nLab Kadeishvili's theorem

Idea

Kadeishvili’s theorem says that every A-∞-algebra (in chain complexes) is equivalent, as an $A_\infty$-algebra, to its cohomology ring.

Statement

Theorem

Let $A$ be an A-∞-algebra in chain complexes and let $H^\bullet(A)$ be the cohomology ring of $A$. There is an $A_\infty$-algebra structure on $H^\bullet(A)$ with $m_1 = 0$ and $m_2$ induced by the multiplication on $A$, constructed from the $A_\infty$-structure of $A$, such that there is a quasi-isomorphism of $A_\infty$-algebras $H^\bullet(A) \to A$ lifting the identity of $H^\bullet(A)$. This $A_\infty$-algebra structure on $H^\bullet(A)$ is unique up to quasi-isomorphism.

This is due to (Kadeishvili). A clear English exposition with applications to Kähler manifolds is in (Merkulov).

Proposition

The $A_\infty$-brackets on $H^\bullet(A)$ by theorem are up to a sign equal to the Massey products on cohomology, whenever the latter are defined.

References for this statement are listed at Massey product – Relation to A-infinity algebra.

References

• T. V. Kadeishvili, On the theory of homology of fiber spaces, (Russian) International Topology Conference (Moscow State Univ., Moscow, 1979). Uspekhi Mat. Nauk 35 (1980), no. 3 (213), 183–188. Translated in Russ. Math. Surv. 35 3 (1980) 231–238 [doi:10.1070/RM1980v035n03ABEH001842, arXiv:math/0504437]

• S. A. Merkulov, Strong homotopy algebras of a Kähler manifold, Internat. Math. Res. Notices 1999 3 (1999) 153-164 [doi:10.1155/S1073792899000070, arXiv:math/9809172]