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 $S^n$ 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

{{cite book|last=Milnor|first= John W.|authorlink=John Milnor|title=Topology from the Differentiable Viewpoint|publisher=Princeton University Press|year=1997|isbn=978-0-691-04833-8}}

{{Topology-stub}}

Category:Theorems in differential topology