Degree of a Map

One integer per self-map of a sphere — the winding number, computed by homology.

The idea

The degree of a map assigns to every continuous $f \colon S^{n} \to S^{n}$ an integer, and that integer classifies the map up to homotopy.

The circle shows what the integer measures. Regard $S^{1}$ as the unit circle in $\mathbb{C}$ and let $f(z) = z^{k}$. As $z$ travels once around the circle, its image travels $k$ times around: backwards when $k$ is negative, and not at all when $k = 0$. A homotopy moves the image path gradually, and a count of complete turns cannot change gradually, so no deformation changes the count.

Homology defines the same integer in every dimension. $H_{n}(S^{n})$ is infinite cyclic, and a homomorphism from an infinite cyclic group to itself is multiplication by one integer.

Definition (Degree of a map).

Let $f \colon S^{n} \to S^{n}$ be continuous, with $n \ge 1$. The induced map $f_{} \colon H_{n}(S^{n}) \to H_{n}(S^{n})$ is multiplication by a single integer, the degree of $f$, written $\deg f$: $f_{}(\alpha) = (\deg f)\,\alpha \qquad \text{for every } \alpha \in H_{n}(S^{n}).$

The degree distinguishes exactly homotopy classes.

Theorem.

For $n \ge 1$, two maps $S^{n} \to S^{n}$ are homotopic if and only if their degrees agree.

The set of self-maps of the sphere up to deformation is thereby a copy of $\mathbb{Z}$, and questions about maps become questions about integers.

Ways to work on it

• Walkthrough. Winding on the circle, the homological definition, and the arithmetic of degrees: composites, the antipodal map, fixed points and local degrees.
• Proof. Why a reflection has degree -1 and the antipodal map of S^n has degree (-1)^n+1.
• Practice. Degrees of composites and antipodal maps, homotopy and fixed-point decisions, and degrees computed from local data.
• Hardest. Determine which spheres can be combed — whether a nowhere-zero tangent vector field exists.

Not sure where to start? Take the ten-question placement test.