# nLab Kadeishvili's theorem

### Context

#### Higher algebra

higher algebra

universal algebra

cohomology

# Contents

## 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 (1980), no. 3, 231–238.
• S. A. Merkulov, Strong homotopy algebras of a Kähler manifold, Internat. Math. Res. Notices 1999, no. 3, 153–164.

Last revised on December 13, 2012 at 18:07:27. See the history of this page for a list of all contributions to it.