degree mod 2 of a mapping

Suppose that M and N are two differentiable manifolds of dimension n (without boundary) with M compact and N connected and suppose that f:MN is a differentiable mapping. If yN is a regular value of f, then we denote by #f-1(y) the number of points in M that map to y.


Let yN be a regular value, then we define the degree mod 2 of f by


It can be shown that the degree mod 2 does not depend on the regular value y that we pick so that deg2f is well defined.

This is similar to the Brouwer degreeMathworldPlanetmath but does not require oriented manifolds. In fact deg2f=degf(mod2).


  • 1 John W. Milnor. . The University Press of Virginia, Charlottesville, Virginia, 1969.
Title degree mod 2 of a mapping
Canonical name DegreeMod2OfAMapping
Date of creation 2013-03-22 14:52:39
Last modified on 2013-03-22 14:52:39
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 6
Author jirka (4157)
Entry type Definition
Classification msc 57R35
Synonym degree mod 2
Synonym degree modulo 2
Related topic BrouwerDegree