# Divisibility in the homotopy groups of spheres?

Consider the $$k$$-sphere (I'm particularly interested in $$k=3$$). For each positive integer $$n$$ let $$P_{n}$$ be the set of primes that divide the order of $$\pi_{i}S^{k}$$ for some $$i\leq n$$.

Does the cardinality of $$P_{n}$$ tend to infinity with $$n$$?

Yes. In fact $$\bigcup_{n\geq 1} P_n$$ is the set of all primes. Serre proved that, for each odd prime $$p$$, there is some very predictable $$p$$-torsion in $$\pi_{k+(p-1)}(S^k)$$, for example.

There is a very nice theorem of McGibbon and Neisendorfer (proving, I think, a conjecture of Serre) that implies and vastly generalizes this.

Theorem (McGibbon & Neisendorfer). Let $$X$$ be a simply-connected finite complex such that $$\widetilde{H}_*(X; \mathbb{Z}/p) \neq 0$$. Then $$\pi_n X$$ contains $$p$$-torsion for infinitely many values of $$n$$.

• Taking $Q_n$ be the set of primes that divide the order of $\pi_nS^3$, do we also know whether the cardinality of $Q_n$ goes to infinity? – Lennart Meier Nov 25 '19 at 7:26
• I guess it is assumed that $k\ge 2$. Is there any other implicit assumption on $k$? – YCor Nov 25 '19 at 10:12
• For sure $k\neq 1$, but all other $k$ are ok. – Jeff Strom Nov 25 '19 at 15:32