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

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

2010-03-10

arxiv.org/abs/1003.2215v3

Mathematics

Differential Geometry

v1: 9 Figures. In this version, we improve the exposition of our arguments in the earlier arXiv version. v2: added one more gr

Scientific paper

10.1007/s12220-010-9169-5

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

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