Degree (Map)

Let be a Map between two compact, connected, oriented -D Manifolds withoutboundary. Then induces a Homeomorphism from the Homology Groups to, both canonically isomorphic to the Integers, and so can be thought of as aHomeomorphism of the Integers. The Integer to which the number 1 gets sent iscalled the degree of the Map .

There is an easy way to compute if the Manifolds involved are smooth. Let , andapproximate by a smooth map Homotopic to such that is a ``regular value'' of (which existand are everywhere by Sard's Theorem). By the Implicit Function Theorem, each point in has aNeighborhood such that restricted to it is a Diffeomorphism. If the Diffeomorphism is orientationpreserving, assign it the number , and if it is orientation reversing, assign it the number . Add up all the numbersfor all the points in , and that is the , the degree of . One reason why the degree of a map is importantis because it is a Homotopy invariant. A sharper result states that two self-maps of the -sphere are homotopicIff 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 degreeof any representation.

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

• Balanced ANOVA
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