  We dualise the classical fact that an operad with multiplication leads to
cohomology groups which form a Gerstenhaber algebra to the context of
cooperads: as a result, a cooperad with comultiplication induces a homology
theory that is endowed with the structure of a Gerstenhaber coalgebra, that is,
it comes with a graded cocommutative coproduct which is compatible with a
coantisymmetric cobracket in a dual Leibniz sense. As an application, one
obtains Gerstenhaber coalgebra structures on Tor groups over bialgebras or Hopf
algebras, as well as on Hochschild homology for Frobenius algebras.
