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

We will simplify the earlier proofs of Perelman’s collapsing theorem of 3-manifolds given by Shioya-Yamaguchi [SY00]-[SY05] and Morgan-Tian [MT08]. A version of Perelman’s collapsing theorem states that: “Let {M3 i } be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and diam(M3 i ) ≥ c0 > 0. Suppose that all unit metric balls in M3 i have very small volume at most vi → 0 as i → ∞ and suppose that either M3 i is closed or it has possibly convex incompressible tori boundary. Then M3 i must be a graph-manifold for sufficiently large i”. Among other things, we use Perelman’s semi-convex analysis of distance functions to construct the desired local Seifert fibration structure on collapsed 3-manifold M3 i . 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-maniflds. 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.