|
|
|
|
|
In the following we will assume that the term ``smooth'' implies just $C^1$ (once continuously differentiable). By smooth homotopy we will mean that the homotopy mapping is itself continuously differentiable
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 2 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.
- 1
- John W. Milnor. Topology From The Differentiable Viewpoint. The University Press of Virginia, Charlottesville, Virginia, 1969.
|
"Hopf theorem" is owned by jirka.
|
|
(view preamble | get metadata)
Cross-references: degree mod 2, loop, closed, orientation, orientable, Brouwer degree, homotopic, dimension, boundary, smooth manifold, connected, mapping, homotopy, smooth, continuously differentiable, implies, term
There is 1 reference to this entry.
This is version 5 of Hopf theorem, born on 2004-12-10, modified 2005-03-07.
Object id is 6553, canonical name is HopfTheorem.
Accessed 3499 times total.
Classification:
| AMS MSC: | 57R35 (Manifolds and cell complexes :: Differential topology :: Differentiable mappings) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
