Collapsing to Alexandrov spaces with isolated mild singularities

Abstract

Let $M_j$ 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 $(k,\delta )$-strained and that the space of directions at each exceptional point contains $k+1$ directions making obtuse angles with each other. We prove that $M_j$ 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 $\delta$.

Added a simple proof of Theorem 1.1

References in corpus (4)

• Margulis lemma and Hurewicz fibration Theorem on Alexandrov spaces
• A fibration theorem for collapsing sequences of Alexandrov spaces
• Application of good coverings to collapsing Alexandrov spaces
• Homotopy lifting property of an $e^{\epsilon }$-Lipschitz and co-Lipschitz map