Locally Volume Collapsed 4-Manifolds with Respect to a Lower Sectional Curvature Bound by Thunwa Theerakarn A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor John Lott, Chair Professor Ian Agol Professor Lawrence Craig Evans Professor Kam-Biu Luk Summer 2018 Locally Volume Collapsed 4-Manifolds with Respect to a Lower Sectional Curvature Bound Copyright 2018 by Thunwa Theerakarn 1 Abstract Locally Volume Collapsed 4-Manifolds with Respect to a Lower Sectional Curvature Bound by Thunwa Theerakarn Doctor of Philosophy in Mathematics University of California, Berkeley Professor John Lott, Chair Perelman stated without proof that a 3-dimensional compact Riemannian manifold which is locally volume collapsed, with respect to a lower curvature bound, is a graph manifold. The theorem was used to complete his Ricci flow proof of Thurston's geometrization conjecture. Kleiner and Lott gave a proof of the theorem as a part of their presentation of Perelman's proof. In this dissertation, we generalize Kleiner and Lott's version of Perelman's theorem to 4- dimensional closed Riemannian manifolds. We show that under some regularity assumptions, if a 4-dimensional closed Riemannian manifold is locally volume collapsed then it admits an F -structure or a metric of nonnegative sectional curvature. i Contents Contents i List of Figures iv List of Tables v 1 Introduction 1 1.1 Statement of result . 2 1.2 Outline of the proof . 4 1.3 Notation and conventions . 11 2 Preliminaries 14 2.1 Pointed Gromov-Hausdorff approximations . 14 2.2 CK -convergence . 15 2.3 Alexandrov Spaces . 15 2.4 Topology of Riemannian 4-manifolds with nonnegative sectional curvature . 16 2.5 S1 and T 2-actions . 18 2.6 Plumbing . 21 2.7 F -structures . 22 2.8 Cloudy submanifolds . 24 2.9 Approximate splittings and adapted coordinates . 25 3 Standing Assumptions 30 4 The modified volume scale r 32 5 Stratifications 37 6 The local geometry of the 3-stratum 38 6.1 Adapted coordinates, cutoff functions, and local topology near 3-stratum points 38 6.2 Selection of 3-stratum balls . 39 7 The local geometry of the 2-stratum 40 7.1 Edge 2-stratum points and associated structures . 40 7.2 Regularization of the distance function dE0 ................... 41 7.3 Adapted coordinates tangent to the edge . 42 ii 7.4 The topology of the edge region . 43 7.5 Selection of the edge balls . 43 7.6 Additional cutoff functions . 44 7.7 Adapted coordinates, cutoff functions, and local topology near slim 2-stratum points . 44 7.8 Selection of slim 2-stratum balls . 44 8 The local geometry of the 1-stratum 45 8.1 The good annulus lemma for 1-stratum points . 45 8.2 Adapted coordinates, cutoff functions, and local topology near 1-stratum points 47 8.3 Selection of ridge 1-stratum balls . 51 8.4 Selection of slim 1-stratum balls . 52 9 The local geometry of the 0-stratum 53 9.1 The good annulus lemma . 53 9.2 The radial function near 0-stratum point . 54 9.3 Selecting the 0-stratum balls . 54 10 Mapping into Euclidean Space 56 10.1 Definition of the map E 0 : M ! H ....................... 56 10.2 The image of E 0 .................................. 58 10.3 Structure of E 0 near the 3-stratum . 58 10.4 Structure of E 0 near the edge 2-stratum . 64 10.5 Structure of E 0 near the slim 2-stratum . 65 10.6 Structure of E 0 near the ridge 1-stratum . 66 10.7 Structure of E 0 near the slim 1-stratum . 72 10.8 Structure of E 0 near the 0-stratum . 73 11 Adjusting the map to Euclidean space 74 11.1 Overview of the proof of Proposition 11.1 . 75 11.2 Adjusting the map near the 3-stratum . 75 11.3 Adjusting the map near the edge 2-stratum . 76 11.4 Adjusting the map near the slim 2-stratum . 78 11.5 Adjusting the map near the ridge 1-stratum . 79 11.6 Adjusting the map near the slim 1-stratum . 82 11.7 Proof of Proposition 11.1 . 84 12 Extracting a good decomposition of M 86 12.1 The definition of M 0-stratum ............................ 87 12.2 The definition of M 1-slim ............................. 88 12.3 The definition of M 1-ridge ............................. 89 12.4 The definition of M 2-slim ............................. 91 iii 12.5 The definition of M 2-edge ............................. 