Beta families arising from a $v_2^9$ self map on $S/(3,v_1^8)$

Abstract

We show that $v_2^9$ is a permanent cycle in the 3-primary Adams-Novikov spectral sequence computing ${\pi }_{\ast }(S/(3,v_1^8))$, 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 $t\ge 0$. 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 ${\pi }_{\ast }S\to {\pi }_{\ast }tmf$ at $p=3$.

Version to appear in A&GT

References in corpus (1)

• The 2-primary Hurewicz image of tmf

Cited by in corpus (1)

• Equivariant $v_{1,\vec{0}}$-self maps