Steenrod operations and A-module extensions

TL;DR

This paper presents explicit methods for constructing extensions representing cocycles in Ext groups over the Steenrod algebra, aiding in the calculation of Steenrod operations and Adams spectral sequence differentials.

Contribution

It introduces explicit extension constructions for cocycles in Ext groups, enhancing computational techniques for Steenrod operations and spectral sequence analysis.

Findings

Explicit cocycle extensions facilitate Steenrod operation calculations.

Method improves identification of cocycles in minimal resolutions.

Assists in determining differentials in the Adams spectral sequence.

Abstract

Explicit extensions representing cocycles $x \in Ext_{A}^{s,t}(F_2,F_2)$ are useful in calculating Steenrod operations $Sq^i : Ext_{A}^{s,t}(F_2,F_2) \longrightarrow Ext_{A}^{s+i,2t}(F_2,F_2)$ by a method devised by the second author. This can be used to identify explicit cocycles in the minimal resolutions produced by the first author's computer programs, and this information is useful in determining differentials in the Adams spectral sequence.