Hopf theorem

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.