Freudenthal suspension theorem (${\pi }_k(X)⟶{\pi }_{k+1}(\Sigma X)$)

If $X$ is an $(n-1)$-connected based CW complex, $n\ge 2$, suspension induces an isomorphism ${\pi }_k(X)\to {\pi }_{k+1}(\Sigma X)$ for $k\le 2n-2$ and a surjection for $k=2n-1$. For a sphere, $\Sigma S^n\cong S^{n+1}$, giving the stable range ${\pi }_k(S^n)\cong {\pi }_{k+1}(S^{n+1})$.

Write the suspension of an $(n-2)$-connected based CW complex as two cones meeting in the original space. Each cone pair is $(n-1)$-connected because its relative groups shift the original homotopy groups by one. Homotopy excision theorem compares the cone pair with the suspension relative to the other cone, giving the suspension homomorphism as an isomorphism for $i<2n-2$ and a surjection at $i=2n-2$ in the indexing ${\pi }_{i-1}X\to {\pi }_i\Sigma X$.

The integral homology of the loop space of a sphere implies that the unit $S^{r-1}\to \Omega S^r$ has vanishing relative homology through degree $2r-3$. Both spaces are simply connected for $r\ge 3$. The absolute Hurewicz theorem first identifies their degree-two homotopy groups with homology; the relative Hurewicz theorem then rules out a first nonzero relative homotopy group in degrees three through $2r-3$. The relative homotopy sequence makes the unit an isomorphism on ${\pi }_i$ through degree $2r-4$. Loop-suspension adjunction identifies this map with suspension, yielding ${\pi }_i(S^{r-1})\cong {\pi }_{i+1}(S^r)$ in that range.