Steenrod operations on bar complex

TL;DR

This paper constructs a chain map involving the bar complex and an $E_$-operad, showing Steenrod operations correspond to those on the cohomology of based loop spaces.

Contribution

It introduces a new chain map linking the bar complex with an $E_$-operad, elucidating Steenrod operations in this context.

Findings

Steenrod operations match those on based loop space cohomology

Chain map explicitly constructed using the sequence operad

Provides a new perspective on operations in algebraic topology

Abstract

We define a chain map of the form $\E(k)\otimes BA^{\otimes k}\longrightarrow BA$, where $\E$ is a combinatorial $E_\infty$-operad called the sequence operad, and $BA$ is the bar complex of an $\E$-algebra $A$. We see that Steenrod-type operations derived from the chain map are equal to the corresponding operations on the cohomology of the based loop space under an isomorphism.