On the $$\varpi_n$$ -related elements in the stable homotopy groups of spheres

Abstract

In this paper, we prove the existence of two nontrivial families of homotopy elements \(\beta_{1} \varpi_{n}\) and \(\gamma_{s}+3\varpi_n\) in the stable homotopy groups of spheres \(\pi_{*}\) (S), where \(\varpi_n \in \pi_{q (p^{n}+2p+1)-3} (S)\) was constructed by X. G. Liu, p is a prime number greater than five, \(n \geq 4, 0 \leq s < p–4, q = 2 (p-1)\). The elementary method of proof is by explicit combinatorial analysis of the May spectral sequence.