Hopf theorem
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Short description: Topological degree is the only homotopy invariant of continuous maps to spheres
The Hopf theorem (named after Heinz Hopf) 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 connected oriented manifold and [math]\displaystyle{ S^n }[/math] the n-sphere and [math]\displaystyle{ f,g\colon M\to S^n }[/math] be continuous. Then [math]\displaystyle{ \deg(f)=\deg(g) }[/math] if and only if f and g are homotopic.
References
- Milnor, John W. (1997). Topology from the Differentiable Viewpoint. Princeton University Press. ISBN 978-0-691-04833-8.
![]() | Original source: https://en.wikipedia.org/wiki/Hopf theorem.
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