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 by 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.

References

• Milnor, J.W. (1997). Topology from the Differentiable Viewpoint. Princeton University Press. ISBN 978-0691048338.
• Enrique Outerelo, Jesús M. Ruiz (2009). Mapping degree theory. AMS. ISBN 978-0821849156.