Collapsing to Alexandrov spaces with isolated mild singularities

TL;DR

This paper proves that sequences of Riemannian manifolds with bounded sectional curvature collapsing to Alexandrov spaces exhibit a locally trivial fibration structure under mild singularity conditions, extending understanding of collapsing phenomena.

Contribution

It establishes a new fibration theorem for collapsing Riemannian manifolds with isolated mild singularities in the limit space.

Findings

Existence of locally trivial fibrations over the limit space for large sequences

Fibration structure persists under mild singularity conditions

Results extend collapsing theory to spaces with isolated 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 $\delta$.