DEGREE OF A CONTINUOUS MAPPING

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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. It is a generalization of winding number. In physics, the degree of a continuous map is usually called a topological quantum number.

Contents

From a circle to itself

Between manifolds

Calculating degree

Properties

From a circle to itself

The simplest and most important case is the degree of a continuous map from the circle to itself (this is called the winding number):
:$fcolon\; S^1\; o\; S^1.\; ,$
There is a projection
:$mathbb\; R\; o\; S^1=\; mathbb\; R/\; mathbb\; Z\; ,$, $xmapsto\; [x],$
where $[x]$ is the equivalence class of $x$ modulo1 (i.e. $xsim\; y$ if and only if $x-y$ 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 $f([z])\; =\; [F(z)]\; ,$. 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 $fcolon\; X\; o\; Y\; ,$ be a continuous map, $X$ and $Y$ closed oriented $m$-dimensional manifolds.
Then the 'degree' of $f$ is the degree of the map on top homology groups:
:$f\_mcolon\; H\_m(X)\; cong\; mathbf\{Z\}\; o\; H\_m(Y)\; cong\; mathbf\{Z\}.$
This map is a homomorphism between two copies of the integers $mathbf\{Z\}$, and thus is multiplication by some integer $d$, which is the degree.
In terms of fundamental classes,
:$f\_m([X])=deg(f)[Y]$
where $H\_m(X)$ is generated by $[X]$, likewise for $Y$.
In terms of differential forms, this says that if you pull back a volume form on $Y$ to $X$ and integrate, you get the degree: this is using the map on forms, and the induced map on cohomology.
Recall that topologically, an orientation is a choice of fundamental class; in other words, an identification of the top homology group with the integers (otherwise you can't tell positive from negative).

Calculating degree

One way to calculate the degree is that a (smooth) degree $d$ map is generically d-to-1, counting orientation (away from singular values).
Concretely, 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 (it's a covering map), so it is either orientation preserving or orientation reversing. If $r$ is the number of orientation preserving and $s$ is the number of orientation reversing locations, then $deg(f)=r-s\; ,$.
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 $deg(f)\; =\; deg(g)$.
In other words, degree is an isomorphism $[S^n,S^n]=pi\_n\; S^n\; o\; mathbf\{Z\}$.