Lie models of homotopy automorphism monoids and classifying fibrations

TL;DR

This paper develops algebraic models using complete differential graded Lie algebras to describe the rational homotopy types of classifying fibrations associated with homotopy automorphism monoids of finite nilpotent simplicial sets.

Contribution

It introduces explicit algebraic models in terms of cdgl's for the rational homotopy types of classifying fibrations related to homotopy automorphisms, extending previous understanding.

Findings

Provides algebraic models for classifying fibrations in rational homotopy theory.

Describes the Malcev Q-completion of automorphism groups using cdgl's.

Characterizes the rational homotopy types of classifying spaces of automorphism groups.

Abstract

Given $X$ a finite nilpotent simplicial set, consider the classifying fibrations $$ X\to Baut_G^*(X)\to Baut_G(X),\qquad X\to Z\to Baut_{\pi}^*(X), $$ where $G$ and $\pi$ 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 $\pi_*(X)$. 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 $Der^G L$ and $Der^{\pi} L$ of the Lie algebra $Der L$ of derivations of $L$ such that the geometrical realization of the sequences of cdgl morphisms $$ L\stackrel{ad}{\to} Der^G L\to Der^G L\widetilde\times sL,\qquad L\to L\widetilde\times Der^{\pi} L\to Der^{\pi} L $$ have the rational homotopy type of the above classifying fibrations. Among the consequences we also describe in cdgl terms the Malcev $Q$-completion of $G$ and $\pi$ together with the rational homotopy type of the classifying spaces $BG $ and $B\pi$.