Homotopy algebras and noncommutative geometry

TL;DR

This paper explores cohomology theories of strongly homotopy algebras, establishing Hodge decompositions and invariance properties, and extends classical algebraic structures to a homotopy-invariant setting with applications to string topology.

Contribution

It generalizes Hochschild and cyclic cohomology decompositions to $C_ olinebreak_ ext{infty}$-algebras and demonstrates homotopy invariance of string topology operations.

Findings

Hodge decomposition of Hochschild and cyclic cohomology for $C_ ext{infty}$-algebras

Extension of invariant inner product $C_ ext{infty}$-algebras to symplectic $C_ ext{infty}$-algebras

Homotopy invariance of string topology operations

Abstract

We study cohomology theories of strongly homotopy algebras, namely $A_\infty, C_\infty$ and $L_\infty$-algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of $C_\infty$-algebras thus generalising previous work by Loday and Gerstenhaber-Schack. These results are then used to show that a $C_\infty$-algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_\infty$-algebra (an $\infty$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). As another application, we show that the `string topology' operations (the loop product, the loop bracket and the string bracket) are homotopy invariant and can be defined on the homology or equivariant homology of an arbitrary Poincare duality space.