Hirsch-Smale theory
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− | {{beginthm|Definition}} Let $\left(f,f^\prime\right):S^k\rightarrow{\mathbb R}^n$ be an ${\mathbb R}^q$-immersion. The obstruction to extending $ | + | {{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}} |
Revision as of 12:49, 28 July 2011
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1 Introduction
An immersion is a map of manifolds which is locally an embedding, i.e. such that for each there exists an open neighbourhood with and an embedding. A regular homotopy of immersions is a homotopy such that each () 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 and a manifold , a pair is called an -immersion if
- is an immersion,
- is a linear bundle 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].
1 Applications
Theorem 4.1. Let be a smooth manifold of dimension . Then the following assertions are equivalent:
(i) can be immersed into ,
(ii) there exists a -equivariant map , where is the -frame bundle and is the Stiefel manifold,
(iii) the bundle associated to with fiber 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 using Theorem 3.9 (which can be adapted from to ) for the inductive step. The equivalence between (ii) and (iii) is a general fact from the theory of fiber bundles.
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