Hopf theorem

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 oriented manifold and $S^{n}$ the n-sphere and $f,g\colon M\to S^{n}$ be continuous. Then $\deg(f)=\deg(g)$ if and only if f and g are homotopic.

gollark: Maybe it does work by word somehow.

gollark: > I don't have patience to mess with this much, but I was suspecting that the autogenerated "I consider it interesting" quotes on the root page above the auction give random pairs of words that are equal in His eyes.That is... interesting.

gollark: I can't believe that He likes apiogollarioforms more than semiapiosteganohazards.

gollark: It says "you." after it, thus insult.

gollark: Yes, it was an insult directed at you.

References

Milnor, John W. (1997). Topology from the Differentiable Viewpoint. Princeton University Press. ISBN 978-0-691-04833-8.

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