degree (map of spheres)
Given a nonnegative integer $n$, let ${S}^{n}$ denote the $n$dimensional sphere. Suppose $f:{S}^{n}\to {S}^{n}$ is a continuous map. Applying the ${n}^{th}$ reduced homology functor ${\stackrel{~}{H}}_{n}(\mathrm{\_})$, we obtain a homomorphism^{} ${f}_{*}:{\stackrel{~}{H}}_{n}({S}^{n})\to {\stackrel{~}{H}}_{n}({S}^{n})$. Since ${\stackrel{~}{H}}_{n}({S}^{n})\approx \mathbb{Z}$, it follows that ${f}_{*}$ is a homomorphism $\mathbb{Z}\to \mathbb{Z}$. Such a map must be multiplication^{} by an integer $d$. We define the degree of the map $f$, to be this $d$.
0.1 Basic Properties

1.
If $f,g:{S}^{n}\to {S}^{n}$ are continuous, then $\mathrm{deg}(f\circ g)=\mathrm{deg}(f)\cdot \mathrm{deg}(g)$.

2.
If $f,g:{S}^{n}\to {S}^{n}$ are homotopic, then $\mathrm{deg}(f)=\mathrm{deg}(g)$.

3.
The degree of the identity map^{} is $+1$.

4.
The degree of the constant map is $0$.

5.
The degree of a reflection^{} through an $(n+1)$dimensional hyperplane^{} through the origin is $1$.

6.
The antipodal map, sending $x$ to $x$, has degree ${(1)}^{n+1}$. This follows since the map ${f}_{i}$ sending $({x}_{1},\mathrm{\dots},{x}_{i},\mathrm{\dots},{x}_{n+1})\mapsto ({x}_{1},\mathrm{\dots},{x}_{i},\mathrm{\dots},{x}_{n+1})$ has degree $1$ by (4), and the compositon ${f}_{1}\circ \mathrm{\cdots}\circ {f}_{n+1}$ yields the antipodal map.
0.2 Examples
If we identify ${S}^{1}\subset \u2102$, then the map $f:{S}^{1}\to {S}^{1}$ defined by $f(z)={z}^{k}$ has degree $k$. It is also possible, for any positive integer $n$, and any integer $k$, to construct a map $f:{S}^{n}\to {S}^{n}$ of degree $k$.
Using degree, one can prove several theorems, including the socalled ’hairy ball theorem’, which that there exists a continuous nonzero vector field on ${S}^{n}$ if and only if $n$ is odd.
Title  degree (map of spheres) 

Canonical name  DegreemapOfSpheres 
Date of creation  20130322 13:22:12 
Last modified on  20130322 13:22:12 
Owner  drini (3) 
Last modified by  drini (3) 
Numerical id  12 
Author  drini (3) 
Entry type  Definition 
Classification  msc 55M25 
Defines  degree 