Quillen Spectral Sequences in Homology and Rational Homotopy of Cofibrations

We construct Quillen type spectral sequences in homology and rational homotopy for coobration sequences which are Eckmann-Hilton dual to analogous ones for bration sequences. These spectral sequences are constructed by direct ltrations of the Adams cobar construction. We also prove various collapsing theorems generalizing results of Clark and Smith in the case of a wedge of 1-connected nicely pointed spaces. 1. Introduction The purpose of this paper is to study the relation between the rational homotopy groups of topological spaces in coobration sequences. In his pathbreaking paper on rational homotopy theory Quillen derives a Lie algebra spectral sequence relating the rational homotopy Lie algbras of spaces in a coobration. This spectral sequence is a perfect Eckmann-Hilton dual to the coalgebra spectral sequence of Serre relating the rational homology coalgebras of spaces in a bration. 13] Quillens approach is via homotopical algebra viewing rational ho-motopy theory in terms of both equivalent homotopy theories of the closed model categories of diierential graded Lie algebras and diier-ential graded coalgebras. Instead of using models we construct the Quillen spectral sequence by deening an appropriate ltration of the cobar construction functor and use a theorem of Adams, which gives an isomorphism between the homology of the loops of a space and the homology of the cobar construction applied to the normalized chain complex on the space, explicitely we have an isomorphism of graded connected k-algebras 1]