Homotopy properties of horizontal loop spaces and applications to closed sub-riemannian geodesics

Given a manifold M and a proper sub-bundle Δ⊂ TM, we investigate homotopy properties of the horizontal free loop space Λ, i.e., the space of absolutely continuous maps γ: S1→M whose velocities are constrained to Δ (for example: legendrian knots in a contact manifold). In the first part of the paper