In the following we will assume that the term “smooth” implies just (once continuously differentiable). By smooth homotopy we will that the homotopy mapping is itself continuously differentiable
When 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.
Suppose that is not orientable, connected smooth manifold without boundary of dimension , and suppose are smooth mappings to the -sphere. Then and are smoothly homotopic if and only if and have the same degree mod 2.
- 1 John W. Milnor. . The University Press of Virginia, Charlottesville, Virginia, 1969.
|Date of creation||2013-03-22 14:52:34|
|Last modified on||2013-03-22 14:52:34|
|Last modified by||jirka (4157)|