Hirsch-Smale theory
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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]]. | 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]]. | ||
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+ | == Results == | ||
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+ | {{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 | ||
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+ | - $f:A\rightarrow N$ is an immersion, | ||
+ | |||
+ | - $f^\prime: T{\mathbb R}^q\mid_{A}\rightarrow TN$ is a linear map, and | ||
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+ | - 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 $\tau\left(f,f^\prime\right)\in \pi_{k-1}\left(V_{n,q}\right)$ 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}} | ||
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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. | ||
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+ | 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}} | ||
+ | {{cite|Hirsch1959}}, Theorem 3.9. | ||
==References== | ==References== |
Revision as of 14:50, 5 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 ofTex syntax errorin and an immersion such that and .
Definition 2.2.Let be an -immersion. The obstruction to extending , denoted by with the Stiefel manifold of -frames in , is the homotopy class of
Theorem 2.3. Let be a smooth -immersion.
If and , then can be extended to an -immersion .[Hirsch1959], Theorem 3.9.
References
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603