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 $x_i$.


2 Answers 2


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).

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    $\begingroup$ @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 $\mu_n(x_1,\dots,x_n)=0$? $\endgroup$
    – Steve
    Mar 27, 2012 at 15:44
  • $\begingroup$ I don't think I know of a result like that. You could look at our proof and see if you can get anything out of it. $\endgroup$ Mar 27, 2012 at 20:48
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    $\begingroup$ 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. $\endgroup$ Apr 3, 2019 at 23:50

This is in Loday/Vallette's new book on operads, in particular sections 9.4.10 to 9.4.12.

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    $\begingroup$ Alternatively see arxiv.org/abs/1202.3245 from item 1.4 on... $\endgroup$ Mar 26, 2012 at 23:28
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    $\begingroup$ The Loday & Vallette results are strange, since they don't seem to deal with indeterminacies in the Massey products. But maybe I'm just reading it wrong. $\endgroup$ Mar 27, 2012 at 1:31
  • $\begingroup$ I agree it's confusing how they are sweeping this under the rug. In their definition of Massey product on top of p.282, <x,y,z> depends on choices of a,b which don't appear in the notation. Then lemma 9.4.11 talks about a particular choice of a,b which is made in the body of the proof. The actual statement of the lemma only can be deduced after reading the proof. It then says that for these particular choices of a,b the said equality holds. Steve's statement from the question above (with n=3) follows. $\endgroup$ Mar 28, 2012 at 0:12

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