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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 3-manifolds

Proving Thurston's  hyperbolization for Haken 3-manifolds 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 3-manifolds [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 3-manifolds 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 non-obtuse dihedral angles. The theorem is  essential for proving Thurston's hyperbolization theorem for Haken 3-manifolds (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 Fenchel-Nielsen 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

 

 

 

 

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