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.
The obstruction class
( denotes the Stiefel manifold and its homotopy group ) is defined as the homotopy class of
for and the standard base of .
If the unit sphere can continue to immersion , then is . So one can see it as an obstacle to the continuation of immersion.
The Hirsch-Smale theory deals with the question of whether, conversely, the continuability of immersion follows.
Deer smale set
If is, then each immersion can be continued with to an immersion .
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:
- can be immersed in the.
- There is an equivalent mapping of the frame bundle into the Stiefel manifold .
With the Hirsch-Smale theorem, this equivalence follows a triangulation of by induction on the dimension of subsimplices .
The corollaries of this theorem include the following:
- Parallelizable -dimensional manifolds can be immersed in the.
- Compact 3-manifolds can be immersed in the.
- Exotic 7 spheres can be immersed in the.
Code dimension zero
Hirsch-Smale's theorem does not apply to .
For precise conditions for continuability of are known.
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
- ^ Theorem 3.9 in Hirsch, op. Cit.
- ^ Theorem 6.1 in Hirsch, op. Cit.
- ^ Samuel Joel Blank, Extending Immersions and regular Homotopies in Codimension 1 , PhD Thesis Brandeis University, 1967.
- ^ V. Poénaru, Extension des immersions en codimension 1 (d'aprés Samuel Blank) , Séminaire Bourbaki, Vol. 10, Soc. Math. France (1995), Exp. 42, 473-505.
- ↑ Dennis Frisch, Classification of Immersions which are bounded by Curves in Surfaces , PhD Thesis TU Darmstadt, 2010.