\(A_\infty\) structures and Massey products (Q2295438)

Massey products were introduced by \textit{W. S. Massey} [J. Knot Theory Ramifications 7, No. 3, 393--414 (1998; Zbl 0911.57009)], with higher order versions in [\textit{W. S. Massey}, in: Sympos. Int. Topologia Algebraica 145--154 (1958; Zbl 0123.16103)]. These were originally defined for differential graded (associative) algebras (DGAs) \((C^{\ast},d)\), taking value in their cohomology rings \(H^{\ast}(C^{\ast})\). They have a certain indeterminacy -- that is, a Massey product is actually a subset \(H^{n}(C^{\ast})\), and is only defined if all lower-order Massey products vanish. A more flexible structure is that of an {\(A_{\infty}\)-algebra}, first introduced by \textit{J. D. Stasheff} [Trans. Am. Math. Soc. 108, 275--292, 293--312 (1963; Zbl 0114.39402)] under the name of an {sha} (i.e., strongly homotopy associative) {algebra}. This is again a graded \(K\)-module \(A^{\ast}\) equipped with a sequence of linear maps \(m_{k}:A^{\otimes k}\to A\), where \(d=m_{1}\) is a differential, \(m_{2}\) a multiplication map, and the others providing (higher) assciativity (modulo \(d\)). In [Russ. Math. Surv. 35, No. 3, 231--238 (1980; Zbl 0521.55015); translation from Usp. Mat. Nauk 35, No. 3(213), 183--188 (1980)], \textit{T. V. Kadeishvili} showed that any DGA \(C^{\ast}\) has a minimal \(A_{\infty}\)-structure on its cohomology which is quasi-isomorphic (as an \(A_{\infty}\)-algebra) to \(C^{\ast}\). In this paper the authors show that, given a DGA \(C=(C^{\ast},d)\), any particular value \(a\) of an \(n\)-th order Massey product \(\langle a_{1},\dotsc,a_{n}\rangle\) in \(H=H^{\ast}(C^{\ast})\) can be {recovered} by an appropriate \(A_{\infty}\)-structure on \(H\), in the sense that \(m_{n}(a_{1},\dotsc,a_{n})=a\). They also describe how this is related to an appropriate contraction of \(C\) onto \(A\) (as \(A_{\infty}\)-algebras), and use their analysis to make more precise a statement in [\textit{D. M. Lu} et al., J. Pure Appl. Algebra 213, No. 11, 2017--2037 (2009; Zbl 1231.16008)].