BOUNDING SIZE OF HOMOTOPY GROUPS OF SPHERES

Let p be prime. We prove that, for n odd, the p-torsion part of pi(q)(S-n) has cardinality at most p(21/(p-1)(q-n+3-2p)) and hence has rank at most 2(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(2q-n+1) and 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.