THE UNIVERSITY OF CHICAGO THE CLASSIFICATION OF FIVE-DIMENSIONAL GEOMETRIES A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS BY ANDREW LIANG LI GENG CHICAGO, ILLINOIS JUNE 2016 Copyright c 2016 by Andrew Liang Li Geng All Rights Reserved To my sister June, who is working on the other half of our MD-PhD. Four-dimensional hypergeometric eigenschlumpkins! — Alar Toomre Wait, is this going to be on the test? — Arka Dhar TABLE OF CONTENTS LIST OF FIGURES . viii LIST OF TABLES . ix ACKNOWLEDGMENTS . x ABSTRACT . xi I SUMMARY 1 INTRODUCTION . 2 2 SALIENT EXAMPLES . 7 2.1 New phenomena . .7 2.2 Examples that highlight tools . .8 2.3 Examples that clarify how the classification is organized . 10 3 OVERVIEW OF METHOD . 12 4 RELATED WORK . 17 II THE NON-FIBERED GEOMETRIES 5 OVERVIEW . 21 6 GENERAL BACKGROUND: GEOMETRIES . 26 7 CLASSIFICATION OF ISOTROPY REPRESENTATIONS . 29 7.1 Proof of the classification of isotropy representations . 30 7.2 Standard facts about representations of lower-dimensional groups . 33 8 THE CASE OF IRREDUCIBLE ISOTROPY . 36 8.1 The list of candidates . 36 8.2 Maximality and the existence of compact quotients . 38 9 THE CASE OF TRIVIAL ISOTROPY: SOLVABLE LIE GROUPS . 40 9.1 Notations . 41 9.2 Reduction to semidirect products . 42 9.2.1 The general extension problem . 43 9.2.2 Nilradicals . 44 9.2.3 The elimination of non-split extensions . 47 3 2 9.3 The R o R geometry . 52 9.4 Classification of semidirect sums with R .................... 54 v 4 9.4.1 R semidirect sums . 56 9.4.2 n4 semidirect sums . 62 9.4.3 R ⊕ n3 semidirect sums . 65 9.5 Maximality and distinctness . 72 III THE FIBERED GEOMETRIES 10 OVERVIEW . 79 11 BACKGROUND . 84 11.1 Geometries, products, and isotropy . 84 11.2 Lie algebra extensions . 86 11.3 Notations . 88 12 FIBER BUNDLE STRUCTURES ON GEOMETRIES . 89 12.1 Invariant distributions and foliations . 90 12.2 Complementary distributions and conformal actions . 92 12.3 Proof of the fibering description (Proposition 12.3) . 96 12.3.1 Case (i): over dimension 4 . 97 12.3.2 Case (ii): over dimension 3 . 98 12.3.3 Case (iii): over dimension 2 . 99 13 GEOMETRIES FIBERING OVER 4D GEOMETRIES . 103 13.1 Reconstructing geometries from base and curvature information . 104 13.2 Classifying and verifying geometries . 106 14 GEOMETRIES FIBERING OVER 3D GEOMETRIES . 111 14.1 Geometries fibering isometrically are products . 112 14.2 Case study: The geometry fibering essentially conformally . 115 14.3 Groups acting on constant-curvature spaces . 119 15 GEOMETRIES FIBERING OVER 2D GEOMETRIES . 124 15.1 Setting up the extension problem . 125 15.2 Geometries fibering essentially . 129 15.2.1 Over the sphere . 130 15.2.2 Over the plane . 131 15.3 Geometries requiring non-split extensions . 142 15.3.1 Data for the extension problem . 144 15.3.2 Reduction to one extension problem . 148 15.3.3 The geometries . 150 4 1 1;2 15.4 Line bundles over F and T E (semidirect-product isometry groups) . 154 15.5 Product geometries and associated bundles . 160 15.5.1 Associated bundle geometries . 161 15.5.2 Proof of the classification . 169 15.5.3 Explicit enumeration of product geometries . 178 vi 15.6 Deferred maximality proofs for isometrically fibering geometries . 180 REFERENCES . 187 vii LIST OF FIGURES 3.1 Closed connected subgroups of SO(5). (Duplicate of 7.2) . 12 3.3 Flowchart of the classification. 13 3.4 Identification key for solvable geometries. (Duplicate of 5.3) . 14 3.6 Geometries fibering over 2-D spaces. (Duplicate of 15.2) . 16 5.3 Identification key for geometries G = G=f1g................... 24 7.2 Closed connected subgroups of SO(5), with inclusions. 30 11.4 Closed connected subgroups of SO(5). (Duplicate of 7.2) . 86 15.2 Classification strategy for geometries fibering over 2-D spaces. 125 viii LIST OF TABLES 3.2 Non-product geometries listed by point stabilizer. 13 3.5 Geometries with irreducible 4-dimensional isotropy. (Duplicate of 13.3) . 15 5.5 Names from [PSWZ76, Tables I, II] for geometries with trivial isotropy. 25 10.2 Non-product fibering geometries by isotropy group Gp (see also Fig. 11.4) . 82 10.3 Non-contractible non-product fibering geometries by diffeomorphism type . 82 13.3 Candidate geometries with irreducible 4-dimensional isotropy summand . 104 15.5 3-dimensional model geometries with SO(2) isotropy. 128 15.14 Centers and traceless outer derivation algebras. 144 15.19 Nontrivial semidirect products covering candidate isometry groups G ..... 156 4 15.21 Finite-volume manifolds ΓnG=K modeled on G=K fibering over F ...... 159 15.24 Groups covering the isometry group G of geometries in this section . 161 15.29 Descriptions of associated bundle geometries, assuming nonzero a and b ... 