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Hopf theorem

From Wikipedia, the free encyclopedia

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

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Let M be an n-dimensional compact connected oriented manifold and the n-sphere and be continuous. Then if and only if f and g are homotopic.

References

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