[Paper Review] Koszul A-infinity algebras and free loop space homology

Alexander Berglund, Kaj Börjeson|arXiv (Cornell University)|Nov 17, 2017

Homotopy and Cohomology in Algebraic Topology8 references3 citations

TL;DR

This paper introduces Koszul $A_{ u}$-algebras as a generalization of Priddy's Koszul algebras, enabling small $A_{ u}$-algebra models for Hochschild cochains. It establishes that free loop space homology of simply connected manifolds—whether formal or coformal—can be computed via twisted convolution $A_{ u}$-algebras on Koszul dual structures, yielding new computations for $\mathbb{C}P^n$ and a 7-manifold over $\mathbb{Z}$, generalizing prior results.

ABSTRACT

We introduce a notion of Koszul A-infinity algebra that generalizes Priddy's notion of a Koszul algebra and we use it to construct small A-infinity algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples.

Motivation & Objective

• To extend the theory of Koszul algebras to $A_{\infty}$-algebras for applications in algebraic topology.
• To provide a framework for computing free loop space homology $H_*(LM)$ of manifolds that are formal or coformal, not necessarily both.
• To construct small $A_{\infty}$-algebra models for Hochschild cochains using Koszul duality.
• To generalize previous results on $H_*(LM)$ by relaxing the joint formal/coformal condition.

Proposed method

• Introduce a new notion of Koszul $A_{\infty}$-algebra via weight-graded $A_{\infty}$-algebras with specific homogeneity conditions on structure maps.
• Establish a characterization: an $A_{\infty}$-algebra is quasi-isomorphic to a Koszul $A_{\infty}$-algebra iff its bar construction is formal as a dg coalgebra.
• Construct small $A_{\infty}$-models for Hochschild cochains on Koszul $A_{\infty}$-algebras using twisted convolution algebras.
• Use a twisting morphism $\kappa: H_*(M) \to H_*(\Omega M)$ to define a twisted convolution $A_{\infty}$-algebra quasi-isomorphic to Hochschild cochains on $C_*(\Omega M)$.
• Apply the framework to compute $H_{*+d}(LM)$ via the isomorphism $H_{*+d}(LM) \cong H_*\operatorname{Hom}^\kappa(H_*(M), H_*(\Omega M))$.
• Verify the construction on two examples: $\mathbb{C}P^n$ (formal, non-coformal) and a 7-manifold (coformal, non-formal) over $\mathbb{Z}$.

Experimental results

Research questions

• RQ1 Can the theory of Koszul algebras be extended to $A_{\infty}$-algebras to handle more general manifolds in free loop space homology computations?
• RQ2 How can Hochschild cochains on $C_*(\Omega M)$ be modeled efficiently when $M$ is formal or coformal but not both?
• RQ3 What is the role of the twisting morphism $\kappa: H_*(M) \to H_*(\Omega M)$ in relating $H_*(LM)$ to $A_{\infty}$-structures on Koszul duals?
• RQ4 Can the Chas-Sullivan loop product on $H_*(LM)$ be recovered from a twisted convolution $A_{\infty}$-algebra structure?
• RQ5 What new computations of $H_*(LM)$ become accessible using this framework beyond the formal/coformal intersection?

Key findings

• The paper establishes that $M$ is formal over $\Bbbk$ if and only if $H_*(\Omega M;\Bbbk)$ admits a minimal Koszul $A_{\infty}$-algebra structure quasi-isomorphic to $C_*(\Omega M;\Bbbk)$, with $H_*(M;\Bbbk) \cong H_*(\Omega M;\Bbbk)^{<}$.
• The paper shows that $M$ is coformal over $\Bbbk$ if and only if $H_*(M;\Bbbk)$ admits a minimal Koszul $A_{\infty}$-coalgebra structure quasi-isomorphic to $C_*(M;\Bbbk)$, with $H_*(\Omega M;\Bbbk) \cong H_*(M;\Bbbk)^!$.
• For a $d$-dimensional manifold $M$ that is formal or coformal over $\Bbbk$, there is an isomorphism of graded algebras $H_{*+d}(LM;\Bbbk) \cong H_*\operatorname{Hom}^\kappa(H_*(M;\Bbbk), H_*(\Omega M;\Bbbk))$, where the left side carries the Chas-Sullivan loop product.
• The framework provides a streamlined computation of the Chas-Sullivan algebra of $\mathbb{C}P^n$, viewed as a chain-level refinement of the Cohen-Jones-Yan spectral sequence.
• The paper computes $H_{*+7}(LM;\mathbb{Z})$ for a 7-manifold $M$ that is coformal but not formal over $\mathbb{Z}$, a result previously inaccessible with earlier methods.
• The Hochschild cohomology $HH^*(U,U)$ is presented as a graded commutative algebra over $\mathcal{Z}(U)$ with explicit generators and relations, and there is an isomorphism $H_{*+7}(LM) \cong HH^*(U,U)$.