The Adams spectral sequence for the image-of-$J$ spectrum

TL;DR

This paper analyzes the Adams spectral sequence for the connective image-of-$J$ spectrum, relating differentials to Yoneda composition, and completes a calculation initiated in 1975.

Contribution

It introduces a method to relate Adams spectral sequence differentials to Yoneda composition and completes the calculation for the connective image-of-$J$ spectrum.

Findings

Relation between $d_2$-differential and Yoneda composition established

Complete analysis of the Adams spectral sequence for the connective image-of-$J$ spectrum achieved

Builds on and finalizes a calculation started by D. Davis in 1975

Abstract

We show that if we factor the long exact sequence in cohomology of a cofiber sequence of spectra into short exact sequences, then the $d_2$-differential in the Adams spectral sequence of any one term is related in a precise way to Yoneda composition with the 2-extension given by the complementary terms in the long exact sequence. We use this to give a complete analysis of the Adams spectral sequence for the connective image-of-$J$ spectrum, finishing a calculation that was begun by D. Davis in 1975.