[Paper Review] André-Quillen cohomology and rational homotopy of function spaces

Jonathan Block, Andrey Lazarev|ArXiv.org|Jun 28, 2003

Homotopy and Cohomology in Algebraic Topology11 references4 citations

TL;DR

This paper establishes a direct link between André-Quillen cohomology of Sullivan-de Rham models and the homotopy groups of function spaces and self-equivalence spaces in rational homotopy theory. It shows that the homotopy groups of mapping spaces $ F(X,Y) $, and the group of homotopy self-equivalences $ hAut(X_{\mathbb{Q}}) $, are computed by André-Quillen cohomology of the rational models of $ X $ and $ Y $, with the Lie algebra of $ hAut(X_{\mathbb{Q}}) $ identified as $ H^{0}_{AQ}(A^{*}(X), A^{*}(X)) $, providing a conceptual framework for Sullivan's arithmeticity results.

ABSTRACT

We develop a simple theory of André-Quillen cohomology for commutative differential graded algebras over a field of characteristic zero. We then relate it to the homotopy groups of function spaces and spaces of homotopy self-equivalences of rational nilpotent $CW$-complexes. This puts certain results of Sullivan in a more conceptual framework.

Motivation & Objective

• To develop a direct, computable theory of André-Quillen cohomology for commutative differential graded algebras (dga’s) over a field of characteristic zero.
• To relate André-Quillen cohomology to the homotopy groups of function spaces $ F(X,Y) $ between rational nilpotent $ CW $-complexes.
• To reprove and conceptualize Sullivan’s result that $ hAut(X_{\mathbb{Q}}) $ is an arithmetic group by identifying its Lie algebra as $ H^{0}_{AQ}(A^{*}(X), A^{*}(X)) $.
• To extend the computation of higher homotopy groups of function spaces to the unpointed case using dga models.

Proposed method

• Define André-Quillen cohomology via an explicit cochain complex $ C^{*}_{AQ}(A,M) $ for a dga $ A $ and dg $ A $-module $ M $.
• Establish equivalence of this complex to derived functors and show homotopy invariance using model category structures on dga’s and dg modules.
• Introduce the Gerstenhaber bracket on $ C^{*}_{AQ}(A,A) $, showing it induces a Lie algebra structure on cohomology.
• Use the Sullivan-de Rham functor to translate topological mapping spaces into dga morphism spaces in the category of dga’s over $ A^{*}(Y) $.
• Apply the homotopy category anti-equivalence between rational finite-type spaces and dga’s to relate $ \pi_n(F(X,Y)) $ to $ H^{-n}_{AQ}(A^{*}(Y), A^{*}(X)) $.
• Utilize the structure of $ A^{*}(X \times S^n) \simeq A^{*}(X) \ltimes A^{*}(X)[n] $ to model the homotopy fibre sequence and compute homotopy groups.

Experimental results

Research questions

• RQ1 How can André-Quillen cohomology be systematically computed for commutative dga’s over a field of characteristic zero?
• RQ2 What is the precise relationship between the homotopy groups of function spaces $ F(X,Y) $ and André-Quillen cohomology of their rational models?
• RQ3 How is the Lie algebra of the group $ hAut(X_{\mathbb{Q}}) $ of rational self-equivalences realized in cohomological terms?
• RQ4 Can the higher homotopy groups of unpointed function spaces be computed via André-Quillen cohomology?

Key findings

• The $ n $-th homotopy group of the function space $ F(X,Y) $ at a basepoint $ f $ is isomorphic to $ H^{-n}_{AQ}(A^{*}(Y), A^{*}(X)) $, with an abelian group isomorphism for $ n \geq 2 $.
• The Lie algebra of $ hAut(X_{\mathbb{Q}}) $ is isomorphic to $ H^{0}_{AQ}(A^{*}(X), A^{*}(X)) $, with the Gerstenhaber bracket inducing the Lie bracket.
• When $ f: X \to Y $ is null-homotopic, $ H^{*}_{AQ}(A^{*}(Y), A^{*}(X)) \cong H^{*}_{AQ}(A^{*}(Y), \mathbb{Q}) \hat{\otimes} H^{*}(X) $, recovering a known result in rational homotopy theory.
• For simply-connected $ X $ and $ Y $, $ \pi_n(F(X,Y)) \cong H^{1-n}(L(X), L(Y)) $, where $ L(X) $ and $ L(Y) $ are Quillen models, suggesting a generalization beyond finiteness assumptions.
• The cohomology $ H^{*}_{AQ}(A^{*}(Y), A^{*}(X)) $ computes the homotopy groups of $ F(X,Y) $ via the anti-equivalence between rational spaces and dga’s over $ A^{*}(Y) $.