Title of Paper:
On the action of $\beta_1$ in the stable homotopy of spheres at the prime $3$
Author:
Katsumi Shimomura
AMS Classification numbers:
55Q45, 55Q52
Address of Author:
Department of Mathematics,
Faculty of Science,
Kochi University,
Kochi, 780-8520,
Japan
Email address of Author:
katsumi@math.kochi-u.ac.jp
Text of Abstruct:
The element $\beta_1$ is the generator of the stable homotopy group
$\pi_{10}(S^0)$. Here $S^0$ denotes the $3$-localized sphere spectrum.
Toda showed that $\beta_1^5\neq 0$ and $\beta_1^6=0$.
Here we generalize it to $\beta_1^4\beta_{9t+1}\neq 0$ and $\beta_1^5\beta_{9t+1}= 0$ for $\beta_{9t+1}\in\pi_{144t+10}(S^0)$ with $t\ge 0$.
In particular, $\beta_1^4\beta_{10}\neq 0$ and $\beta_1^5\beta_{10}= 0$ for
$\beta_{10}$ shown to exist by Oka.
This is proved by determining subgroups of $\pi_*(L_2S^0)$, where $L_2$
denotes the Bousfield localization functor with respect to $v_2^{-1}BP$.