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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
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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 MSC55M25 (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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