# Lie models for nilpotent spaces

## TL;DR

This paper establishes a criterion for when a differential graded Lie algebra model of a nilpotent space is quasi-isomorphic to its completion, linking algebraic properties to topological models in rational homotopy theory.

## Contribution

It proves that the injection into the completion is a quasi-isomorphism if and only if the homology is a finite type pronilpotent Lie algebra, providing a new criterion for Lie models of nilpotent spaces.

## Key findings

- Injection into completion is a quasi-isomorphism iff homology is finite type pronilpotent
- Establishes an equivalence between graded Lie models and nilpotent spaces in rational homotopy theory
- Provides algebraic conditions for models of topological spaces

## Abstract

Let $(L,d)$ be a differential graded Lie algebra, where $L=L(V)$ is free as graded Lie algebra and $V=V_{\geq 0}$ is a finite type graded vector space. We prove that the injection of $(L,d)$ into its completion $(\widehat{L},d)$ is a quasi-isomorphism if and only if $H(L,d)$ is a finite type pronilpotent graded Lie algebra. As a consequence, we obtain an equivalence between graded Lie models for nilpotent spaces in rational homotopy theory.