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 \( {\displaystyle f,g\colon M\to S^{n}} \) be continuous. Then \( \deg(f)=\deg(g) \) 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.

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

Hellenica World - Scientific Library

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