  The double cobar construction of a double suspension comes with a
Connes-Moscovici structure, that is a homotopy G-algebra (or
Gerstenhaber-Voronov algebra) structure together with a particular BV-operator
up to a homotopy. We show that the homology of the double cobar construction of
a double suspension is a free BV-algebra. In characteristic two, a similar
result holds for the underlying $2$-restricted Gerstenhaber algebra. These
facts rely on a formality theorem for the double cobar construction of a double
suspension.
