Someone asked me if *all* finite abelian groups arise as homotopy groups of spheres. I strongly doubted it, and I bet ten bucks that $\mathbb{Z}_5$ is not $\pi_k(S^n)$ for any $n,k$. But I don't know how to prove it's not.

Which finite abelian groups are known to *not* arise as homotopy groups of spheres?

I conjecture that $\mathbb{Z}_5$ is the smallest one. From some tables we can see that all smaller groups do actually arise:

$$ \begin{array}{ccl} \pi_1(S^2) &\cong & 1 \\ \pi_4(S^3) &\cong& \mathbb{Z}_2 \\ \pi_9(S^3) &\cong& \mathbb{Z}_3 \\ \pi_8(S^4) &\cong& \mathbb{Z}_2 \times \mathbb{Z}_2 \\ \pi_{122}(S^{62}) &\cong& \mathbb{Z}_4. \end{array} $$

In fact, I conjecture that for no odd prime $p \gt 3$ is $\mathbb{Z}_p$ isomorphic to $\pi_k(S^n)$ for any $n,k$.

odd orderappearing. (But the unstable part is really outside my wheelhouse...) $\endgroup$7more comments