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) and $\mathrm{diam}(M^{3}_{i})\ge c_{0}>0$ . Suppose that all unit metric balls in $M^{3}_{i}$ have very small volume, at most v _i →0 as i→∞, and suppose that either $M^{3}_{i}$ 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.