Hopf theorem

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

Let M be an n-dimensional compact oriented manifold and Sⁿ the n-sphere and $f,g: M\to S^n$ be continuous. Then $\deg(f)=\deg(g)$ if and only if f and g are homotopic.