The secondary periodic element ${\beta }_{p^2/p^2-1}$ and its applications

Abstract

In this paper we prove that ${\beta }_{p^2/p^2-1}$ survives to $E_{\infty }$ in the Adams-Novikov spectral sequence for $p⩾5$. As an easy consequence we prove that ${\beta }_{s{p}^2/j}$ are perminent cycles for all $s⩾1$, $j⩽p^2-1$. From the Thom map $\Phi :Ex{t}_{B{P}_{\ast }BP}^{s,t}(B{P}_{\ast },B{P}_{\ast })⟶Ex{t}_A^{s,t}(\mathbb{Z}/p,\mathbb{Z}/p)$, we also see that $h_0{h}_3$ survives to $E_{\infty }$ in the classical Adams spectral sequence.

15 pages, 2 figures