Hirsch-Smale theory

From Manifold Atlas
Revision as of 10:26, 6 July 2011 by Kuessner (Talk | contribs)
Jump to: navigation, search

This page has not been refereed. The information given here might be incomplete or provisional.

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 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
\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].

References

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

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox