Degree (Map)

Let $f: M \mapsto N$ be a Map between two compact, connected, oriented $n$-D Manifolds without boundary. Then $f$ induces a Homeomorphism $f_*$ from the Homology Groups $H_n(M)$ to $H_n(N)$, both canonically isomorphic to the Integers, and so $f_*$ can be thought of as a Homeomorphism of the Integers. The Integer $d(f)$ to which the number 1 gets sent is called the degree of the Map $f$.

There is an easy way to compute $d(f)$ if the Manifolds involved are smooth. Let $x\in\Bbb{N}$, and approximate $f$ by a smooth map Homotopic to $f$ such that $x$ is a ``regular value'' of $f$ (which exist and are everywhere by Sard's Theorem). By the Implicit Function Theorem, each point in $f^{-1}(x)$ has a Neighborhood such that $f$ restricted to it is a Diffeomorphism. If the Diffeomorphism is orientation preserving, assign it the number $+1$, and if it is orientation reversing, assign it the number $-1$. Add up all the numbers for all the points in $f^{-1}(x)$, and that is the $d(f)$, the degree of $f$. One reason why the degree of a map is important is because it is a Homotopy invariant. A sharper result states that two self-maps of the $n$-sphere are homotopic Iff they have the same degree. This is equivalent to the result that the $n$th Homotopy Group of the $n$-Sphere is the set $\Bbb{Z}$ of Integers. The Isomorphism is given by taking the degree of any representation.

One important application of the degree concept is that homotopy classes of maps from $n$-spheres to $n$-spheres are classified by their degree (there is exactly one homotopy class of maps for every Integer $n$, and $n$ is the degree of those maps).