1. GEOMETRIZATION OF THREE-MANIFOLDS 3
REMARK
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
S1
x
V2
by
a cyclic group action. But Epstein showed later [39] that if a compact 3-
manifold is foliated by
Sl
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
M3
is Haken if it is prime and contains an incompressible
surface other than
S2.
In [122], Thurston proved the important result that if
M3 is Haken (in particular, if M3 admits a nontrivial torus decomposition)
then
M3
admits a canonical decomposition into finitely many pieces J\ff such
that each possesses a unique geometric structure. For now, the reader should
regard a geometric structure as a complete locally homogeneous Riemannian
metric gi on Mf. 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
A/*3
be an irreducible orientable closed 3-manifold. If its fundamental group
7Ti(A/r3)
contains a subgroup isomorphic to the fundamental group Z © Z of
the torus, then either
ni(Af3)
has a nontrivial center or else
A/*3
contains an
incompressible torus. (See [114] and also [124].) If
7Ti(A/'3)
has a nontrivial
center, then results of Casson-Jungreis [22] and Gabai [42] imply that
A/"3
is a Seifert fiber space, all of which are known to be geometrizable. On the
other hand, if
A/*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
A/"3
be an irreducible orientable
closed 3-manifold of finite fundamental group ni
(A/*3). ThenM3
is diffeo-
morphic to a quotient
S3/Y
of the 3-sphere by a finite subgroup TofO (4). In
particular,
J\f3
admits a Riemannian metric of constant positive curvature.
CONJECTURE
1.4 (Hyperbolization).
LetM3
be an irreducible orientable
closed 3-manifold of infinite fundamental group
TT\
(AT3)
such that
TT^A/"3)
contains no subgroup isomorphic to Z © Z. Then
Af3
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-
ture has come from topology. For example, Gabai has proved [43] that if
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