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This article is about the term "degree" as used in algebraic topology. For alternate meanings, see degree (mathematics) or degree.

In topology, the term degree is applied to continuous maps between manifolds of the same dimension.

[edit] From a circle to itself

The simplest and most important case is the degree of a continuous map

<math>f\colon S^1\to S^1 \,</math>.

There is a projection

<math>\mathbb R \to S^1= \mathbb R/ \mathbb Z \,</math>, <math>x\mapsto [x]</math>,

where <math>[x]</math> is the equivalence class of <math>x</math> modulo1 (i.e. <math>x\sim y</math> if and only if <math>x-y</math> is an integer).

If

<math>f : S^1 \to S^1 \,</math>

is continuous then there exists a continuous

<math>F : \mathbb R \to \mathbb R</math>,

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

<math>deg(f)= F(x + 1)-F(x) \,</math>.

Note that

<math>F(x + 1)-F(x) \,</math>

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

[edit] Between manifolds

Let <math>f:X\to Y \,</math> be a continuous map, <math>X</math> and <math>Y</math> closed oriented <math>m</math>-dimensional manifolds. Then the degree of <math>f</math> is an integer such that

<math>f_m([X])=\deg(f)[Y]. \,</math>

Here <math>f_m</math> is the map induced on the <math>m</math> dimensional homology group, <math>[X]</math> and <math>[Y]</math> denote the fundamental classes of <math>X</math> and <math>Y</math>.

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

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

One can also define degree modulo 2 (deg₂(f)) the same way as before but taking the fundamental class in Z₂ homology. In this case deg₂(f) is element of Z₂, the manifolds need not be orientable and if <math>f^{-1}(p)=\{x_1,x_2,..,x_n\} \,</math> as before then deg₂(f) is n modulo 2.

[edit] 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 <math>f,g:S^n\to S^n \,</math> are homotopic if and only if deg(f) = deg(g).ru:Степень отображения it:Grado topologico