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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
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 (S n ) 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:
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。
更新日期:2020-11-05
中文翻译:
Spheres同伦群的边界大小
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