Hopf theorem explained

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

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.

References

• Book: Milnor, John W.. John Milnor

. John Milnor. Topology from the Differentiable Viewpoint. Princeton University Press. 1997. 978-0-691-04833-8.