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
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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