Any singular foliation F hides a Q-manifold, built on a resolution of vector fields tangent to it. It is unique up to homotopy, and a universal object in a caterogy of Q-manifolds inducing a sub-foliation of F. It deserves therefore to be called "universal Q-manifold of F".
I will explain the geometric meaning of this Q-manifold and how it allows to define first return (higher) maps for singular leaves.
Joint works with Sylvain Lavau Thomas Strobl and with Leonid Ryvkin