# Massey Products vs. $A_\infty$-Structures

Does anyone know a good reference for a proof of the fact that given a dga $A$, an $A_\infty$-structure on $HA$ is ''the same'' as coherent choices for all of the higher Massey products of $HA$? More concretely the fact I am looking for is something like the following.

When defining the Massey product $\langle x_1,\dots, x_n\rangle$ there are multiple non-canonical choices that need to be made, which in turn give multiple cycles that could be called the Massey product of $x_1,\dots, x_n$. If $M(x_1,\dots, x_n)$ is the set of all possible resulting Massey products of $x_1,\dots, x_n$, and $\mu_n$ is the $n$-th $A_\infty$ structure map (on $HA$), then $$\mu_n(x_1\otimes\cdots\otimes x_n)\in M(x_1,\dots, x_n)$$ for all $n$ and $x_i$.

When $n=3$, this is in Stasheff's H-spaces from a homotopy point of view, Chapter 12. For general $n$, it is in a paper of mine with Lu, Wu, and Zhang, "$A_\infty$-structures in Ext algebras, J. Pure Appl. Alg. 213 (2009), 2017--2037 (Theorem 3.1 and Corollary A.5).
• @John: this is more or less the kind of result I was looking for. Do you know if we can say anything about $\mu_n(x_1,\dots,x_n)$ if the product $\langle x_1,\dots, x_n\rangle$ is not defined? For example, does this force $\mu_n(x_1,\dots,x_n)=0$? – Steve Mar 27 '12 at 15:44
• Maybe worth pointing out, and hopefully somebody can clear up the issue: In Section 3 of Buijs, Moreno-Fernandez, Murillo "$A_{\infty}$ structures and Massey products" (arxiv.org/abs/1801.03408) the authors state there is a "small gap in the proof" of Theorem 3.1 in the above paper for $n\geq 4$; they then give a modified statement and proof. – Aleksandar Milivojevic Apr 3 '19 at 23:50