2 results
Bounding size of homotopy groups of Spheres
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 63 / Issue 4 / November 2020
- Published online by Cambridge University Press:
- 05 November 2020, pp. 1100-1105
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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.
A Relation Between S1 and S3-Invariant Homotopy In The Stable Range
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- Journal:
- Canadian Mathematical Bulletin / Volume 35 / Issue 1 / 01 March 1992
- Published online by Cambridge University Press:
- 20 November 2018, pp. 75-80
- Print publication:
- 01 March 1992
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For any X and any q > 0, one has natural inclusions
where the groups S1 and S3 act on S4q-1 in the standard way and 
are the G-invariant homotopy subsets, G = S1 or G = S3. It is proved here that for any space X of the homotopy type of a CW-complex and for π4q-1 (X) in the c3 cl stable range, the inclusion 
is m fact an equality when localized away from the prime 2.
where the groups S
are the 
is m fact an equality when localized away from the prime 2.