Good coverings of Alexandrov spaces

TL;DR

This paper introduces the concept of good coverings for Alexandrov spaces with curvature bounds, proves their existence and homotopy equivalence to nerves, and establishes nerve stability in non-collapsing cases, including a new version of Perelman's fibration theorem.

Contribution

It defines good coverings for Alexandrov spaces, proves their existence and homotopy properties, and extends Perelman's fibration theorem to this context.

Findings

Every Alexandrov space admits a good covering.

Good coverings have the same homotopy type as their nerve.

Nerves of good coverings are stable in non-collapsing cases.

Abstract

In the present paper, we define a notion of good coverings of Alexandrov spaces with curvature bounded below, and prove that every Alexandrov space admits such a good covering and that it has the same homotopy type as the nerve of the good covering. We also prove the stability of the isomorphism classes of the nerves of good coverings in the non-collapsing case. In the proof, we need a version of Perelman's fibration theorem, which is also proved in this paper.