A simple proof of Perelman's collapsing theorem for 3-manifolds

TL;DR

This paper provides a simplified, accessible proof of Perelman's collapsing theorem for 3-manifolds, crucial for Thurston's Geometrization Conjecture, using Alexandrov space theory and Perelman's critical point techniques.

Contribution

It offers a more straightforward, self-contained proof of Perelman's collapsing theorem, extending the implicit function theorem to 3-manifold collapse scenarios.

Findings

Constructed local Seifert fibrations on collapsed 3-manifolds

Simplified proof accessible to non-experts and students

Confirmed the collapsing theorem as a key step in Geometrization

Abstract

We will simplify earlier proofs of Perelman's collapsing theorem for 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's critical point theory (e.g., multiple conic singularity theory and his fibration theory) for Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3-manifolds. The verification of Perelman's collapsing theorem is the last step of Perelman's proof of Thurston's Geometrization Conjecture on the classification of 3-manifolds. Our proof of Perelman's collapsing theorem is almost self-contained, accessible to non-experts and advanced graduate students. Perelman's collapsing theorem for 3-manifolds can be viewed as an extension of implicit function theorem