# homotopy groups of spheres. [closed]

I not sure that my question has a research level so feel free to remove it, I'll not be offended. Let $S^{n}$ be a sphere of dimension $n>1$ and $p<q$ two prime numbers. Is there always a minimal integer $i_{p}$ such that $\pi_{i_{p}}S^{n}$ has $p$-torsion ? and is true that $i_{p}\leq i_{q}$ if $p<q$ ?

Thanks.

• The answer to this question is contained in the Wikipedia article: en.wikipedia.org/wiki/…. Jan 2, 2015 at 12:09
• @QiaochuYuan Thanks for the reference! I sill don't see a comprehensive and a natural explanation for the fact that $p$-torsion appears before $q$-torsion when $p<q$. It seems very mysterious at this level. Jan 2, 2015 at 14:33

There is an obvious map $i\colon S^n\to K(\mathbb{Z},n)$, with fibre $F$ say. The homotopy groups of $F$ are essentially the same as those of $S^n$. The mod $p$ cohomology of $K(\mathbb{Z},n)$ is polynomial tensor exterior, with a generator $u$ in degree $n$, and other generators obtained by applying Steenrod operations to $u$, which means they have dimension at least $n+2p-2$. Using this and the Serre spectral sequence we see that the mod $p$ cohomology of $F$ starts roughly in dimension $n+2p-2$, and it follows that the same is true of the $p$-torsion in the homotopy groups. A more careful argument along these lines gives the result stated in Wikipedia.