Homology operations and cosimplicial iterated loop spaces

TL;DR

This paper investigates the algebraic structures on the homology of cosimplicial $E_{n+1}$ spaces, demonstrating that the spectral sequence converging to their homology admits compatible Dyer-Lashof and Browder operations.

Contribution

It proves that the spectral sequence for the homology of Tot(X) admits compatible algebraic operations, extending understanding of homology operations in cosimplicial $E_{n+1}$ spaces.

Findings

Spectral sequence admits compatible Dyer-Lashof and Browder operations.

Homology of Tot(X) has structured algebraic operations.

Positive answer to the compatibility of operations in the spectral sequence.

Abstract

If X is a cosimplical $E_{n+1}$ space then Tot(X) is an $E_{n+1}$ space and its mod 2 homology $H_*(Tot(X))$ has Dyer-Lashof and Browder operations. It's natural to ask if the spectral sequence converging to $H_*(Tot(X))$ admits compatible operations. In this paper I give a positive answer to this question.