Collapsing to Alexandrov spaces with isolated mild singularities

Abstract

Let Mj be a sequence of Riemannian manifolds with sectional curvature bound below collapsing to a compact Alexandrov space X of dimension k. Suppose that all but finitely many points of X are (k,δ)-strained and that the space of directions at each exceptional point contains k+1 directions making obtuse angles with each other. We prove that Mj admits a structure of locally trivial fibration over X for sufficiently large j. The same is true for collapsing sequences of Alexandrov spaces such that the infimum of the volume of the spaces of directions is sufficiently large relative to δ.