92 12.6 The definition of M 3-stratum ............................ 93 12.7 Proof of Proposition 12.1 . 93 13 Decomposing M into fiber bundle components 94 13.1 Fiber bundle components without boundary . 94 13.2 Fiber bundle components with boundary . 95 13.3 Compatibility of fibers . 96 13.4 Notation . 97 13.5 Fiber bundle components with fibers S2 or T 2 . 98 3 2 1 2 2 13.6 Fiber bundle components with fibers D ;S ×Z2 I;S × D , or T ×Z2 I . 101 13.7 Fiber bundle components with fibers D2 or S1 . 104 4 13.8 Fiber bundle components (D ;::: ! Mi ! pt) . 106 3 13.9 Fiber bundle components (S =Γ;::: −! Mi −! (I; @I)) . 135 14 Gluing fiber bundle components into building blocks 136 4 3 14.1 Gluing (D ;::: ! Mj ! pt) with (S =Γ;::: ! Mi ! (I; @I)) . 136 14.2 Overview of the gluing strategy . 137 14.3 Elementary building block of type (2;S2) . 139 14.4 Elementary building block of type (2;T 2) . 144 14.5 Elementary building block of type (2;D2) . 156 14.6 Elementary building block of type (1;S1 × D2) . 166 14.7 Combining elementary building blocks of type (2;D2) and (1;S1 × D2) . 176 14.8 Combining elementary building blocks of type (2;S2) and (2;D2) . 180 F 1 3 3 14.9 Connected components of M − `(S ! M` ! (X ; @X )) that contains 2 2 2 (T ! Mi ! (Σ ;@Σ )) . 185 15 Proof of Theorem 1.4 196 15.1 M contains a fiber bundle component without boundary . 196 15.2 M does not contain a component with 2 or 3-dimensional base . 196 2 2 2 15.3 Boundary components of (S ! Mi ! (Σ ;@Σ )) . 197 2 2 2 15.4 Boundary components of (T ! Mi ! (Σ ;@Σ )) . 199 1 3 3 15.5 Boundary components of (S ! Mi ! (X ; @X )) . 202 4 3 15.6 Replacing (D ;::: ! M0 ! pt) with a boundary component of (S =Γ;::: ! Mi ! (I; @I)) . 203 15.7 The gluing instruction . 206 15.8 Satisfying the constraints . 206 Bibliography 207 iv List of Figures ∗ 2 2 ∗ 2.1 Orbit space Y! of a circle action on S ×! D with 2 exceptional orbits on @(Y! ) 19 ∗ 2 2 ∗ 2.2 Orbit space Y! of a circle action on S ×! D with 1 exceptional orbit on @(Y! ) 20 ∗ 2 2 ∗ 2.3 Orbit space Y! of a circle action on S ×! D with no exceptional orbits on @(Y! ) 20 2 2 13.1 A decomposition of a fiber of (S ! Ni ! @Σ ) in Lemma 13.2 . 99 13.2 The decomposition of @M0 in Lemma 13.10 . 107 13.3 Example of B1 ⊂ @M0 in Lemma 13.30 . 114 13.4 Example of a decomposition of @M0 in Lemma 13.41. @M0 = B1 [@ A [@ C. 118 13.5 The decomposition of @M0 in Lemma 13.50 case 1 when @M0 = A1 [@ B1 . 123 13.6 The decomposition of @M0 in Lemma 13.50 case 1 when @M0 = (A1 t A2) [@ B1 124 13.7 The decomposition of @M0 in Lemma 13.50 case 2 . 124 13.8 The decomposition of @M0 in Lemma 13.50 case 3 . 125 13.9 Example of a decomposition of @M0 in Lemma 13.64. 131 13.10 Example of a decomposition of @M0 in Lemma 13.67. 134 14.1 A representation of an elementary of building block of type (2;S2) . 139 14.2 A representation of an elementary building block of type (2;T 2) . 144 14.3 Illustration in the proof of Lemma 14.14 (2) case (a)-(S1 × D3) . 147 14.4 Illustration in the proof of Lemma 14.14 (2) case (a)-(S2 × D2) . 148 14.5 Illustration in the proof of Lemma 14.14 (2) case (a)-(S2 ×D2) in terms of plumbing149 14.6 Illustration in the proof of Lemma 14.14 (2) case (a)-(D4) in terms of plumbing 151 2 1 14.7 Illustration in the proof of Lemma 14.14 (2) case (a)-((S ×eS )×eI) in terms of double cover . 152 14.8 A representation of an elementary building block of type (2;D2) . 157 14.9 Example of the configuration of an elementary building block of type (2;D2) . 161 14.10 An edge in the representation of an elementary building block of type (2;D2) . 162 14.11 A representation of an elementary building block of type (2;D2) . 166 14.12 A representation of an elementary building block of type (1;S1 × D2) . 167 14.13 A representation of a building block of type (2;D2) + (1;S1 × D2) . 176 14.14 A representation of a building block of type (2;D2) + (1;S1 × D2) . 179 14.15 A representation of a building block of type (2;S2) + (2;D2) . 181 14.16 Example of an elementary building block of type (2;S2) + (2;D2) . 184 14.17 Example of a representation of a building block of type (2;S2)+(2;D2)+(1;S1×D2)185 14.18 A representation of a building block of type (2;S2) + (2;D2) + (1;S1 × D2) .