Hopf theorem

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 {\displaystyle S^{n}} the n-sphere and f , g : M → S n {\displaystyle f,g\colon M\to S^{n}} be continuous. Then deg ⁡ ( f ) = deg ⁡ ( g ) {\displaystyle \deg(f)=\deg(g)} 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.
  • v
  • t
  • e