Perelman's collapsing theorem for 3-manifolds

Details Perelman's collapsing theorem for 3-manifolds Perelman's collapsing theorem for 3-manifolds

2009-08-22

arxiv.org/abs/0908.3229v5

Mathematics

Differential Geometry

v1: 5 pictures v2: 2 pictures added v3: 9 pictures total v4: added one more graph, This final version was refereed and accepte

Scientific paper

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.

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