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It is known from Serre's classical result that every p-torsion occurs in the homotopy groups of every sphere. Is it known: do elements of every order occur in homotopy groups of spheres?

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    $\begingroup$ I'm aware of this question, thanks, but I'm asking a much weaker question. There may not be a group isomorphic to $Z_5$, but it is easy to prove that every sphere has homotopy groups containing elements of order $5$. $\endgroup$ Jul 25 at 14:12
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    $\begingroup$ Theorem 5.30, page 44 pi.math.cornell.edu/~hatcher/AT/ATch5.pdf $\endgroup$ Jul 25 at 14:19
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    $\begingroup$ I always have a little bit of difficulty parsing "any" as a quantifier. I think "every" might be clearer here. (Unless that's not what's meant!) $\endgroup$
    – LSpice
    Jul 25 at 14:48
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    $\begingroup$ Sure. Let n be a positive integer. Let N be the product of 2n and Euler phi(2n). Then for each prime p that divides n, the number p-1 divides 2N. So for each prime divisor p of n, the denominator of the Bernoulli number B_N is div'l by p. So none of the prime factors of n cancel with factors in the numerator of B_N/N. So denom(B_N/N) is divisible by n. Now denom(B_N/N) or 2*denom(B_N/N) is the order of a cyclic summand in the image of the J-homomorphism in the (2N-1)st stable homotopy group of S^0. So by Freudenthal, there is an unstable homotopy group of a sphere with an element of order n. $\endgroup$
    – A.S.
    Jul 25 at 15:38
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    $\begingroup$ @A.S. This seems more like an answer than a comment. $\endgroup$
    – Mark Grant
    Jul 26 at 7:57

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As others suggested, I am posting my earlier comment as an answer:


$\DeclareMathOperator\denom{denom}$Sure. Let $n$ be a positive integer. Let $N$ be the product of $2n$ and Euler $\phi(2n)$. Then for each prime $p$ that divides $n$, the number $p-1$ divides $N$. So for each prime divisor $p$ of $n$, the denominator of the Bernoulli number $B_N$ is divisible by $p$. So none of the prime factors of $n$ cancel with factors in the numerator of $B_N/N$. So $\denom(B_N/N)$ is divisible by $n$. Now $\denom(B_N/N)$ or $2\denom(B_N/N)$ is the order of a cyclic summand in the image of the J-homomorphism in the $(2N-1)$st stable homotopy group of $S^0$. So by Freudenthal, there is an unstable homotopy group of a sphere with an element of order $n$.

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