Brouwer degree

Suppose that M and N are two oriented differentiable manifolds of dimension n (without boundary) with M compact and N connected and suppose that f:MN is a differentiable mapping. Let Df(x) denote the differentialMathworldPlanetmath mapping at the point xM, that is the linear mapping Df(x):Tx(M)Tf(x)(N). Let signDf(x) denote the sign of the determinant of Df(x). That is the sign is positive if f preserves orientation and negative if f reverses orientation.


Let yN be a regular value, then we define the Brower degree (or just degree) of f by


It can be shown that the degree does not depend on the regular value y that we pick so that degf is well defined.

Note that this degree coincides with the degree ( as defined for maps of spheres.


  • 1 John W. Milnor. . The University Press of Virginia, Charlottesville, Virginia, 1969.
Title Brouwer degreeMathworldPlanetmath
Canonical name BrouwerDegree
Date of creation 2013-03-22 14:52:37
Last modified on 2013-03-22 14:52:37
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Definition
Classification msc 57R35
Synonym degree
Related topic DegreeMod2OfAMapping