A note on products in stable homotopy groups of spheres via the classical Adams spectral sequence

In recent years, Liu and his collaborators found many non-trivial products of generators in the homotopy groups of the sphere spectrum. In this paper, we show a result which not only implies most of their results, but also extends a result of theirs

Beta families arising from a v29v_2^9 self map on S/(3,v18)S/(3,v_1^8)

We show that $v_2^9$ is a permanent cycle in the 3-primary Adams-Novikov spectral sequence computing $\pi_*(S/(3,v_1^8))$, and use this to conclude that the families $\beta_{9t+3/i}$ for $i=1,2$, $\beta_{9t+6/i}$ for $i=1,2,3$, $\beta_{9t+9/i}$ for $i=1,\dots,8$, $\alpha_1\beta_{9t+3/3}$, and $\alpha_1\beta_{9t+7}$ are permanent cycles in the 3-primary Adams-Novikov spectral sequence for the sphere for all $t\geq 0$. We use a computer program by Wang to determine the additive and partial multiplicative structure of the Adams-Novikov $E_2$ page for the sphere in relevant degrees. The $i=1$ cases recover previously known results of Behrens-Pemmaraju and the second author. The results about $\beta_{9t+3/3}$, $\beta_{9t+6/3}$ and $\beta_{9t+8/9}$ were previously claimed by the second author; the computer calculations allow us to give a more direct proof. As an application, we determine the image of the Hurewicz map $\pi_*S \to \pi_*tmf$ at $p=3$.Comment: Version to appear in A&G

On the homotopy groups of E(n)-local spectra with unusual invariant ideals

Let E(n) and T(m) for nonnegative integers n and m denote the Johnson-Wilson and the Ravenel spectra, respectively. Given a spectrum whose E(n)_*-homology is E(n)_*(T(m))/(v_1,...,v_{n-1}), then each homotopy group of it estimates the order of each homotopy group of L_nT(m). We here study the E(n)-based Adams E_2-term of it and present that the determination of the E_2-term is unexpectedly complex for odd prime case. At the prime two, we determine the E_{infty}-term for pi_*(L_2T(1)/(v_1)), whose computation is easier than that of pi_*(L_2T(1)) as we expect.Comment: This is the version published by Geometry & Topology Monographs on 18 April 200