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 quasiisomoprhic CDGA, and $\hat{A}$ and $\hat{B}$ be their PD models respectively, are $\hat{A}$ and $\hat{B}$ quasiisomorphic as PD algebras?
I guess it depends what you mean by "quasiisomorphic 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 quasiisomorphism 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 2connected and homologically 3connected (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 quasiisomorphism 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 $y \in A^{nk}$ 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.
Another plausible notion of "quasiisomorphic as Poincaré duality algebras" would be a zigzag $\hat{A} \xleftarrow{f} R \xrightarrow{g} \hat{B}$ of quasiisomorphisms 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.)

$\begingroup$ Yes, by quasiisomorphic PD algebras, I strictly meant to say zigzag of quasiisomorphisms with intermediary PDalgebras that commutes with fixed orientation. Thanks. $\endgroup$ – Arun Oct 18 '18 at 14:12

$\begingroup$ @Arun I see, then in this case I think it's open in general. I talked with Pascal Lambrechts about it and he seemed to think so. (I also talked with Don Stanley and I think we mentioned this, I would remember if he had told me the problem had been resolved, in any case.) $\endgroup$ – Najib Idrissi Oct 18 '18 at 14:19
By the work of Hamilton and Lazarev, cyclic $C_{\infty}$algebras are uniquely determined up to cyclic $C_{\infty}$quasiisomorphism 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 $C_{\infty}$quasiisomorphic and the induced isomorphism in cohomology is one of PD algebras. Then there exists a cyclic $C_{\infty}$quasiisomorphism 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 $C_{\infty}$quasiisomorphism 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 $C_{\infty}$quasiisomorphism 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?