Foliations and the geometry of 3-manifolds

This webpage contains some information about my book, "Foliations and the geometry of 3-manifolds", published by Oxford University Press as part of their Mathematical Monograph series.

The purpose of this book is to give an exposition of the "pseudo-Anosov" theory of foliations of 3-manifolds. This theory generalizes Thurston's theory of surface automorphisms, and reveals an intimate connection between dynamics, geometry and topology in 3 dimensions. This book is not meant to be an introduction to either the theory of foliations in general, nor to the geometry and topology of 3-manifolds. The books of Candel-Conlon, Hempel, Jaco, Thurston and Scott are recommended as background.

There are a number of themes which are significant and are repeated again and again throughout the book. One is the importance of geometry, especially the hyperbolic geometry of surfaces. Another is the importance of monotonicity, especially in 1-dimensional and 1-codimensional dynamics. A third theme is combinatorial approximation, using finite combinatorial objects such as train-tracks, branched surfaces and hierarchies to carry more complicated continuous objects.

A principal aim of the book is to expose the idea of universal circles for taut foliations and other dynamical objects in 3-manifolds. Many sources feed into this idea, and I have tried to collect and present some of them and to explain how they work and fit together. Some of these sources (Dehn, Moore, Poincare) are very old; others are very new. Their continued vitality reflects the multiplicity of contexts in which they arise. This diversity is celebrated, and there are many loose threads in the book for the reader to tease out and play with.

Oxford University Press has graciously consented to allow me to make a .pdf of the book freely available. However, I strongly encourage you to buy a copy (eg if you find the electronic version useful).

Download uncorrected version of monograph

Oxford University Press page promoting the book

Zentralblatt MATH review by Athanase Papadopoulos

MathSciNet Review by Thilo Kuessner

EMS newsletter March 2009 review

Buy the book from Amazon

Table of contents: Preface Chapter 1: Surface bundles Surfaces and mapping class groups Geometric structures on manifolds Automorphisms of tori PSL(2,Z) and Euclidean structures on tori Geometric structures on mapping tori Hyperbolic geometry Geodesic laminations Train tracks Singular foliations Quadratic holomorphic differentials Pseudo-Anosov automorphisms of surfaces Geometric structures on general mapping tori Peano curves Laminations and pinching Chapter 2: The topology of S1 Laminations of S1 Monotone maps Pushout of monotone maps Pushforward of laminations Left-invariant orders Circular orders Homological characterization of circular groups Bounded cohomology and Milnor-Wood Commutators and uniformly perfect groups Rotation numbers and Ghys' theorem Homological characterization of laminations Laminar groups Groups with simple dynamics Convergence groups Examples Analytic quality of groups acting on I and S1 Chapter 3: Minimal surfaces Connections, curvature Mean curvature Minimal surfaces in R3 The second fundamental form Minimal surfaces and harmonic maps Stable and least area surfaces Existence theorems Compactness theorems Monotonicity and barrier surfaces Chapter 4: Taut foliations Definition of foliations Foliated bundles and holonomy Basic constructions and examples Volume-preserving flows and dead-ends Calibrations Novikov's theorem Palmeira's theorem Branching and distortion Anosov flows Foliations of circle bundles Small Seifert fibered spaces Chapter 5: Finite depth foliations Addition of surfaces The Thurston norm on homology Geometric inequalities and fibered faces Sutured manifolds Decomposing sutured manifolds Constructing foliations from sutured hierarchies Corollaries of Gabai's existence theorem Disk decomposition and fibered links Chapter 6: Genuine laminations Abstract laminations Essential laminations Branched surfaces Sink disks and Li's theorem Dynamic branched surfaces Pseudo-Anosov flows Push-pull Product-covered flows Genuine laminations Small volume examples Chapter 7: Universal circles Candel's theorem Circle bundle at infinity Separation constants Markers Leaf pocket theorem Universal circles Leftmost sections Turning corners, and special sections Circular orders Examples Special sections and cores Chapter 8: Constructing transverse laminations Minimal quotients Laminations of the universal circle Branched surfaces and branched laminations Straightening interstitial annuli Genuine laminations and Anosov flows Chapter 9: Slitherings and other foliations Slitherings Eigenlaminations Uniform and nonuniform foliations The product structure on the cylinder at infinity Moduli of quadrilaterals Constructing laminations Foliations with one-sided branching Long markers Complementary polygons Pseudo-Anosov flows Chapter 10: Peano curves The Hilbert space H1/2 Universal Teichmüller space Spaces of maps Constructions and Examples Moore's theorem Quasigeodesic flows Endpoint maps and equivalence relations Construction of laminations Quasigeodesic pseudo-Anosov flows Pseudo-Anosov flows without perfect fits Further directions References Index