A Toda bracket in the stable homotopy groups of spheres

Liu, Xiugui

doi:10.2140/agt.2009.9.221

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A Toda bracket in the stable homotopy groups of spheres

Xiugui Liu

Algebraic & Geometric Topology 9 (2009) 221–236

DOI: 10.2140/agt.2009.9.221

Abstract

Let $p$ be a prime number greater than five. In the $p$ –local stable homotopy groups of spheres, H Toda and J Lin, respectively, constructed the elements

\begin{matrix} γ_{s} \in π_{2 s p^{3} - 2 p^{2} - 2 p - 2 s + 1} (S), \\ ω_{m, n} \in π_{2 p^{n + 1} - 2 p^{n} + 2 p^{m + 1} - 2 p^{m} + 2 p - 6} (S) \end{matrix}

of order $p$ . In this paper, we show the nontriviality of the Toda bracket $〈 γ_{s}, p, ω_{m, n} 〉$ in the stable homotopy groups of spheres, where $n \geq m + 2 > 6$ , $3 \leq s < p$ .

Keywords

stable homotopy groups of sphere, Toda bracket, Adams spectral sequence, May spectral sequence

Mathematical Subject Classification 2000

Primary: 55Q45, 55T15

Secondary: 55S10

References

Publication

Received: 24 May 2007

Revised: 10 December 2008

Accepted: 13 December 2008

Published: 3 February 2009

Authors

Xiugui Liu
School of Mathematical Sciences and LPMC Nankai University Tianjin 300071 PR China

Volume 9, issue 1 (2009)