For an $(n-1)$-connected CW complex $X$, suspension $\pi_i(X)\to\pi_{i+1}(SX)$ is an isomorphism for $i<2n-1$ and surjective for $i=2n-1$.Suggest a slogan
The suspension map \[ \pi_i(S^n) \to \pi_{i+1}(S^{n+1}) \] is an isomorphism for \(i < 2n-1\) and a surjection for \(i = 2n-1\). More generally, for an \((n-1)\)-connected CW complex \(X\), the suspension map \(\pi_i(X)\to\pi_{i+1}(SX)\) is an isomorphism for \(i < 2n-1\) and a surjection for \(i = 2n-1\) [@Hat02].
In particular, \(\pi_{n+k}(S^n)\) is independent of \(n\) for \(n > k+1\); this common value is the \(k\)th stable homotopy group of spheres.