Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces

Abstract

This paper establishes that the homotopy category of rational differential graded commutative coalgebras is equivalent to the homotopy category of rational differential graded Lie algebras which have a nilpotent completion as homology. This generalizes a result which Quillen proved in the simply connected case. When combined with Sullivan's work on rational homotopy theory, our result shows that the homotopy category of rational differential graded Lie algebras with nilpotent finite type homology is equivalent to the rational homotopy category of nilpotent topological spaces with finite type rational homology.