Hirsch-Smale theory
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{{beginthm|Theorem |}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be a smooth ${\mathbb R}^q$-immersion. | {{beginthm|Theorem |}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be a smooth ${\mathbb R}^q$-immersion. | ||
− | If $k<n$ 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}} | + | 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. | {{cite|Hirsch1959}}, Theorem 3.9. | ||
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+ | This Theorem does not hold for $n=k+1$. | ||
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+ | 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== | ==References== |
Revision as of 10:26, 6 July 2011
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 and a manifold , a pair is called an -immersion if
- is an immersion,
- is a linear map, and
- there exists an open neighborhood of in and an immersion such that and .
Theorem 2.3. Let be a smooth -immersion.
If and , then can be extended to an -immersion .[Hirsch1959], Theorem 3.9.
This Theorem does not hold for .
If , then conditions for the extendibility of 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.
- [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