Hirsch-Smale theory

Revision as of 12:49, 28 July 2011

1 Introduction

An immersion ${{Stub}} == Introduction == <wikitex>; An immersion $f:A\rightarrow N$ is a map of manifolds which is locally an embedding, i.e. such that for each $a \in A$ there exists an open neighbourhood $U \subseteq A$ with $a \in U$ and $f\vert:U \to N$ an embedding. A regular homotopy of immersions $f_0,f_1:A \rightarrow N$ is a homotopy $h:f_0 \simeq f_1:A \rightarrow N$ such that each $h_t:A \rightarrow N$ ($t \in I$) is an immersion. 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 bundle 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 $\left(f,f^\prime\right)$, 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]]f:A\rightarrow N$ is a map of manifolds which is locally an embedding, i.e. such that for each $a \in A$ there exists an open neighbourhood $U \subseteq A$ with $a \in U$ and $f\vert:U \to N$ an embedding. A regular homotopy of immersions $f_0,f_1:A \rightarrow N$ is a homotopy $h:f_0 \simeq f_1:A \rightarrow N$ such that each $h_t:A \rightarrow N$ ($t \in I$) is an immersion.

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 2.2. Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be an ${\mathbb R}^q$-immersion. The obstruction to extending $\left(f,f^\prime\right)$, 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

$$\displaystyle x\rightarrow f^\prime\left(e_1\left(x\right),\ldots,e_q\left(x\right)\right).$$

Theorem 2.3. Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ 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 $f:D^{k+1}\rightarrow {\mathbb R}^n$.

[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 [Blank1967], more details are worked out in [Frisch2010].

1 Applications

[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.

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

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

2 References

• [Blank1967] Samuel Joel Blank, Extending Immersions and regular Homotopies in Codimension 1, PhD Thesis Brandeis University, 1967.

• [Frisch2010] Dennis Frisch, Classification of Immersions which are bounded by Curves in Surfaces, PhD Thesis TU Darmstadt, 2010.

• [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603