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Degree of map
Redirect to:
Degree as the winding number
The degree is initially defined for maps of circles: $$f:{\bf S}^1 \to {\bf S}^1$$ as the number of times the first circle is wraps around the second. It is also defined this way for maps $$f:{\bf S}^1 \to {\bf R}^2 - \{0\}$$ as the winding number.
The degree is homotopy invariant: If two map are homotopic, then their degrees are equal: $$f\sim g\colon {\bf S}^1 \to {\bf S}^1 \Rightarrow \deg(f) = \deg(g).$$
The idea of degree of a map can be generalized in two ways. First, one can look at the homology maps of spheres and, further, orientable manifolds. Or one can pursue the meaning of the degree in the fundamental group context and, further, homotopy groups.
Homology
More generally, suppose $f\colon {\bf S}^n \to {\bf S}^n$ is a continuous map the $n$-dimensional sphere. Then there is a homomorphism $f_*\colon {H}_n({\bf S}^n) \to {H}_n({\bf S}^n)$ between the homology groups. But since both groups are isomorphic to ${\bf Z}$, we know from group theory that this homomorphism $f_*$ has to be multiplication by an integer $d$. This integer is called the degree of $f$.
Properties are interesting...
Homotopy invariance. If $f,g\colon {\bf S}^n \to {\bf S}^n$ are homotopic, then $$\deg(f) = \deg(g).$$
Examples.
- The degree of the constant map is $0$.
- The degree of the identity map is $1$.
- The degree of a reflection through an $(n+1)$-dimensional hyperplane containing $0$ is $-1$.
- The the degree antipodal map is $(-1)^{n+1}$.
Multiplicativity. If $f,g\colon {\bf S}^n \to {\bf S}^n$ are maps, then $$\deg(fg) = \deg(f)\cdot\deg(g).$$
Surjectivity. A map of nonzero degree is onto.
The Hairy Ball Theorem follows from these properties. And so does the Fundamental Theorem of Algebra.
Since ${H}_n(M^n)={\bf Z}$, the concept can be generalized to a map between any two $n$-dimensional (compact connected) manifolds.
