Hopf degree theorem

Remarks

One direction is the homotopy invariance of degree. For the other, one may take $f$ and $g$ smooth and use a regular value. By the Pontryagin–Thom construction, the homotopy class of a map $M \to S^n$ is determined by the framed cobordism class of the preimage of a regular value. This preimage is a finite set of signed points, and in dimension $0$ such a class is determined by the sum of the signs. In other words, the degree gives a bijection from the set $[M, S^n]$ of homotopy classes to $\Z$.

Example: for $M = S^1$ and $n = 1$, the theorem says that loops in the circle are classified up to homotopy by their winding number, which recovers $\pi_1(S^1) \cong \Z$.

The dimensions must agree: the Hopf map $S^3 \to S^2$ is not null-homotopic, though $H_3(S^2; \Z) = 0$ so that no degree can distinguish it from a constant map. If $M$ is not orientable, the same holds with the degree replaced by the degree modulo $2$.

Going deeper

For a student

The theorem computes $[M, S^n] \cong \Z \cong H^n(M; \Z)$ for closed connected oriented $n$-manifolds. It is a first case of the representability of cohomology by Eilenberg–MacLane spaces. The reason is that a map $S^n \to K(\Z, n)$ representing a generator induces isomorphisms on $\pi_i$ for $i \le n$, which is enough for maps out of an $n$-dimensional $M$. It is used to define indices of vector fields and to prove the Poincaré–Hopf theorem, and it is the base case of obstruction theory for maps into spheres.