Brouwer Degree
Let 
be a map between two compact, connected, oriented 
-dimensional manifolds without boundary. Then 
induces a homomorphism 
from the homology groups 
to 
, both canonically isomorphic to the integers, and so 
can be thought of as a homomorphism of the integers. The integer 
to which the number 1 gets sent is called the degree of the map 
.
There is an easy way to compute 
if the manifolds involved are smooth. Let 
, and approximate 
by a smooth map homotopic to 
such that 
is a "regular value" of 
(which exist and are everywhere dense by Sard's theorem). By the implicit function theorem, each point in 
has a neighborhood such that 
restricted to it is a diffeomorphism. If the diffeomorphism is orientation preserving, assign it the number 
, and if it is orientation reversing, assign it the number 
. Add up all the numbers for all the points in 
, and that is the 
, the Brouwer degree of 
. One reason why the degree of a map is important is because it is a homotopy invariant. A sharper result states that two self-maps of the 
-sphere are homotopic iff they have the same degree. This is equivalent to the result that the 
th homotopy group of the 
-sphere is the set 
of integers. The isomorphism is given by taking the degree of any representation.
One important application of the degree concept is that homotopy classes of maps from 
-spheres to 
-spheres are classified by their degree (there is exactly one homotopy class of maps for every integer 
, and 
is the degree of those maps).
REFERENCES:
Drábek, P. and Milota, J. "Brouwer Degree." §4.3D in Methods of Nonlinear Analysis: Applications to Differential Equations. Basel, Switzerland: Birkhäuser, pp. 228-248, 2007.
Milnor, J. W. Topology from the Differentiable Viewpoint. Princeton, NJ: Princeton University Press, pp. 27-31, 1965.
