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Bounding size of homotopy groups of Spheres

Part of: Homotopy groups

Published online by Cambridge University Press:  05 November 2020

Guy Boyde
Guy Boyde*
Affiliation:
Mathematical Sciences, University of Southampton, SouthamptonSO17 1BJ, UK (gb7g14@soton.ac.uk)
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Abstract

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.

Keywords

homotopy groups of spheresEHP sequence

MSC classification

Primary: 55Q40: Homotopy groups of spheres
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 4 , November 2020 , pp. 1100 - 1105
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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