Let p be prime. We prove that, for n odd, the p-torsion part of πq(Sn) has cardinality at most $p^{2^{{1}/({p-1})(q-n+3-2p)}}$ and hence has rank at most 21/(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 2q−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.

中文翻译：

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Spheres同伦群的边界大小

让p成为素数。我们证明，对于n奇怪的是p- π 的扭转部分q(小号n) 至多有基数$p^{2^{{1}/({p-1})(q-n+3-2p)}}$因此最多排名 21/(p-1)(q-n+3−2p). 为了p= 2，这些结果也适用于n甚至。现有文献中证明的最佳界限是$p^{2^{q-n+1}}$和 2q-n+1，分别是由于 Hans-Werner Henn。因此，我们的结果的要点是，对于较大的素数，界限增长得更慢。作为 Henn 工作的推论，我们对更广泛的空间类别的同伦群获得了类似的结果。