$ is the Stiefel manifold,

(iii) the bundle associated to $T_k\left(M\right)$ with fiber $V_{n,k}$ has a cross section.

Corollary 4.2. Parallelizable $k$-manifolds can be immersed into ${\mathbb R}^{k+1}$.

Corollary 4.3. Exotic $7$-spheres can be immersed into ${\mathbb R}^8$.