Bounding size of homotopy groups of spheres

Abstract

Let $p$ be prime. We prove that, for $n$ odd, the $p$-torsion part of ${\pi }_q(S^n)$ has cardinality at most $p^{2^{\frac{1}{p-1}(q-n+3-2p)}}$, and hence has rank at most $2^{\frac{1}{p-1}(q-n+3-2p)}$. For $p=2$ these results also hold for $n$ even. The best bounds proven in the existing literature are $p^{2^{q-n+1}}$ and $2^{q-n+1}$ respectively, both due to Hans-Werner Henn. The main point of our result is therefore that the bound grows more slowly for larger primes. As a corollary of work of Henn, we obtain a similar result for the homotopy groups of a broader class of spaces.

5 pages. Two corrections in v2: First, Lemma 3.1 in v1 was wrong, and has been replaced with a different lemma which makes the same point. The error was the sentence beginning 'We will not do so here, but one can show....'. Second, since submitting v1 I discovered the paper of Iriye, which gives a result similar to Theorem 1.1 without proof. The introduction has been updated to acknowledge this