Given a manifold $M$ and a proper sub-bundle $Deltasubset TM$, we studyhomotopy properties of the horizontal base-point free loop space $Lambda$,i.e. the space of absolutely continuous maps $gamma:S^1o M$ 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$F:Lambda o M$ (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 $pi_k(Lambda)simeq pi_k(M) ltimespi_{k+1}(M)$ for all $kgeq 0.$ 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 $Lambda$).
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