Definition:Degree of Sphere Self-Mapping

Definition

Let $\Bbb S^n$ denote the $n$-sphere for $n \ge $.

Let $f: S^n \to S^n$ be a continuous mapping from $S^n$ to itself.

Let $\map {H_n} {S^n}$ denote the homology group of $S^n$.

From Homology Group is Infinite and Homology Group is Cyclic, the induced homomorphism $f*$ satisfies:

$\forall x \in \map {H_n} {S^n}: \map {f*} x = d \cdot x$

for some integer $d$.

Then $d$ is known as the degree of $f$.

Also see

• Results about homological algebra can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): degree: 4. (of a map of a sphere to itself)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): degree: 4. (of a map of a sphere to itself)