# Do elements of every order occur in homotopy groups of spheres?

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?

• 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 $5$. Jul 25 at 14:12
• Theorem 5.30, page 44 pi.math.cornell.edu/~hatcher/AT/ATch5.pdf Jul 25 at 14:19
• 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!) Jul 25 at 14:48
• 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.
– A.S.
Jul 25 at 15:38
• @A.S. This seems more like an answer than a comment. Jul 26 at 7:57

$$\DeclareMathOperator\denom{denom}$$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 $$n$$.