Gerstenhaber duality in Hochschild cohomology

Abstract

Let C be a differential graded chain coalgebra, Ω ¯ C the reduced cobar construction on C and C ∨ 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 * ( C ∨ , C ∨ ) and HH * ( Ω ¯ C ; Ω ¯ C ) . 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.