Naturality of PD model of a CDGA

Question

In the paper "Poincaré duality and commutative differential graded algebras", Lambrechts and Stanley constructed PD model for cdga with simply connected cohomology. My question is: if $A$ and $B$ are two quasi-isomoprhic CDGA, and $\hat{A}$ and $\hat{B}$ be their PD models respectively, are $\hat{A}$ and $\hat{B}$ quasi-isomorphic as PD algebras?

Answer 1

I guess it depends what you mean by "quasi-isomorphic as Poincaré duality algebras". I suppose you mean: does there exist a zigzag $\hat{A} \gets \cdot \to \cdot \gets \cdot \to \hat{B}$ such that all the intermediary algebras are Poincaré duality algebras, and each map is a quasi-isomorphism that commutes with the fixed orientations?

In the paper of Lambrechts & Stanley, they show (Theorem 7.1) that this is true for Poincaré duality algebras of dimension $\ge 7$ which are 2-connected and homologically 3-connected (i.e. $A^0 = \Bbbk$, $A^1 = A^2 = 0$, $H^3(A) = 0$). They explicitly build a third Poincaré duality algebra $\hat{C}$ and the zigzag $\hat{A} \to \hat{C} \gets \hat{B}$.

To my knowledge, the question remains open in general. Note that this is a difficult question. Indeed a quasi-isomorphism of (connected for simplicity) Poincaré duality algebras $f : A \to B$ is necessarily injective, for example. If $A^n = \Bbbk\langle\omega_A\rangle$ and $B^n = \Bbbk\langle\omega_B\rangle$ are the respective top classes, then $f(\omega_A) = \omega_B$. Now if $x \in A^k$ is any nonzero element, there exists $y \in A^{n-k}$ such that $xy = \omega_A$, so $f(x) f(y) = \omega_B \neq 0$ so $f(x) \neq 0$. This severely restrict your possibilities to build the zigzag, as it's hard to take quotients etc.

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Another plausible notion of "quasi-isomorphic as Poincaré duality algebras" would be a zigzag $\hat{A} \xleftarrow{f} R \xrightarrow{g} \hat{B}$ of quasi-isomorphisms of cdgas such that $\varepsilon_A \circ f = \varepsilon_B \circ g$ (where $\varepsilon_A : \hat{A}^n \to \Bbbk$ (resp $\varepsilon_B$) is the fixed orientation). In other words you don't require a Poincaré duality algebra in the middle but you still have some compatibility.

I showed in my thesis in Proposition 2.4.34 that in characteristic zero, for simply connected Poincaré duality algebras, you can always find such a zigzag. Despite what's written there, the argument is purely algebraic and $\hat{A}$, $\hat{B}$ don't have to be models of some closed manifold. (The statement is also going to appear in the corresponding paper but I'm waiting on a referee report to update the arXiv version.)

Answer 2

By the work of Hamilton and Lazarev, cyclic $C_{\infty}$-algebras are uniquely determined up to cyclic $C_{\infty}$-quasi-isomorphism by the cyclic (PD) algebra structure on their cohomology. It is a beautiful result. More precisely:

Suppose $A$ and $B$ are simply connected cyclic $C_{\infty}$-algebras which are $C_{\infty}$-quasi-isomorphic and the induced isomorphism in cohomology is one of PD algebras. Then there exists a cyclic $C_{\infty}$-quasi-isomorphism between $A$ and $B$. The statement does not hold for $A_{\infty}$ and $L_{\infty}$-algebras.

Actually, something a bit stronger is true. It is the following version of the transfer theorem:

Suppose A is a cyclic $C_{\infty}$-algebra. Then there exists a cyclic $C_{\infty}$-algebra structure on the cohomology $H(A)$ such that:

1) the pairing in $H(A)$ is induced by the pairing in $A$

2) the $C_{\infty}$-algebra structure on $H(A)$ in particular has structure maps $m_1=0$, $m_2=$ the product induced by the product of $A$.

3) there is a cyclic $C_{\infty}$-quasi-isomorphism between $A$ and $H(A)$

4) the cyclic $C_{\infty}$-algebra structure on $H(A)$ is unique up to cyclic $C_{\infty}$-ISOMORPHISM.

PD CDGA's are particular examples of cyclic $C_{\infty}$-algebras. Note that this does not give a positive answer to the question since having a cyclyc $C_{\infty}$-quasi-isomorphism between two PD CDGA's is weaker than having a zig zag with intermediate PD CDGAs. However, this does provide certain uniqueness for the PD models which, for example, is useful for studying the invariance of certain string topology operations at the level of cohomology.

Hamilton and Lazarev wrote several papers about this. They called cyclic $C_{\infty}$-algebras "symplectic" $C_{\infty}$-algebras. A relevant paper is "Symplectic $C_{\infty}$-algebras" which appeared in Moscow Mathematical Journal in 2008.

@Najib Idrissi, I wonder if this implies your result?