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degree (map of spheres) (Definition)

Given a non-negative integer $n$, let $S^n$ denote the $n$-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 $f_*$ is a homomorphism $\mathbbmss{Z}\to \mathbbmss{Z}$. Such a map must be multiplication by an integer $d$. We define the degree of the map $f$, to be this $d$.

Basic Properties

  1. If $f,g\colon S^n \to S^n$ are continuous, then $\deg(f \circ g) = \deg(f)\cdot\deg(g)$.
  2. If $f,g\colon S^n \to S^n$ are homotopic, then $\deg(f) = \deg(g)$.
  3. The degree of the identity map is $+1$.
  4. The degree of the constant map is $0$.
  5. The degree of a reflection through an $(n+1)$-dimensional hyperplane through the origin is $-1$.
  6. The antipodal map, sending $x$ to $-x$, has degree $(-1)^{n+1}$. This follows since the map $f_i$ sending $(x_1,\ldots,x_i,\ldots,x_{n+1}) \mapsto (x_1,\ldots,-x_i,\ldots,x_{n+1})$ has degree $-1$ 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 $f(z) = z^k$ has degree $k$. It is also possible, for any positive integer $n$, and any integer $k$, to construct a map $f\colon S^n \to S^n$ of degree $k$.

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 $S^n$ if and only if $n$ is odd.



"degree (map of spheres)" is owned by drini. [ full author list (2) | owner history (2) ]
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Cross-references: odd, field, non-zero vector, hairy ball theorem, positive, antipodal map, origin, hyperplane, reflection, constant map, identity map, homotopic, multiplication, map, homomorphism, functor, homology, reduced, continuous map, sphere, integer
There are 11 references to this entry.

This is version 9 of degree (map of spheres), born on 2003-01-16, modified 2004-04-24.
Object id is 3897, canonical name is DegreeMapOfSpheres.
Accessed 7123 times total.

Classification:
AMS MSC: 55M25 (Algebraic topology :: Classical topics :: Degree, winding number)
Pending Errata and Addenda
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more general manifolds? by mike on 2004-01-14 10:46:04
Maybe it could be a good idea to include also maps f:N->M, where M and N are arbitrary compact oriented manifolds, with M also connected, and mention definition of deg(f) in this case.
Otherwise I'll (someday) write a new entry for this.

\Mike


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