degree (map of spheres)

Given a non-negative integer n, let Sn denote the n-dimensional sphere. Suppose f:SnSn is a continuous map. Applying the nth reduced homology functor H~n(_), we obtain a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath f*:H~n(Sn)H~n(Sn). Since H~n(Sn), it follows that f* is a homomorphism . Such a map must be multiplicationPlanetmathPlanetmath by an integer d. We define the degree of the map f, to be this d.

0.1 Basic Properties

  1. 1.

    If f,g:SnSn are continuous, then deg(fg)=deg(f)deg(g).

  2. 2.

    If f,g:SnSn are homotopic, then deg(f)=deg(g).

  3. 3.

    The degree of the identity mapMathworldPlanetmath is +1.

  4. 4.

    The degree of the constant map is 0.

  5. 5.

    The degree of a reflectionMathworldPlanetmath through an (n+1)-dimensional hyperplaneMathworldPlanetmathPlanetmath through the origin is -1.

  6. 6.

    The antipodal map, sending x to -x, has degree (-1)n+1. This follows since the map fi sending (x1,,xi,,xn+1)(x1,,-xi,,xn+1) has degree -1 by (4), and the compositon f1fn+1 yields the antipodal map.

0.2 Examples

If we identify S1, then the map f:S1S1 defined by f(z)=zk has degree k. It is also possible, for any positive integer n, and any integer k, to construct a map f:SnSn of degree k.

Using degree, one can prove several theorems, including the so-called ’hairy ball theorem’, which that there exists a continuous non-zero vector field on Sn if and only if n is odd.

Title degree (map of spheres)
Canonical name DegreemapOfSpheres
Date of creation 2013-03-22 13:22:12
Last modified on 2013-03-22 13:22:12
Owner drini (3)
Last modified by drini (3)
Numerical id 12
Author drini (3)
Entry type Definition
Classification msc 55M25
Defines degree