Back to the index. Or to the chambers

This article has 21 links. View as Cloud or List.

Loading ...

Planetmath Browser (2008—2009)

BSD licence | A django site

All articles taken from PlanetMath.org snapshot under CC-BY-SA licence.

→ The original article on PlanetMath.org

Other Formats: LaTeX

New! You can click on formulas to copy the LaTeX source to your clipboard. | Math Videos

Degree Map Of Spheres

Given a non-negative integer , let denote the -dimensional sphere. Suppose $f\colon S^n \to S^n$ is a continuous map. Applying the $n^{th}$ reduced homology functor $\widetilde{H}_n(\_)$ , we obtain a homomorphism $f_*\colon \widetilde{H}_n(S^n) \to \widetilde{H}_n(S^n)$ . Since $\widetilde{H}_n(S^n) \approx \mathbbmss{Z}$ , it follows that is a homomorphism $\mathbbmss{Z}\to \mathbbmss{Z}$ . Such a map must be multiplication by an integer . We define the degree of the map , to be this .

Basic Properties

If $f,g\colon S^n \to S^n$ are continuous, then $\deg(f \circ g) = \deg(f)\cdot\deg(g)$ .
If $f,g\colon S^n \to S^n$ are homotopic, then $\deg(f) = \deg(g)$ .
The degree of the identity map is .
The degree of the constant map is .
The degree of a reflection through an -dimensional hyperplane through the origin is .
The antipodal map, sending to , has degree $(-1)^{n+1}$ . This follows since the map sending $(x_1,\ldots,x_i,\ldots,x_{n+1}) \mapsto (x_1,\ldots,-x_i,\ldots,x_{n+1})$ has degree by (4), and the compositon $f_1\circ\cdots\circ f_{n+1}$ yields the antipodal map.

Examples

If we identify $S^1 \subset \mathbbmss{C}$ , then the map $f : S^1 \to S^1$ defined by

has degree

. It is also possible, for any positive integer

, and any integer

, to construct a map $f\colon S^n \to S^n$ of degree

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

1	Types Of Homomorphisms
1	Functor
1	Euclidean Transformation
1	Continuous
1	Field
1	Positive
1	Function
1	Even Number
1	Identity Map
1	Integer
1	Multiplication
2	Origin
2	Root System
3	Constant Function
4	Linear Manifold
4	Homology Topological Space
4	Homotopy Of Maps
4	Sphere Metric Space
5	Non Zero Vector
5	Antipodal
16	Hairy Ball Theorem