# Hirsch-Smale theory

## 1 Introduction

An immersion $f:A\rightarrow N$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}f:A\rightarrow N$ is a map of manifolds which is locally an embedding, i.e. such that for each $a \in A$$a \in A$ there exists an open neighbourhood $U \subseteq A$$U \subseteq A$ with $a \in U$$a \in U$ and $f\vert:U \to N$$f\vert:U \to N$ an embedding. A regular homotopy of immersions $f_0,f_1:A \rightarrow N$$f_0,f_1:A \rightarrow N$ is a homotopy $h:f_0 \simeq f_1:A \rightarrow N$$h:f_0 \simeq f_1:A \rightarrow N$ such that each $h_t:A \rightarrow N$$h_t:A \rightarrow N$ ($t \in I$$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 2.1. For a submanifold $A\subset{\mathbb R}^q$$A\subset{\mathbb R}^q$ and a manifold $N$$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 bundle 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 $\left(f,f^\prime\right)$$\left(f,f^\prime\right)$, denoted by $\tau\left(f,f^\prime\right)\in \pi_{k}\left(V_{n,q}\right)$$\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 $k+1$k+1 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.

This theorem does not hold for $n=k+1$$n=k+1$.

If $n=k+1=2$$n=k+1=2$, then conditions for the extendibility of $\left(f,f^\prime\right)$$\left(f,f^\prime\right)$ are given in [Blank1967] (see also [Poénaru1995]), more details are worked out in [Frisch2010].

## 3 Applications

Theorem 3.1. Let $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1xgrgY$$M$ be a smooth manifold of dimension $k$k. Then the following assertions are equivalent:

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

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

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

[Hirsch1959, Theorem 6.1] The equivalence between (i) and (ii) is proved by induction over the dimension of subsimplices in a triangulation of $M$$M$ using Theorem 3.9 (which can be adapted from $\left(D^k,S^{k-1}\right)$$\left(D^k,S^{k-1}\right)$ to $\left(\Delta^k,\partial \Delta^k\right)$$\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 3.2. Parallelizable $k$$k$-manifolds can be immersed into ${\mathbb R}^{k+1}$${\mathbb R}^{k+1}$.
Corollary 3.3. Compact $3$$3$-manifolds can be immersed into ${\mathbb R}^4$${\mathbb R}^4$.
Corollary 3.4. Exotic $7$$7$-spheres can be immersed into ${\mathbb R}^8$${\mathbb R}^8$.

## 4 References

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