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In Topology , the term degree is applied to Continuous Maps between Manifold s of the same Dimension .


From a circle to itself


The simplest and most important case is the degree of a Continuous Map

:f\colon S^1 o S^1 \,.

There is a projection

:\mathbb R o S^1= \mathbb R/ \mathbb Z \,, x\mapsto {Link without Title} ,

where {Link without Title} is the Equivalence Class of x Modulo 1 (i.e. x\sim y iff x-y is an integer).

If

:f : S^1 o S^1 \,

is continuous then there exists a continuous

:F : \mathbb R o \mathbb R,

called a ''lift'' of f to \mathbb R, such that f( = [F(z) \,. Such a lift is unique up to an additive integer constant and

:deg(f)= F(x + 1)-F(x) \,.

Note that

:F(x + 1)-F(x) \,

is an integer and it is also continuous with respect to x; Locally Constant functions on the real line must be constant. Therefore the definition does not depend on choice of x.


Between manifolds


Let f:X o Y \, be a continuous map, X and Y closed Oriented m-dimensional Manifold s.
Then the degree of f is an integer such that

:f_m( {Link without Title} )=\deg(f) {Link without Title} . \,

Here f_m is the map induced on the m dimensional Homology Group , and [Y denote the Fundamental Class es of X and Y.

Here is the easiest way to calculate the degree: If f is smooth and p is a regular value of f then f^{-1}(p)=\{x_1,x_2,..,x_n\} \, is a finite number of points. In a neighborhood of each the map f is a Homeomorphism to its image, so it might be orientation preserving or orientation reversing. If m is the number of orientation preserving and k is the number of orientation reversing locations, then deg(f)=m-k \,.

The same definition works for compact manifolds with Boundary but then f should send the boundary of X to the boundary of Y.

One can also define degree modulo 2 (deg2(''f'')) the same way as before but taking the ''fundamental class'' in '''Z'''2 homology. In this case deg2(''f'') is element of '''Z'''2, the manifolds need not be orientable and if f^{-1}(p)=\{x_1,x_2,..,x_n\} \, as before then deg2(''f'') is ''n'' modulo 2.


Properties


The degree of map is a Homotopy invariant; moreover for continuous maps from the Sphere to itself it is a ''complete'' homotopy invariant, i.e. two maps f,g:S^n o S^n \, are homotopic if and only if deg(''f'') = deg(''g'').