Perelman's Stability Theorem

TL;DR

This paper provides a proof of Perelman's stability theorem, demonstrating that noncollapsing sequences of Alexandrov spaces with curvature bounds converge homeomorphically to their limit space.

Contribution

The paper offers a new proof of Perelman's stability theorem for Alexandrov spaces, confirming topological stability under Gromov-Hausdorff convergence.

Findings

Noncollapsing sequences of Alexandrov spaces are homeomorphic to their limit.

The stability theorem holds for spaces with curvature bounded below.

Convergence preserves topological structure in Alexandrov spaces.

Abstract

We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence $X_i$ of Alexandrov spaces with curvature bounded below Gromov-Hausdorff converging to a compact Alexandrov space $X$, $X_i$ is homeomorphic to $X$ for all large $i$.