Some infinite elements in the Adams spectral sequence for the sphere spectrum

Abstract

In the stable homotopy group $\pi_{p^nq+(p+1)q-1}(V(1))$ of the Smith-Toda spectrum $V(1)$, the author constructed an essential element $\varpi_n$ for $n\geq 3$ at the prime greater than three. Let $\beta_s^{\ast}\in[V(1), S]_{spq+(s-1)q-2}$ denote the dual of the generator $\beta_{s}^{\prime\prime} \in \pi_{s(p+1)q}(V(1))$, which defines the $\beta$-element $\beta_s$. In this paper, the author shows that the composite $\alpha_1 \beta_1 \xi_s \in \pi_{p^nq+(s+1)pq+sq-6}(S)$ for $1<s<p-2$ is non-trivial, where $\xi_s=\beta_{s-1}^{\ast} \varpi_n \in \pi_{p^nq+spq+(s-1)q-3}(S)$ and $q=2(p-1)$. As a corollary, $\xi_s$, $\alpha_1 \xi_s$ and $\beta_1 \xi_s$ are also non-trivial for $1<s<p-2$.