Serre's finiteness theorem

Remarks

Since $\pi_n(S^n) \cong \Z$, the theorem says exactly which homotopy groups of spheres are infinite: $\pi_n(S^n)$, and $\pi_{2n-1}(S^n)$ for even $n$. The proof uses the Serre spectral sequence and Serre's theory of classes of abelian groups. For odd $n$, it shows that a map $S^n \to K(\Z, n)$ representing a generator induces isomorphisms on $\pi_k \otimes \Q$. For even $n$, the Whitehead product $[\iota_n, \iota_n] \in \pi_{2n-1}(S^n)$ has Hopf invariant $\pm 2$, so it has infinite order. Example: $\pi_3(S^2) \cong \Z$ is the case $n = 2$, $k = 3$, with $F = 0$, and $\pi_7(S^4) \cong \Z \oplus \Z/12$. In particular, every stable homotopy group $\pi_k^s$ with $k \ge 1$ is finite.

Going deeper

For a student

The theorem says that the rational homotopy groups of spheres are very simple, even though the integral groups are not. It is a starting point of rational homotopy theory. It reduces the study of homotopy groups of spheres to their $p$-primary parts, one prime at a time.