Gerstenhaber duality in Hochschild cohomology

Let C be a differential graded chain coalgebra, image the reduced cobar construction on C and Clogical or the dual algebra. We prove that for a large class of coalgebras C there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies HH*(Clogical or,Clogical or) and image. This result yields to a Hodge decomposition of the loop space homology of a closed oriented manifold, when the field of coefficients is of characteristic zero.