An extension in the Adams spectral sequence in dimension 54

TL;DR

This paper identifies a previously unresolved hidden extension in the Adams spectral sequence at dimension 54, demonstrating the power of synthetic spectra for complex computations in stable homotopy theory.

Contribution

It provides the first proof of a specific hidden extension at dimension 54 using synthetic spectra, completing the analysis up to dimension 80.

Findings

Resolved the last unknown hidden 2-extension in the Adams spectral sequence up to dimension 80

Showcased the effectiveness of $\ ext{F}_2$-synthetic spectra in homotopy computations

Enhanced understanding of the structure of stable homotopy groups of spheres

Abstract

We establish a hidden extension in the Adams spectral sequence converging to the stable homotopy groups of spheres at the prime 2 in the 54-stem. This extension is exceptional in that the only proof we know proceeds via Pstragowski's category of synthetic spectra. This was the final unresolved hidden 2-extension in the Adams spectral sequence through dimension 80. We hope this provides a concise demonstration of the computational leverage provided by $\mathbb{F}_2$-synthetic spectra.