Detection of some elements in the stable homotopy groups of spheres

TL;DR

This paper constructs a new nontrivial family in the stable homotopy groups of spheres, represented by specific elements in the Adams spectral sequence, revealing novel elements and structures for odd primes.

Contribution

It introduces a new family of elements in the stable homotopy groups of spheres and detects a related family in the homotopy of V(1), advancing understanding of these groups.

Findings

Constructed a new nontrivial family in _{p^nq+2pq+q-3}S

Detected a new family of homotopy elements in _*V(1)

Identified specific representatives in the Adams spectral sequence

Abstract

In this paper we constructs a new nontrivial family in the stable homotopy groups of spheres $\pi_{p^nq+2pq+q-3}S$ which is of order $p$ and is represented by $k_0h_{n} \in Ext_A^{3,p^nq+2pq+q}(\mathbb{Z}_p,\mathbb{Z}_p)$ in the Adams spectral sequence, where $p\geq 5$ is an odd prime, $n\geq 3$ and $q=2(p-1)$. In the course of the proof, a new family of homotopy elements in $\pi_{\ast}V(1)$ which is represented by $\beta_{\ast}{i^{\prime}}_{\ast}i_{\ast}({h}_n)\in Ext_A^{2,p^nq+(p+1)q+1}(H^{\ast}V(1),\mathbb{Z}_p)$ in the Adams sequence is detected.