Abstract
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.
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Ginot, G. (2010). On the Hochschild and Harrison (co)homology of C ∞-algebras and applications to string topology. In: Abbaspour, H., Marcolli, M., Tradler, T. (eds) Deformation Spaces. Vieweg+Teubner. https://doi.org/10.1007/978-3-8348-9680-3_1
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DOI: https://doi.org/10.1007/978-3-8348-9680-3_1
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