Hirsch-Smale Theory

The investigation of the regular homotopy classes of immersions is called the Hirsch-Smale theory (after the mathematicians Morris William Hirsch and Stephen Smale ) in the mathematical field of differential topology .

A known application is the " eversion of the sphere " (Engl .: sphere eversion ), which is illustrated in the popular video "Outside In".

Continuability of Immersions: The Obstacle

Be an immersion that can be continued as an immersion on an environment of in , and be its differential. ${\displaystyle f\colon S^{k}\to \mathbb {R} ^{n}}$ ${\displaystyle U}$ ${\displaystyle S^{k}}$ ${\displaystyle \mathbb {R} ^{k+1}}$ ${\displaystyle f^{\prime }\colon T\mathbb {R} ^{k+1}\mid _{S^{k}}\to T\mathbb {R} ^{n}}$

The obstruction class

${\displaystyle \tau (f)\in \pi _{k}V_{n,k+1}}$

( denotes the Stiefel manifold and its homotopy group ) is defined as the homotopy class of ${\displaystyle V_{n,k+1}}$ ${\displaystyle \pi _{k}}$

${\displaystyle x\mapsto f^{\prime }(e_{1}(x),\ldots ,e_{k+1}(x))}$

for and the standard base of . ${\displaystyle x\in S^{k}}$ ${\displaystyle e_{1}(x),\ldots ,e_{k+1}(x)}$ ${\displaystyle T_{x}\mathbb {R} ^{k+1}}$

If the unit sphere can continue to immersion , then is . So one can see it as an obstacle to the continuation of immersion. ${\displaystyle f}$ ${\displaystyle B^{k+1}\to \mathbb {R} ^{n}}$ ${\displaystyle \tau (f)=0}$ ${\displaystyle \tau (f)}$

The Hirsch-Smale theory deals with the question of whether, conversely, the continuability of immersion follows. ${\displaystyle \tau (f)=0}$

Deer smale set

If is, then each immersion can be continued with to an immersion . ${\displaystyle k+1<n}$ ${\displaystyle f\colon S^{k}\to \mathbb {R} ^{n}}$ ${\displaystyle \tau (f)=0}$ ${\displaystyle B^{k+1}\to \mathbb {R} ^{n}}$

This sentence is considered to be one of the first examples of an h-principle .

Applications

Theorem: For a smooth manifold of dimension the following conditions are equivalent: ${\displaystyle k<n}$

• ${\displaystyle M}$can be immersed in the. ${\displaystyle \mathbb {R} ^{n}}$
• There is an equivalent mapping of the frame bundle into the Stiefel manifold . ${\displaystyle GL(k,\mathbb {R} )}$ ${\displaystyle F_{k}M}$ ${\displaystyle V_{n,k}}$

With the Hirsch-Smale theorem, this equivalence follows a triangulation of by induction on the dimension of subsimplices . ${\displaystyle M}$

The corollaries of this theorem include the following:

• Parallelizable -dimensional manifolds can be immersed in the. ${\displaystyle k}$ ${\displaystyle \mathbb {R} ^{k+1}}$
• Compact 3-manifolds can be immersed in the. ${\displaystyle \mathbb {R} ^{4}}$
• Exotic 7 spheres can be immersed in the. ${\displaystyle \mathbb {R} ^{8}}$

Code dimension zero

Hirsch-Smale's theorem does not apply to . ${\displaystyle k+1=n}$

For precise conditions for continuability of are known. ${\displaystyle k+1=n=2}$ ${\displaystyle f}$

literature

• Morris W. Hirsch, Immersions of manifolds , Trans. Amer. Math. Soc. 93: 242-276 (1959). Online (accessed January 1, 2017)

Web links

• Hirsch-Smale theory (Manifold Atlas)
• J. Francis: The h-principle: The Hirsch-Smale theorem
• M. Weiss: Immersion theory for homotopy theorists
• Outside In (Geometry Center Video Productions)

Individual evidence

1. ^ Theorem 3.9 in Hirsch, op. Cit.
2. ^ Theorem 6.1 in Hirsch, op. Cit.
3. ^ Samuel Joel Blank, Extending Immersions and regular Homotopies in Codimension 1 , PhD Thesis Brandeis University, 1967.
4. ^ V. Poénaru, Extension des immersions en codimension 1 (d'aprés Samuel Blank) , Séminaire Bourbaki, Vol. 10, Soc. Math. France (1995), Exp. 42, 473-505.
5. ↑ Dennis Frisch, Classification of Immersions which are bounded by Curves in Surfaces , PhD Thesis TU Darmstadt, 2010.