NON-TRIVIALITY OF TWO HOMOTOPY ELEMENTS IN π*S

Title & Authors
NON-TRIVIALITY OF TWO HOMOTOPY ELEMENTS IN π*S
Liu Xiugui;

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 $\small{{\alpha}_1{\jmath}{\xi}_n\;(p{\geq}5,n\;{\geq}\;3)\;and\;{\gamma}_s{\alpha}_1{\jmath}{\xi}_n\;(p\;{\geq}\;7,\;n\;{\geq}\;4)}$ are detected in the stable homotopy groups of spheres, where $\small{{\xi}_n\;{\in}\;{\pi}_{p^nq+pq-2}M}$ is obtained in [2]. The new ones are of degree 2(p - 1)($\small{p^n+p+1}$) - 4 and 2(p - 1)($\small{p^n+sp^2}$ + sp + (s - 1)) - 7 and are represented up to nonzero scalar by $\small{b_0h_0h_n,\;b_0h_0h_n\tilde{\gamma}_s\;{\neq}\;0\;{\in}\;Ext^{*,*}_A^(Z_p,\;Z_p)}$ in the Adams spectral sequence respectively, where $3\;{\leq}\;s\;<\;p-2$.
Keywords
stable homotopy groups of spheres;Adams spectral sequence;Toda-Smith spectrum;May spectral sequence;
Language
English
Cited by
1.
A Four-filtrated may spectral sequence and its applications, Acta Mathematica Sinica, English Series, 2008, 24, 9, 1507
2.
Detection of some elements in the stable homotopy groups of spheres, Chinese Annals of Mathematics, Series B, 2008, 29, 3, 291
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