# Hopf theorem

In the following we will assume that the term “smooth” implies just $C^{1}$ (once continuously differentiable). By smooth homotopy we will that the homotopy mapping is itself continuously differentiable

###### Theorem.

Suppose that $M$ is a connected, oriented (http://planetmath.org/Orientation2) smooth manifold without boundary of dimension $m$ and suppose $f,g\colon M\to S^{m}$ are smooth mappings to the $m$-sphere. Then $f$ and $g$ are smoothly homotopic if and only if $f$ and $g$ have the same Brouwer degree.

When $M$ is not orientable, then we can always “flip” the orientation by following a closed loop on the manifold and one can then prove the following result.

###### Theorem.

Suppose that $M$ is not orientable, connected smooth manifold without boundary of dimension $m$, and suppose $f,g\colon M\to S^{m}$ are smooth mappings to the $m$-sphere. Then $f$ and $g$ are smoothly homotopic if and only if $f$ and $g$ have the same degree mod 2.

## References

• 1 John W. Milnor. . The University Press of Virginia, Charlottesville, Virginia, 1969.
Title Hopf theorem HopfTheorem 2013-03-22 14:52:34 2013-03-22 14:52:34 jirka (4157) jirka (4157) 8 jirka (4157) Theorem msc 57R35