Hopf theorem

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The Hopf theorem is a statement in differential topology, saying that the topological degree is the only homotopy invariant of continuous maps to spheres.

Formal statement[edit]

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


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