l. GEOMETRIZATION OF THREE-MANIFOLDS 3REMARK 1.2. Seifert's original definition [116] of a fiber space required
the existence of a fiber-preserving diffeomorphism of a tubular neighborhood
of each fiber to a neighborhood of a fiber in some quotient of 51 x V^2 by
a cyclic group action. But Epstein showed later [39] that if a compact 3-
manifold is foliated by 51 fibers, then each fiber must possess a tubular
neighborhood with the prescribed property. Hence the simpler definition we
have given above is equivalent to the original.
One says M^3 is Haken if it is prime and contains an incompressible
surface other than 52. In [122], Thurston proved the important result that if
M^3 is Haken (in particular, if M^3 admits a nontrivial torus decomposition)
then M^3 admits a canonical decomposition into finitely many pieces ~^3 such
that each possesses a unique geometric structure. For now, the reader should
regard a geometric structure as a complete locally homogeneous Riemannian
metric 9i on ~^3. A more thorough discussion of geometric structures will
be found in Sections 2 and 3 below.
Thurston's Geometrization Conjecture asserts that a geometric de-
composition holds for non-Haken manifolds as well, namely that every closed
3-manifold can be canonically decomposed into pieces such that each admits
a unique geometric structure. In light of the Torus Decomposition Theo-
rem and more recent results from topology, one can summarize what is
currently known about the Geometrization Conjecture as follows. Let N^3
be an irreducible orientable closed 3-manifold. If its fundamental group
7r 1 (N^3 ) contains a subgroup isomorphic to the fundamental group ZEB Z of
the torus, then either 7r 1 (N^3 ) has a nontrivial center or else N^3 contains an
incompressible torus. (See [114] and also [124].) If 7r 1 (N^3 ) has a nontrivial
center, then results of Casson- Jungreis [22] and Gabai [42] imply that N^3
is a Seifert fiber space, all of which are known to be geometrizable. On the
other hand, if N^3 contains an incompressible torus, then it is Haken, hence
geometrizable by Thurston's result. Thus only the following two cases of
Thurston's Conjecture are open today:
CONJECTURE 1.3 (Elliptization). Let N^3 be an irreducible orientable
closed 3-manifold of finite fundamental group 7r1 (N^3 ). Then N^3 is diffeo-
morphic to a quotient 53 /r of the 3-sphere by a finite subgroup r of 0 ( 4). In
particular, N^3 admits a Riemannian metric of constant positive curvature.CONJECTURE 1.4 (Hyperbolization). LetN^3 be an irreducible orientable
closed 3-manifold of infinite fundamental group 7r1 (N^3 ) such that 7r1 (N^3 )
contains no subgroup isomorphic to ZEB Z. Then N^3 admits a complete
hyperbolic metric of finite volume.
REMARK 1.5. A complete proof of the Elliptization Conjecture would
imply the Poincare Conjecture, which asserts that any homotopy 3-sphere
is actually a topological 3-sphere. (See Chapter 6.)
REMARK 1.6. Noteworthy progress toward the Hyperbolization Conjec-