Journal of the
Korean Mathematical Society JKMS

Let $A$ be the mod $p$ Steenrod algebra for $p$ an arbitrary odd prime and $S$ the sphere spectrum localized at $p$. In this paper, some useful propositions about the May spectral sequence are first given, and then, two new nontrivial homotopy elements $\alpha_1 j\xi_n$ ($p \geq 5$,$n \geq 3$) and $\gamma_s\alpha_1 j\xi_n$ ($p \geq 7$, $n \geq 4$) are detected in the stable homotopy groups of spheres, where $\xi_n \in \pi_{p^nq+pq-2}M$ is obtained in [2]. The new ones are of degree $2(p-1)(p^n+p+1)-4$ and $2(p-1)(p^n+sp^2+sp+(s-1))-7$ and are represented up to nonzero scalar by $b_0h_0h_n$, $b_0 h_0 h_n \tilde{\gamma}_{s}\not=0\in \operatorname{Ext}_A^{\ast,\ast}(Z_p,Z_p)$ in the Adams spectral sequence respectively, where $3 \leq s < p-2$.

Keywords: stable homotopy groups of spheres, Adams spectral sequence, Toda-Smith spectrum, May spectral sequence

MSC numbers: 55Q45