164 15.42 3-dimensional fixed sets of order 2 isometries . 184 ix ACKNOWLEDGMENTS I wish to express my gratitude to my advisor, Benson Farb—for his unwavering guidance through the best and the worst of times; for steadfastly combing through my drafts; and for showing me the way to a wide and beautiful swath of mathematics. I have my second advisor Danny Calegari to thank as well, for enlightening conversations that helped me to see the bigger picture, the still-unanswered questions, and a sane sense of perspective. The ideas in this project have been sharpened by exposure to discussion; and I am thank- ful to Greg Friedman, Christoforos Neofytidis, and Kenneth Knox for giving me chances to speak about and gather reactions to my results. Many conversations have contributed to my progress, including those with Jonathan Hillman about many aspects of the four-dimensional geometries, with Daniel Studenmund about the correct version of an old theorem, with An- drzej Szczepański about symmetry groups, and with Ilya Grigoriev and Nick Salter about a particular family of “super goofy” geometries. Thanks are due to the University of Chicago for providing a nurturing environment and my financial support over the last five years. I thank also my officemates Jonny, Kate, Tom, Fedya, Nir, and Joe, who have immea- surably enriched my time here as a student. I thank my parents Chaoxian and Lan, my sister June, and my wife Chelsea, who have supported me and watched me grow throughout this effort, and whose companionship has kept me happy, healthy, and rooted in the wider world. Finally, I thank my high school math team coach Julie Perreault, who set me on this path in probably more ways than I have yet realized. x ABSTRACT We classify the 5-dimensional homogeneous geometries in the sense of Thurston. The 5- dimensional geometries with irreducible isotropy are the irreducible Riemannian symmetric spaces, while those with trivial isotropy are simply-connected solvable Lie groups of the form 3 2 R o R or N o R where N is nilpotent. The geometries with nontrivial reducible isotropy are mostly products, but a number of interesting examples arise. These include a countably 3 2 infinite family L(a; 1) ×S1 L(b; 1) of inequivalent geometries diffeomorphic to S × S , an 3 uncountable family SLg2×αS in which only a countable subfamily admits compact quotients, and the non-maximal geometry SO(4)= SO(2) realized by two distinct maximal geometries. xi PART I SUMMARY CHAPTER 1 INTRODUCTION By the classification of closed surfaces (see e.g. [Mat02, Thm. 5.11]), every closed surface is 2 2 2 diffeomorphic to a quotient of E , S , or H by a discrete group of isometries. It is a classical result that in dimension 2, these three are the only connected, simply-connected, complete Riemannian manifolds with transitive isometry group (see e.g. [Thu97, Thm. 3.8.2]). The quest for the 3-dimensional generalization that became Thurston’s Geometrization Conjecture led to a version of the following definition. (The equivalence to older definitions is outlined in Part II, Prop. 6.5.) Definition 1.1 (Geometries, following [Thu97, Defn. 3.8.1] and [Fil83, §1.1]). (i)A geometry is a connected, simply-connected homogeneous space M = G=Gp where G is a connected Lie group acting faithfully with compact point stabilizers Gp. (ii) M is a model geometry if there is some lattice Γ ⊂ G that acts freely on M. Then the manifold ΓnG=Gp is said to be modeled on M. 0 0 (iii) M is maximal if it is not G-equivariantly diffeomorphic to any other geometry G =Gp 0 0 0 with G ( G . Any such G =Gp is said to subsume G=Gp. Then a closed 3-manifold is a quotient of at most one maximal model geometry, which can be determined from the fundamental group [Thu97, Thm. 4.7.8] or from the existence of certain bundle structures (usually Seifert bundles) and some topological data (usually two Euler numbers) [Sco83, Thm. 5.3]. Thurston classified the 3-dimensional maximal model geometries and found eight (see [Thu97, Thm. 3.8.4]). In 4 dimensions, Filipkiewicz classified the maximal model geometries in [Fil83]. Though 4-manifolds without geometric decompositions [Hil02, §13.3 #3] indicate there is less hope for a straightforward generalization of geometrization, Filipkiewicz’s classification highlights 2 a few interesting firsts. The list comprises 18 geometries and—for the first time—a countably 4 infinite family, named Solm;n. (See e.g. [Hil02, §7.1] or [Wal86, §1, Table 1] for the names 4 2 currently in use.) One of the eighteen is F = R o SL(2; R)= SO(2), the first geometry to admit finite-volume quotients but no compact quotients.