Massey Product and Twisted Cohomology of A-infinity Algebras

We study the twisted cohomology groups of $A_\infty$-algebras defined by twisting elements and their behavior under morphisms and homotopies using the bar construction. We define higher Massey products on the cohomology groups of general $A_\infty$-algebras and establish the naturality under morphisms and their dependency on defining systems. The above constructions are also considered for $C_\infty$-algebras. We construct a spectral sequence converging to the twisted cohomology groups an show that the higher differentials are given by the $A_\infty$-algebraic Massey products.

💡 Research Summary

The paper develops a comprehensive theory of twisted cohomology for $A_\infty$‑algebras, introduces higher Massey products on their cohomology, and connects these constructions to a spectral sequence whose differentials are precisely the higher Massey products. The authors begin by recalling the bar construction $\Bar(A)=T(sA)$ for an $A_\infty$‑algebra $(A,{m_k})$ and define a twisting element $h\in A^0$ as a degree‑zero element satisfying $m_1(h)=0$ and $m_k(h,\dots,h)=0$ for all $k\ge2$. Inserting $h$ into the bar differential yields a new differential \