  It is shown that every algebra over the chain operad of the little disks
operad gives naturally rise to a Hertling-Manin's F-manifold, that is a smooth
manifold equipped with an integrable graded commutative associative product on
the tangent sheaf. In particular, moduli spaces of extended deformations of
complex/symplectic structures are shown to have a canonical structure of
F-manifold. With the help of the $G_\infty$-operad a strong homotopy version of
the notion of F-manifold is constructed. Among natural examples of
$F_\infty$-manifolds one finds formal manifolds associated with the Hochschild
cohomology of an associative algebra and with the singular cohomology of an
arbitrary compact topological space.
