Nonvanishing of products in $v_2$-periodic families at the prime $3$

TL;DR

This paper demonstrates the nonvanishing of products in $v_2$-periodic families at prime 3 by analyzing Adams operations on topological modular forms and utilizing synthetic spectra to study spectral sequences.

Contribution

It introduces a novel approach combining Adams operations and synthetic spectra to establish nonvanishing results in stable homotopy groups.

Findings

Many products in $v_2$-periodic families do not vanish.

Certain Toda brackets are shown not to contain zero.

The method leverages synthetic spectra and Adams--Novikov spectral sequences.

Abstract

Many products amongst $v_2$-periodic families in the stable homotopy groups of spheres are shown not to vanish and some Toda brackets are shown not to contain zero. This is done by carefully studying the action of Adams operations on topological modular forms. A crucial ingredient is Pstragowski's category of synthetic spectra which affords us the necessary freedom to work with (modified) Adams--Novikov spectral sequences.