Algebraic structures of F-manifolds and pre-Lie algebras

F-manifolds (or weak Frobenius manifolds) were introduced by Hertling and Manin about 20 years
ago. Algebraically, the structure of an F-manifold amounts to a commutative associative product
on the tangent bundle of a manifold which is related to the Lie bracket of vector fields by an algebraic
identity weakening the Poisson identity. I shall explain that the operad controlling this arising algebraic
structure (which to many people appeared really mysterious) is in the same relationship to the operad
of pre-Lie algebras as the operad of Poisson algebras to the operad of associative algebras.
Homotopical computations needed for the proof of this algebraic result exhibit an interesting connection to
Merkulov's supergeometric approach to strong homotopy F-manifolds.