Hopf theorem


In the following we will assume that the term “smooth” implies just C1 (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 manifoldMathworldPlanetmath without boundary of dimension m and suppose f,g:MSm 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 degreeMathworldPlanetmath.

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:MSm 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
Canonical name HopfTheorem
Date of creation 2013-03-22 14:52:34
Last modified on 2013-03-22 14:52:34
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 8
Author jirka (4157)
Entry type Theorem
Classification msc 57R35