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: You probably want to be able to improvise and stuff for emergencies, like in The Martian, and obviously need to be good at repair, but mostly those don't happen much.

gollark: "Oh no! We drove into a potatron warp! We need to reflux the hyperluminar subquantum transistors!"

gollark: Only if you're in a stupid TV show where weird ridiculous novel stuff happens all the time.

gollark: Those are generally called "expert systems" instead.

gollark: Repair is tricky, I guess?

References

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

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