Collapsed Riemannian manifolds with bounded diameter and bounded covering geometry

We study the class of n-Riemannian manifolds in the title such that the torsion elements in the fundamental group have a definite bound on their orders. Our main result asserts the existence of a kind of generalized Seifert fiber structure on Mⁿ, for which the fundamental group of fibers injects into that of Mⁿ. This provides a necessary and sufficient topological condition for a manifold to admit a sufficiently collapsed metric in our class. Among other consequences we obtain a strengthened version of the "gap conjecture" in this context.