Unstable operations in the Bousfield-Kan spectral sequence for simplicial commutative FF₂-algebras

Abstract

In this thesis we study the Bousfield-Kan spectral sequence (BKSS) in the Quillen model category sCom of simplicial commutative FF₂ -algebras. We develop a theory of unstable operations for this BKSS and relate these operations with the known unstable operations on the homotopy of the target. We also prove a completeness theorem and a vanishing line theorem which, together, show that the BKSS for a connected object of sCom converges strongly to the homotopy of that object. We approach the computation of the BKSS by deriving a composite functor spectral sequence (CFSS) which converges to the BKSS E2 -page. In fact, we generalize the construction of this CFSS to yield an infinite sequence of CFSSs, with each converging to the E2-page of the previous. We equip each of these CFSSs with a theory of unstable spectral sequence operations, after establishing the necessary chain-level structure on the resolutions defining the CFSSs. This technique may also yield operations on Blanc and Stover's generalized Grothendieck spectral sequences in other settings. We are able to compute the Bousfield-Kan E2-page in the most fundamental case, that of a connected sphere in sCom, using the structure defined on the CFSSs. We use this computation to describe the Ei-page of a May-Koszul spectral sequence which converges to the BKSS E2-page for any connected object of sCom. We conclude by making two conjectures which would, together, allow for a full computation of the BKSS for a connected sphere in sCom.