  The Colding-Gromov gap theorem asserts that an almost non-negatively Ricci
curved manifold with unit diameter and maximal first Betti number is
homeomorphic to the flat torus. In this paper, we prove a parametrized version
of this theorem, in the context of collapsing Riemannian manifolds with Ricci
curvature bounded below: if a closed manifold with Ricci curvature uniformly
bounded below is Gromov-Hausdorff close to a (lower dimensional) manifold with
bounded geometry, and has the difference of their first Betti numbers equal to
the dimensional difference, then it is diffeomorphic to a torus bundle over the
one with bounded geometry. We rely on two novel technical tools: the first is
an effective control of the spreading of minimal geodesics with initial data
parallel transported along a short geodesic segment, and the second is a Ricci
flow smoothing result for certain collapsing initial data with Ricci curvature
bounded below.
