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| | The '''Hopf theorem''' is a statement in [[differential topology]], saying that the [[degree of a continuous mapping|topological degree]] is the only [[homotopy invariant]] of [[continuous maps]] to [[n-sphere|sphere]]s. |
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| | ==Formal statement== |
| | Let ''M'' be an ''n''-dimensional [[Closed manifold|compact]] [[Manifold#Orientability|oriented manifold]] and ''S''<sup>''n''</sup> the [[n-sphere|''n''-sphere]] and <math>f,g: M\to S^n</math> be continuous. Then <math>\deg(f)=\deg(g)</math> if and only if ''f'' and ''g'' are [[homotopic]]. |
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| | ==References== |
| | * {{cite book|author=Milnor, J. W.|authorlink=John Milnor|title=Topology from the Differentiable Viewpoint|publisher=Princeton University Press|year=1997|isbn=978-0-691-04833-8}} |
| | * {{cite book|author=Enrique Outerelo, Jesús M. Ruiz|title=Mapping degree theory|publisher=AMS|year=2009|isbn=978-0-8218-4915-6}} |
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| | {{Topology-stub}} |
| | [[Category:Theorems in differential topology]] |
The Hopf theorem is a statement in differential topology, saying that the topological degree is the only homotopy invariant of continuous maps to spheres.
Formal statement
Let M be an n-dimensional compact oriented manifold and Sn the n-sphere and be continuous. Then if and only if f and g are homotopic.
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
Template:Topology-stub