We have decided to break up volume 2 of the Teichmüller theory book (Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 2: Four Theorems by William Thurston) into three shorter volumes:
Volume
2: Surface Homeomorphisms and Rational Functions
Volume 3: 3Manifolds that fiber over the circle
Volume 4: Hyberbolization of Haken Manifolds
Our
hope is that Volume 2 will be published by summer 2016, and Volume 2 by
the end of the year. Volume 3 presents greater mathematical
difficulties, and we hesitate to hazard a date.
We plan to use color in the illustrations.
Table of Contents for Volume 2: Surface Homeomorphisms and Rational Functions
Chapter 8 The classification of homeomorphisms of surfaces
8.1 The classification theorem
8.2 Periodic and reducible homeomorphisms
8.3 PseudoAnosov homeomorphisms
8.4 Proof of the classification theorem
8.5 The structure in the reducible case
Chapter 9 Dynamics of polynomials
9.1 Julia sets
9.2 Fixed points
9.3 Green's functions, Böttcher coordinates
9.4 Extending f_0 to S^1
9.5 External rays at rational angles land
Chapter 10 Rational functions
10.1 Introduction
10.1 Thurston mappings
10.2 Thurston mapps associated to spiders
10.3 Thurston obstructions for spider maps and Levy cycles
10.4 Julia sets of quadratic polynomials with superattracting cycles
10.5 Parameter spaces for quadratic polynomials
10.6 The Thurston pullback mapping s_f
10.7 The derivative and coderivative of s_f
10.8 The necessity of the eigenvalue criterion
10.9 Convergence in moduli spaces implies convergence in Teichmüller space
10.10 Asymptotic geometry of Riemann surfaces
10.11 Sufficiency of the eigenvalue criterion
Appendix C1 The PerronFrobenius theorem
Appendix C2 The Alexander trick
Appendix C3 Homotopy implies isotopy
Appendix C4 The mapping class group and outer automorphisms
Appendix C5 Totally real stretch factors
Appendix C6 Irrationally indifferent fixed points
Appendix C7 Examples of Thurston pullback maps
Appendix C8 Branched maps with nonhyperbolic orbifolds
Appendix C9 The Sullivan dictionary
Table of Contents for Volume 3: 3Manifolds that fiber over the circle
Chapter 11 Geometry of hyperbolic space
Chapter 12 Rigidity theorems
Chapter 13 Hyperbolization of 3manifolds that fiber over the circle
Appendix D1 The Nullstellensatz and Selberg's lemma
Appendix D2 Fundamental groups, amalgamated sums, and HNN extensions
Appendix D3 The Margulis lemma: another proof
Appendix D4 Arithmetic Kleinian groups
Appendix D5 Ergodic flow and Mostow rigidity
Appendix D6 Sullivan's rigidity theorem
Appendix D7 Positivity of the Riemann matrix
Table of Contents for Volume 4: Hyberbolization of Haken Manifolds
Chapter 14 Hierarchies in 3manifolds
Chapter 15 Andreev's theorem
Chapter 16 Hyberbolization of Haken manifolds
Appendix E1 Triangulations
Appendix E2 Connected sums
From
the preface to volume 1:
Not
only are the theorems of extraordinary beauty in themselves,
but the methods of proof that Thurston introduced were so novel
and displayed such amazing geometric insight that to this day
they have barely entered the accepted methods of mathematicians
in the field.
The
results sound more or less unrelated, but they are linked by
a common thread: each one goes from topology to geometry. Each
says that either a topological problem has a natural geometry,
or there is an understandable obstruction.
The
proofs are closely related: you use the topology to set up an
analytic mapping from a Teichmüller space to itself; the
geometry arises from a fixed point of this mapping. Thurston
proceeds to show that if there is no fixed point, then some system
of simple closed curves on the surface is an obstruction to finding
a solution.
Thus
the proofs of the theorems are somehow similar, although the
details and difficulty are very different....
