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Bounding size of homotopy groups of Spheres
Proceedings of the Edinburgh Mathematical Society ( IF 0.7 ) Pub Date : 2020-11-05 , DOI: 10.1017/s001309152000036x
Guy Boyde

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)(qn+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 2qn+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:

Spheres同伦群的边界大小

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。
更新日期:2020-11-05
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