Homology operations and cosimplicial iterated loop spaces

Abstract

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