Given a non-negative integer $n$ , let $S^n$ denote the $n$ -dimensional sphere. Suppose $f\colon S^n \to S^n$ is a continuous map. Applying the $n^{th}$ reduced homology functor $\widetilde{H}_n(\_)$ , we obtain a homomorphism $f_*\colon \widetilde{H}_n(S^n) \to \widetilde{H}_n(S^n)$ . Since , it follows that $f_*$ is a homomorphism . Such a map must be multiplication by an integer $d$ . We define the degree of the map $f$ , to be this $d$ .

Basic Properties

1. If $f,g\colon S^n \to S^n$ are continuous, then $\deg(f \circ g) = \deg(f)\cdot\deg(g)$ .
2. If $f,g\colon S^n \to S^n$ are homotopic, then $\deg(f) = \deg(g)$ .
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 $(x_1,\ldots,x_i,\ldots,x_{n+1}) \mapsto (x_1,\ldots,-x_i,\ldots,x_{n+1})$ has degree $-1$ by (4), and the compositon $f_1\circ\cdots\circ f_{n+1}$ yields the antipodal map.

Examples

Using degree, one can prove several theorems, including the so-called 'hairy ball theorem', which states that there exists a continuous non-zero vector field on $S^n$ if and only if $n$ is odd.