A synthetic approach to detecting $v_1$-periodic families

TL;DR

The paper introduces a new synthetic framework and $t$-structure to prove the surjectivity of the unit map from the sphere spectrum to the connective image-of-$J$ spectrum, simplifying detection proofs in stable homotopy theory.

Contribution

It presents a novel synthetic approach and $t$-structure that enable detection results without explicit homology or Ext group calculations.

Findings

Proves surjectivity of the unit map from sphere to j spectrum.

Develops synthetic lifts for $ extbf{F}_p$ and BP spectra.

Creates modified Adams and Adams–Novikov spectral sequences.

Abstract

We provide a simple proof that the unit map from the sphere spectrum to the connective image-of-$J$ spectrum $\mathrm{j}$ is surjective on homotopy groups. This is achieved using a novel $t$-structure on the category of $E$-synthetic spectra and a specific construction of $\mathbf{F}_p$- and BP-synthetic lifts of $\mathrm{j}$. These synthetic lifts then easily produce modified Adams and Adams--Novikov spectral sequences for $\mathrm{j}$ which we use the prove the above detection statement, all without ever calculating $\mathbf{F}_p$- or BP-homology nor the associated Ext groups.