Brouwer degree
Suppose that M and N are two oriented differentiable manifolds
of dimension n (without boundary) with M compact and N connected and suppose that
f:M→N is a differentiable mapping. Let Df(x) denote the
differential mapping at the point x∈M,
that is the linear mapping Df(x):Tx(M)→Tf(x)(N). Let signDf(x) denote the sign
of the determinant of Df(x). That is the sign is positive if f preserves
orientation and negative if f reverses orientation.
Definition.
Let y∈N be a regular value, then we define the Brower degree (or just degree) of f by
degf:= |
It can be shown that the degree does not depend on the regular value that we pick so that is well defined.
Note that this degree coincides with the degree (http://planetmath.org/Degree5) as defined for maps of spheres.
References
- 1 John W. Milnor. . The University Press of Virginia, Charlottesville, Virginia, 1969.
Title | Brouwer degree![]() |
---|---|
Canonical name | BrouwerDegree |
Date of creation | 2013-03-22 14:52:37 |
Last modified on | 2013-03-22 14:52:37 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 7 |
Author | jirka (4157) |
Entry type | Definition |
Classification | msc 57R35 |
Synonym | degree |
Related topic | DegreeMod2OfAMapping |