## Degree (Map)

Let be a Map between two compact, connected, oriented -D Manifolds without boundary. Then induces a Homeomorphism from the Homology Groups to , both canonically isomorphic to the Integers, and so can be thought of as a Homeomorphism of the Integers. The Integer to which the number 1 gets sent is called the degree of the Map .

There is an easy way to compute if the Manifolds involved are smooth. Let , and approximate by a smooth map Homotopic to such that is a regular value'' of (which exist and are everywhere by Sard's Theorem). By the Implicit Function Theorem, each point in has a Neighborhood such that restricted to it is a Diffeomorphism. If the Diffeomorphism is orientation preserving, assign it the number , and if it is orientation reversing, assign it the number . Add up all the numbers for all the points in , and that is the , the degree of . 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 -sphere are homotopic Iff they have the same degree. This is equivalent to the result that the th Homotopy Group of the -Sphere is the set 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 -spheres to -spheres are classified by their degree (there is exactly one homotopy class of maps for every Integer , and is the degree of those maps).