A product involving the $β$-family in stable homotopy theory

Details A product involving the $β$-family in stable homotopy theory A product involving the $β$-family in stable homotopy theory

2009-11-05

arxiv.org/abs/0911.1093v1

Bull. Malays. Math. Sci. Soc. (2) 33 (3) (2010), 411-420

Mathematics

Algebraic Topology

10 pages

Scientific paper

In the stable homotopy groups $\pi_{q(p^n+p^m+1)-3}(S)$ of the sphere spectrum $S$ localized at the prime $p$ greater than three, J. Lin constructed an essential family $\xi_{m,n}$ for $n \geq m + 2 >5$. In this paper, the authors show that the composite $\xi_{m,n}\beta_{s}\in \pi_{q(p^n+p^m+sp+s)-5}(S)$ for $2 \leq s < p$ is non-trivial, where $q=2(p-1)$ and $\beta_s \in \pi_{q(sp+s-1)-2}(S)$ is the known $\beta$-family. We show our result by explicit combinatorial analysis of the (modified) May spectral sequence.

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