Lie models of homotopy automorphism monoids and classifying fibrations

Abstract

Given X a finite nilpotent simplicial set, consider the classifying fibrations X → B aut∗ G(X) → B autG(X) and X → Z → B aut∗ π (X) where G and π denote, respectively, subgroups of the free and pointed homotopy classes of free and pointed self homotopy equivalences of X which act nilpotently on H∗(X) and π∗(X). We give algebraic models, in terms of complete differential graded Lie algebras (cdgl’s), of the rational homotopy type of these fibrations. Explicitly, if L is a cdgl model of X, there are connected sub cdgl’s DerGL and DerΠL of the Lie algebra of derivations of L such that the geometrical realizations of the sequences of cdgl morphisms L ad → DerGL → DerGL ̃×sL and L → L ̃×DerΠL → DerΠL have the rational homotopy type of the above classifying fibrations. Among the consequences we also describe in cdgl *We give algebraic models, in terms of complete differential graded Lie algebras (cdgl's), of the rational homotopy type of these fibrations. Explicitly, if L is a cdgl model of X, there are connected sub cdgl's and of the Lie algebra of derivations of L such that the geometrical realizations of the sequences of cdgl morphisms have the rational homotopy type of the above classifying fibrations. Among the consequences we also describe in cdgl terms the Malcev -completion of G and π together with the rational homotopy type of the classifying spaces BG and Bπ.