Perelman's collapsing theorem for 3-manifolds

TL;DR

This paper simplifies the proof of Perelman's collapsing theorem for 3-manifolds, a key step in proving Thurston's Geometrization Conjecture, making it more accessible to learners.

Contribution

The authors provide a simplified, almost self-contained proof of Perelman's collapsing theorem using semi-convex analysis, enhancing understanding of 3-manifold classification.

Findings

Simplified proof of Perelman's collapsing theorem

Construction of local Seifert fibration structures

Accessible approach for non-experts and students

Abstract

We will simplify the earlier proofs of Perelman's collapsing theorem of 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's semi-convex analysis of distance functions 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. We believe that our proof of this collapsing theorem is accessible to non-experts and advanced graduate students.