Operations in the homology spectral sequence of a cosimplicial infinite loop space

TL;DR

This paper develops external and internal operations within the mod 2 homology spectral sequence of cosimplicial spaces, linking them to classical Araki-Kudo operations on the total space homology.

Contribution

It introduces a construction of external operations in the spectral sequence and demonstrates their compatibility with classical operations for cosimplicial E-infinity spaces.

Findings

External operations are constructed for the spectral sequence.

Internal operations are obtained for cosimplicial E-infinity spaces.

Operations agree with Araki-Kudo on the abutment.

Abstract

Consider the mod 2 homology spectral sequence associated to a cosimplicial space X. We construct external operations whose target is the spectral sequence associated to E\Sigma_2 \times_{\Sigma_2} (X\times X). If X is a cosimplicial E_\infty-space, we couple these external operations with the structure map E\Sigma_2 \times_{\Sigma_2} (X\times X) \to X to produce internal operations in the spectral sequence. In the sequel we show that they agree with the usual Araki-Kudo operations on the abutment H_*(Tot X).