Hopf theorem

Short description: Topological degree is the only homotopy invariant of continuous maps to spheres

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 connected oriented manifold and [math]\displaystyle{ S^n }[/math] the n-sphere and [math]\displaystyle{ f,g\colon M\to S^n }[/math] be continuous. Then [math]\displaystyle{ \deg(f)=\deg(g) }[/math] 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.