Strong minimal model theorem and Massey products

Abstract

Kadeishvili's minimal model theorem establishes the existence of an $A_\infty$-structure, unique up to isomorphism, on the cohomology of a dg associative algebra, which captures its homotopy type. In this note we prove the existence of minimal models that are unique up to isotopy, a stronger result obviously known to T. Kadeishvili and certainly to others, yet seemingly overlooked by mankind. We will explore how this stronger result can help in the study of Massey products. First, we show that the attempts to extract a local information from the ternary operation $\mu_3$ of our minimal model leads directly to the rediscovery of the triple Massey product. The motto is: "The triple Massey product is an invariant manifestation of $\mu_3$." We then prove that, under reasonable assumptions, the higher Massey product $\langle x_1,\ldots,x_n\rangle$ equals the set of all values $\mu_n(x_1,\ldots,x_n)$, where $\mu_n$ runs over the $n$-ary products of our minimal models. We believe that this note will help to elucidate the still somewhat enigmatic relationship between minimal models and Massey products.