  We show that the reduced Hochschild homology of a DG open Frobenius algebra
has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced
cyclic homology has the natural structure of a gravity coalgebra. This gives an
algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced
homology of the free loop space of a simply connected closed oriented manifold,
and a gravity coalgebra structure on the reduced equivariant homology.
