An application of simplicial profinite groups

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Abstract
In this paper we use simplicial pro-p-groups to give quite different proofs of the convergence theorems for the lower central series and p-lower central series spectral sequences of a simplicial group (CURTIS I-2], RECTOR [7]). The proofs are, we believe, more conceptual than the delicate calculations with generators in free groups used in [2]. In addition in the case of the p-lower central series spectral sequence we obtain convergence for certain non-connected simplicial groups, in particular those corresponding to connected H-spaces with finitely generated homology in each dimension. The idea of the proof is as follows. If G is a free simplicial group with finitely many generators in each dimension, then because inverse limits are exact for pro finite groups the p-lower central series spectral sequence of G converges strongly to n (G), where A denotes p-completion. So the weak convergence of the spectral sequence to (G) follows from the formula (re G) ^ -~ n (G), and our main theorem gives conditions under which this holds. The main theorem is proved by a modification of a method of ARTIN and MAZUR, who prove an analogous theorem for pro-p homotopy objects in their work on etale homotopy theory (to appear). The paper is in three sections. In the first we give the statement of the main theorem and deduce the convergence theorems from it. Section 2 is devoted to generalizing to simplicial profinite groups standard properties of the cohomology of simplicial groups such as the Serre spectral sequence and the Whitehead theorems. In the third section we use these results to study the p-completion functor from simplicial groups to simplicial pro-p groups. Using ZEEMAN'S comparison theorem in the way indicated by ARTIN and MAZUR we prove a result on the compatibility of completion and fibrations (Th. 3.6) from which the main theorem follows easily. This paper resulted from putting ideas of EI) CURTIS and MIKE ARTIN and BARRY MAZUR together. I wish to acknowledge the benefit of numerous conversations.

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