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Given a manifold $M$ and a proper sub-bundle $\Delta\subset TM$, we studyhomotopy properties of the horizontal base-point free loop space $\Lambda$,i.e. the space of absolutely continuous maps $\gamma:S^1\to M$ whose velocitiesare constrained to $\Delta$ (for example: legendrian knots in a contactmanifold).  A key technical ingredient for our study is the proof that the base-point map$F:\Lambda \to M$ (the map associating to every loop its base-point) is aHurewicz fibration for the $W^{1,2}$ topology on $\Lambda$. Using this resultwe show that, even if the space $\Lambda$ might have deep singularities (forexample: constant loops form a singular manifold homeomorphic to $M$), itshomotopy can be controlled nicely. In particular we prove that $\Lambda$ (withthe $W^{1,2}$ topology) has the homotopy type of a CW-complex, that itsinclusion in the standard base-point free loop space (i.e. the space of loopswith no non-holonomic constraint) is a homotopy equivalence, and consequentlyits homotopy groups can be computed as $\pi_k(\Lambda)\simeq \pi_k(M) \ltimes\pi_{k+1}(M)$ for all $k\geq 0.$  These topological results are applied, in the second part of the paper, tothe problem of the existence of closed sub-riemannian geodesics. In the generalcase we prove that if $(M, \Delta)$ is a compact sub-riemannian manifold, eachnon trivial homotopy class in $\pi_1(M)$ can be represented by a closedsub-riemannian geodesic.  In the contact case, we prove a min-max result generalizing the celebratedLyusternik-Fet theorem: if $(M, \Delta)$ is a compact, contact manifold, thenevery sub-riemannian metric on $\Delta$ carries at least one closedsub-riemannian geodesic. This result is based on a combination of the abovetopological results with a delicate study of the Palais-Smale condition in thevicinity of abnormal loops (singular points of $\Lambda$).

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