3-primary \(\beta\)-family in stable homotopy of a finite spectrum (Q674605)

For \(n=2\) and \(p>3\), L. Smith showed the existence of the \(\nu_{2}\) map \(\beta : \Sigma^{2p^{2}-2}V(1) \rightarrow V(1)\) with \(l=1\). In the case of the prime \(3\) Toda shows that there is no such \(\nu_{2}\)-map on \(V(1)\) with \(l=1\). This paper shows that there is a \(\nu_{2}\)-map \(\overline{\beta} : \Sigma^{2p^{2}-2}VX \rightarrow VX\) with \(l=1\), where \(VX\) is the smash product of \(V(1)\) with the \(8\)-skeleton of the Brown-Peterson spectrum \(BP\).