Perelman's collapsing theorem for 3-manifolds
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.
Cite
@article{arxiv.0908.3229,
title = {Perelman's collapsing theorem for 3-manifolds},
author = {Jianguo Cao and Jian Ge},
journal= {arXiv preprint arXiv:0908.3229},
year = {2010}
}
Comments
v1: 5 pictures v2: 2 pictures added v3: 9 pictures total v4: added one more graph, This final version was refereed and accepted for publication in "The Journal of Geometric Analysis"