Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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NON-TRIVIALITY OF TWO HOMOTOPY ELEMENTS IN π*S

  • Liu Xiugui (School of Mathematical Sciences and LPMC Nankai University)
  • Published : 2006.07.01
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Abstract

Let A be the mod p Steenrod algebra for p an arbitrary odd prime and S the sphere spectrum localized at p. In this paper, some useful propositions about the May spectral sequence are first given, and then, two new nontrivial homotopy elements ${\alpha}_1{\jmath}{\xi}_n\;(p{\geq}5,n\;{\geq}\;3)\;and\;{\gamma}_s{\alpha}_1{\jmath}{\xi}_n\;(p\;{\geq}\;7,\;n\;{\geq}\;4)$ are detected in the stable homotopy groups of spheres, where ${\xi}_n\;{\in}\;{\pi}_{p^nq+pq-2}M$ is obtained in [2]. The new ones are of degree 2(p - 1)($p^n+p+1$) - 4 and 2(p - 1)($p^n+sp^2$ + sp + (s - 1)) - 7 and are represented up to nonzero scalar by $b_0h_0h_n,\;b_0h_0h_n\tilde{\gamma}_s\;{\neq}\;0\;{\in}\;Ext^{*,*}_A^(Z_p,\;Z_p)$ in the Adams spectral sequence respectively, where $3\;{\leq}\;s\;<\;p-2$.

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References

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Cited by

  1. A Four-filtrated may spectral sequence and its applications vol.24, pp.9, 2008, https://doi.org/10.1007/s10114-008-6219-z
  2. Detection of some elements in the stable homotopy groups of spheres vol.29, pp.3, 2008, https://doi.org/10.1007/s11401-006-0519-3