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I remember coming across this result some time ago but I am having trouble finding a reference for it. It goes something like this:

Let $p$ be a(n odd?) prime, then the $p$-primary component of $\pi^S_k$ is $\Bbb Z_p$ when $k=2l(p-1)-1$ for $l=1,\dots,p-1$ and is trivial for all other $k<2p(p-1)-2$.

This is what I have written down on the back of an envelope. I checked this with Wikipedia's table and it seems to be true.

What is the reference in which it is proved? And if its simple could you overview it as an answer here?

My guess is that it is proven by Toda but his papers are difficult to search through.

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  • $\begingroup$ For $p=3$ your statement says that ${_p\pi_k^s}\simeq 0$ for all $k<12$, right! But, for instance $\pi_{7}^s\simeq\mathbb{Z}/240$ which definitely has a nontrivial $3$-primary part! Or maybe I am missing something!!! $\endgroup$
    – user51223
    Commented Jul 27, 2018 at 18:28
  • $\begingroup$ @user51223 My statement says that for $p=3$ the $3$-primary component of $\pi^S_k$ is trivial for all $k<10$ except $k=3,7$. $\endgroup$
    – Ali Caglayan
    Commented Jul 27, 2018 at 19:28

1 Answer 1

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This follows easily from Theorem 4.4.20 of Ravenel's book, Complex cobordism and stable homotopy groups of spheres". The elements in $Ext^i$ with $i>1$ have total degrees greater than or equal to $2p(p−1)−2$, so we only have to look at $Ext^1$, and we have $$\pi _{ql-1}(S^0)\cong Ext ^{1,ql}\cong Z/p \mbox{ where $q=2(p-1)$}$$ other $Ext^1$ groups vanishing in the degree range you give.

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