The beta elements $\beta_{tp^2/r}$ in the homotopy of spheres

Abstract

In In [Ann. Math. (2) 106 (1977) 469–516], Miller, Ravenel and Wilson defined generalized beta elements in the $E_2$–term of the Adams–Novikov spectral sequence converging to the stable homotopy groups of spheres, and in [Hiroshima Math. J. 7 (1977) 427–447], Oka showed that the beta elements of the form ${\beta }_{t{p}^2∕r}$ for positive integers $t$ and $r$ survive to the homotopy of spheres at a prime $p>3$, when $r\le 2p-2$ and $r\le 2p$ if $t>1$. In this paper, for $p>5$, we expand the condition so that ${\beta }_{t{p}^2∕r}$ for $t\ge 1$ and $r\le p^2-2$ survives to the stable homotopy groups.