Hirsch-Smale theory

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[edit] 1 Introduction

An immersion f:A\rightarrow N is a map of manifolds which is locally an embedding, i.e. such that for each a \in A there exists an open neighbourhood U \subseteq A with a \in U and f\vert:U \to N an embedding. A regular homotopy of immersions f_0,f_1:A \rightarrow N is a homotopy h:f_0 \simeq f_1:A \rightarrow N such that each h_t:A \rightarrow N (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.

[edit] 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 bundle 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 \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] (see also [Poénaru1995]), more details are worked out in [Frisch2010].

[edit] 3 Applications

Theorem 3.1. Let M be a smooth manifold of dimension k<n. Then the following assertions are equivalent:

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

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

(iii) the bundle associated to T_k \left( M \right) with fiber 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 using Theorem 3.9 (which can be adapted from \left(D^k,S^{k-1}\right) to \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-manifolds can be immersed into {\mathbb R}^{k+1}.
Corollary 3.3. Compact 3-manifolds can be immersed into {\mathbb R}^4.
Corollary 3.4. Exotic 7-spheres can be immersed into {\mathbb R}^8.

[edit] 4 References

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

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