Freudenthal suspension theorem

Remarks

The usual proof applies the Blakers–Massey theorem to the two cones that form $\Sigma X$, whose intersection is a copy of $X$. For spheres, $S^n$ is $(n-1)$-connected and $\Sigma S^n = S^{n+1}$, so $\pi_i(S^n) \to \pi_{i+1}(S^{n+1})$ is an isomorphism for $i < 2n - 1$. Hence $\pi_{n+k}(S^n)$ does not depend on $n$ once $n \ge k + 2$, which defines the stable homotopy groups of spheres. The bound is sharp: for $n = 2$ and $i = 3$, the map $\pi_3(S^2) \cong \Z \to \pi_4(S^3) \cong \Z/2$ is onto but not injective. The theorem also gives $\pi_n(S^n) \cong \Z$ for all $n \ge 1$ by induction from $\pi_1(S^1) \cong \Z$, using $\pi_2(S^2) \cong \Z$ from the Hopf fibration at the first step.

Going deeper

For a student

The theorem is the starting point of stable homotopy theory. It shows that the suspension sequence $\pi_{n+k}(S^n) \to \pi_{n+k+1}(S^{n+1}) \to \cdots$ eventually becomes constant, so stable groups and spectra make sense. In the stable range, homotopy groups behave like a homology theory, since excision holds there.