Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 4: Hyberbolization of Haken Manifolds
Our
hope is that Volume 3 will be published by
the end of 2016. Volume 4 presents greater mathematical
difficulties, and we hesitate to hazard a date.
Here
we give the opening paragraphs of each chapter, as they now stand, and
some figures. In the book, labels are included in the figures, using
"puts".
Chapter 14 Hierarchies in 3manifolds
Proving Thurston's hyperbolization for Haken 3manifolds will require three major tools: the existence of hierarchies in Haken manifolds, allowing us to cut a Haken manifold into polyhedra; Andreev's theorem, allowing us to give these polyhedra hyperbolic structures; and the skinning lemma, allowing us to glue the pieces together again. The first is topological, the second geometric, the third analytic.
This chapter proves the existence of hierarchies.
It is much influenced by Allen Hatcher's notes on 3manifolds
[11] and by books by John Hempel [12] and William Jaco
[13].
A hierarchy is a way to cut up a manifold along incompressible surfaces so that the resulting components are all homeomorphic to balls. To simplify the exposition, we will assume, unless otherwise specified, that all surfaces and 3manifolds are connected, compact, and orientable. Their boundaries may be nonempty or they may be empty (in which case the manifolds are said to be "closed'': a closed manifold is compact without boundary).
Chapter 15 Andreev's theorem
Here
we present a beautiful theorem of Andreev
[4], which provides a complete characterization of compact hyperbolic
polyhedra with nonobtuse dihedral
angles. The theorem is essential for proving Thurston's
hyperbolization theorem for Haken 3manifolds (Chapter 16).
Andreev's theorem (Theorem
15.2.4) says that if we make a topological model for a polyhedron,
choosing candidate dihedral angles (angles between the faces)
that are at most p/2, then there
exist simply verifiable conditions that tell us
whether a hyperbolic polyhedron with the assigned
angles exists. Moreover, if such a polyhedron exists, it is unique.
Chapter 16 Hyberbolization of Haken manifolds
In Chapter 14 (Proposition 14.6.2) we showed that a Haken manifold M
can be cut up into polyhedra (depending on how the cutting is done, we
may end up with a single polyhedron or several). In Chapter 15
(Proposition 15.10.2) we showed that these polyhedra, perhaps after
further subdivision, can be given hyperbolic structures with all
dihedral angles right angles. In this chapter we will glue the
polyhedra back together (perhaps gluing a face of one polyhedron to
another face of the same polyhedron) to give M a hyperbolic structure.
One
dimension lower, you might think of cutting up a surface along a
maximal multicurve, to produce trousers. Then give the individual
trousers a hyperbolic structure.
You will only be able to sew
the trousers back together if corresponding boundary components have
the same length; fortunately Theorem 3.5.8 in Volume 1 guarantees that
by choosing the hyperbolic structure appropriately, we can adjust these
lengths any way we like. If we choose corresponding lengths to be
equal, and then choose gluing maps, we arrive at the FenchelNielsen
description of hyperbolic structures on surfaces.
Similarly, unless we are very careful, the faces of the polyhedra to be glued won't fit....
Appendix E1 Triangulations
Appendix E2 Connected sums
