nLab Kadeishvili's theorem

Contents

Context

Higher algebra

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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

Statement

Theorem

Let AA be an A-∞-algebra in chain complexes and let H (A)H^\bullet(A) be the cohomology ring of AA. There is an A A_\infty-algebra structure on H (A)H^\bullet(A) with m 1=0m_1 = 0 and m 2m_2 induced by the multiplication on AA, constructed from the A A_\infty-structure of AA, such that there is a quasi-isomorphism of A A_\infty-algebras H (A)AH^\bullet(A) \to A lifting the identity of H (A)H^\bullet(A). This A A_\infty-algebra structure on H (A)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 A_\infty-brackets on H (A)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 (238) 231–211
  • S. A. Merkulov, Strong homotopy algebras of a Kähler manifold, Internat. Math. Res. Notices 1999, no. 3, 153–164.

Last revised on May 26, 2023 at 08:11:38. See the history of this page for a list of all contributions to it.