Brouwer Degree

Let f:M|->N be a map between two compact, connected, oriented n-dimensional manifolds without boundary. Then f induces a homomorphism 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 homomorphism 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 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 dense 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 Brouwer 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 nth homotopy group of the n-sphere is the set 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).

See also

homology Group, Homomorphism, Homotopy, Manifold, Sard's Theorem

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Drábek, P. and Milota, J. "Brouwer Degree." §4.3D in Methods of Nonlinear Analysis: Applications to Differential Equations. Basel, Switzerland: Birkhäuser, pp. 228-248, 2007.Milnor, J. W. Topology from the Differentiable Viewpoint. Princeton, NJ: Princeton University Press, pp. 27-31, 1965.

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Brouwer Degree

Cite this as:

Weisstein, Eric W. "Brouwer Degree." From MathWorld--A Wolfram Web Resource.

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