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