Freudenthal suspension theorem
= Freudenthal suspension theorem
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{title2=$\pi_k(X)\longrightarrow\pi_{k+1}(\Sigma X)$}
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If $X$ is an $(n-1)$-connected based <CW complex>, $n\geq2$, suspension induces an isomorphism $\pi_k(X)\to\pi_{k+1}(\Sigma X)$ for $k\leq2n-2$ and a surjection for $k=2n-1$. For a <sphere>, $\Sigma S^n\cong S^{n+1}$, giving the stable range $\pi_k(S^n)\cong\pi_{k+1}(S^{n+1})$.