A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds

Abstract

We will simplify earlier proofs of Perelman's collapsing theorem for 3-manifolds given by Shioya-Yamaguchi (J. Differ. Geom. 56:1-66, 2000; Math. Ann. 333: 131-155, 2005) and Morgan-Tian (arXiv:0809.4040v1 [math.DG], 2008). A version of Perelman's collapsing theorem states: "Let {M 3 i } be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) andi is closed or has possibly convex incompressible toral boundary. Then M 3 i must be a graph manifold for sufficiently large i". This result can be viewed as an extension of the implicit function theorem. Among other things, we apply Perelman's critical point theory (i.e., multiple conic singularity theory and his fibration theory) to 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. A version of the geometrization conjecture asserts that any closed 3-manifold admits a piecewise locally homogeneous metric. Our proof of Perelman's collapsing theorem is accessible to advanced graduate students and non-experts.

Geometrisation of 3-Manifolds

“…Shioya-Yamaguchi [SY05] and Cao-Ge [CG09] do not have the hypothesis of curvature control in the sense of Perelman. In counterpart, they use deep results on Alexandrov spaces, such as Perelman's stability theorem [Per91] (see also the paper by V. Kapovitch [Kap07]).…”

“…They use deep results of Alexandrov space theory, including Perelman's stability theorem [Per91] (see also the paper by V. Kapovitch [Kap07]) and a fibration theorem for Alexandrov spaces, proved by Yamaguchi [Yam96]. Other proofs of Perelman's collapsing theorem can be found in papers by Morgan-Tian [MT08], Cao-Ge [CG09], and Kleiner-Lott [KL10]. For more details see Section 13.4.…”

“…In particular, there are by now several texts on Perelman's collapsing theorem or some variant of it: an article by Shioya and Yamaguchi [SY05], the above-mentioned paper by the authors of the present monograph [BBB + 10], an arXiv preprint by Morgan and Tian [MT08], and two more recent preprints by Cao and Ge [CG09], and Kleiner and Lott [KL10].…”

“…Hence it is natural to ask for a proof of the Geometrisation Conjecture which does not use this theorem. This is Perelman's original viewpoint which is developed in [KL08], [MT08] or [CG09].…”

Nonnegatively curved fixed point homogeneous manifolds in low dimensions

“…In addition, when X has nonnegative curvature, we have the following result (cf. [38,4]). In the rest of this subsection we will assume that M is a closed nonnegatively curved Riemannian manifold with a fixed-point homogeneous isometric S 1 -action.…”