In this paper we study a
class of sub-spaces of loop spaces which have appeared in the calculus of variations.
Generalizing a result of Smale, we show that the space of loops tangent to a
distribution satisfying Hörmander’s condition is weakly homotopic to the space of
all loops. If the distribution is fat, we resolve the end point map from the space of
horizontal paths. This resolution has two applications: (1) the proof that the
cut-locus on an analytic fat Carnot-Carathéodory manifold is sub-analytic; (2) a
study of the singularity of the horizontal loop space. At the end we study the
geometry of left-invariant Carnot-Carathéodory metrics on fact nilpotent
groups.
