On the Hochschild and Harrison (co)homology of C _∞- algebras and applications to string topology

We study Hochschild (co)homology of commutative and associative up to homotopy algebras with coefficient in a homotopy analogue of symmetric bimodules. We prove that Hochschild (co)homology is equipped with λ-operations and Hodge decomposition generalizing the results in [GS1] and [Lo1] for strict algebras. The main application is concerned with string topology: we obtain a Hodge decomposition compatible with a non-trivial BV-structure on the homology H _*(LX) of the free loop space of a triangulated Poincaré-duality space. Harrison (co)homology of commutative and associative up to homotopy algebras can be defined similarly and is related to the weight 1 piece of the Hodge decomposition. We study Jacobi-Zariski exact sequence for this theory in characteristic zero. In particular, we define (co)homology of relative A _∞-algebras, i.e., A _∞-algebras with a C _∞-algebra playing the role of the ground ring. We also give a relation between the Hodge decomposition and homotopy Poisson-algebras cohomology.