- INCOMPRESSIBLE SURFACES AND GEOMETRIZATION CONJECTURE 437
and Shalen [224] and Johannson [225] says the following. (For simplicity,
we state the result for closed manifolds only.)
THEOREM 9.3 (Jaco and Shalen, Johannson). Suppose M is a closed,
orientable, and irreducible 3-manifold. Then there exists a possibly empty
disjoint union of incompressible tori in M that decomposes M into pieces
which are either Seifert 3-manifolds or geometrically atoroidal 3-manifolds.
Furthermore, the minimal such collection of tori is unique up to isotopies.
Thurston's work on the geometrization of 3-manifolds addresses the ge-
ometries underlying the Seifert pieces and the geometrically atoroidal pieces.
Thurston proved that if N is a non-Seifert, geometrically atoroidal manifold
appearing in the decomposition above (so that it has nonempty boundary),
then the interior of the manifold N admits a complete hyperbolic metric
of finite volume. Also, it is proved that the interior of any compact Seifert
3-manifold admits a complete, locally homogeneous Riemannian metric of
finite volume. Thus, the remaining issue is the geometry of a closed, irre-
ducible, geometrically atoroidal 3-manifold.
The geometrization conjecture of Thurston for a closed, irreducible,
orientable 3-manifold M states that there is an embedding of a (possibly
empty) disjoint union of incompressible tori in M such that every compo-
nent of the complement is either a Seifert space or else admits a complete
Riemannian metric of constant curvature and finite volume.
Using Thurston's theorem and the torus decomposition theorem of Jaco
and Shalen and of Johannson, one can reduce the geometrization conjecture
to the following form. Suppose M is a closed, irreducible, orientable 3-
manifold without any incompressible tori.
Conjecture I: If the fundamental group of M is infinite and does
not contain any subgroup isomorphic to Z EB Z, then M admits a
hyperbolic metric.
Conjecture II: If the fundamental group of M contains a subgroup
isomorphic to Z EB Z, then M is a Seifert space.
Conjecture III: If the fundamental group of M is finite, then M
admits a spherical (constant positive curvature) metric.
Topologists have made great progress toward resolving these conjectures.
First of all, if the manifold is Haken, Conjecture I was shown to be valid
by Thurston. (See also McMullen [262] and Otal [294].) Conjecture II
for Haken manifolds was shown to be valid by the work of Gordon and Heil
[158], Johannson [225], Jaco and Shalen [224], Scott [318] and Waldhausen
[364]. Conjecture II for non-Haken manifolds was solved affirmatively by
Casson and Jungreis [61] and Gabai [149] in 1992. Furthermore, Gabai,
Myerhoff, and Thurston [150] proved that if a closed, irreducible 3-manifold
Mis homotopic to a hyperbolic 3-manifold N, then Mis homeomorphic to
N. This gives evidence that Conjecture I holds.
Note that Conjecture III implies the Poincare conjecture. Indeed,
if M is a simply-connected 3-manifold, then by Kneser's theorem, we may
