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| A Relation in the Stable Homotopy Groups of Spheres |
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Citation： |
Jianxia BAI,Jianguo HONG.A Relation in the Stable Homotopy Groups of Spheres[J].Chinese Annals of Mathematics B,2015,36(3):413~426 |
| Page view： 1031
Net amount： 871 |
Authors： |
Jianxia BAI; Jianguo HONG; |
Foundation： |
supported by the National Natural Science Foundation of China (Nos. 11071125,
11261062, 11471167) and the Specialized Research Fund for the Doctoral Program of Higher Education
(No. 20120031110025). |
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| Abstract： |
Let $p\geqslant 7$ be an odd prime. Based on the Toda bracket
$\langle\alpha_1\beta_1^{p-1}, \alpha_1\beta_1, p, \gamma_s \rangle
$, the authors show that the relation
$\alpha_1\beta_1^{p-1}h_{2,0}\gamma_s$$=\beta_{p/p-1}\gamma_s $
holds. As a result, they can obtain
$\alpha_1\beta_1^{p}h_{2,0}\gamma_s=0 \in \pi_*(S^0) $ for $2
\leqslant s \leqslant p-2$, even though $\alpha_1h_{2,0}\gamma_s $
and $\beta_1\alpha_1h_{2,0}\gamma_s$ are not trivial. They also
prove that $\beta_1^{p-1}\alpha_1h_{2,0}\gamma_3$ is nontrivial in
$\pi_*(S^0) $ and conjecture that
$\beta_1^{p-1}\alpha_1h_{2,0}\gamma_s$ is nontrivial in $\pi_*(S^0)
$ for $3 \leqslant s \leqslant p-2$. Moreover, it is known that
$\beta_{p/p-1}\gamma_3=0 \in {\rm Ext}^{5,*}_{BP_*BP}(BP_*, BP_*)$,
but $\beta_{p/p-1}\gamma_3$ is nontrivial in $\pi_*(S^0)$ and
represents the element $\beta_1^{p-1}\alpha_1h_{2,0} \gamma_3$. |
Keywords： |
Toda bracket, Stable homotopy groups of spheres, Adams-Novikov
spectral sequence, Method of infinite descent |
Classification： |
55Q10, 55Q45, 55T99 |
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