2
1. THE RICCI FLOW OF SPECIAL GEOMETRIES
REMARK
1.1. In dimension n — 3, the categories
TOP, PL,
and
DIFF
all
coincide. In this volume, any manifold under consideration — regardless of
its dimension — is assumed to be smooth (C°°). Moreover, unless explicitly
stated otherwise, every manifold is assumed to be without boundary, so that
it is closed if and only if it is compact.
1. Geometrization of three-manifolds
To motivate our interest in locally homogeneous metrics on 3-manifolds,
we begin with a heuristic discussion of the Geometrization Conjecture, which
will be reviewed in somewhat more detail in the successor to this volume.
(Good references are [115] and [123].) We begin that discussion with a brief
review of some basic 3-manifold topology. One says an orientable closed
manifold
M3
is prime if
M3
is not diffeomorphic to the 3-sphere
S3
and
if a connected sum decomposition
M3
= M\#M\ is possible only if
M3
or M\
is
itself diffeomorphic to
S3.
One says an orientable closed manifold
M3
is irreducible if every separating embedded 2-sphere bounds a 3-ball.
It is well known that the only orientable 3-manifold that is prime but not
irreducible is
S2
x
Sl.
A consequence of the Prime Decomposition Theorem [85, 97] is
that an orientable closed manifold
M3
can be decomposed into a finite
connected sum of prime factors
M3
« (#,-#/) #
(#fcyfe3)
# (#/
(«S2
x
S1)).
Each
X3
is irreducible with finite fundamental group and universal cover a
homotopy 3-sphere. Each y\ is irreducible with infinite fundamental group
and a contractible universal cover. The prime decomposition is unique up to
re-ordering and orientation-preserving diffeomorphisms of the factors. Prom
the standpoint of topology, the Prime Decomposition Theorem reduces the
study of closed 3-manifolds to the study of irreducible 3-manifolds.
There is a further decomposition of irreducible manifolds, but we must
recall more nomenclature before we can state it precisely. Let
E2
be a two-
sided compact properly embedded surface in a manifold-with-boundary
A/"3.
Assume that S
2
has no components diffeomorphic to the 2-disc P
2
, and that
E2
either lies in
dJ\f3
or intersects
dM3
only in
9E2.
Under these conditions,
one says E2 is incompressible if for each V2 C A/"3 with V2 Pl E2 = dV,
there exists a disc P* C
E2
with dV* = dV.
Let
M3
be irreducible. The Torus Decomposition Theorem [79, 80]
says that there exists a finite (possibly empty) collection of disjoint incom-
pressible 2-tori
T2
such that each component
J\f3
of
M3\UT2
is either geo-
metrically atoroidal or a Seifert fiber space, and that a minimal such collec-
tion {%} is unique up to homotopy. One says an irreducible manifold-with-
boundary
Af3
is geometrically atoroidal if every incompressible torus
T2
C
A/"3
is isotopic to a component of dj\f. One says a compact manifold
A/"3
is a Seifert fiber space if it admits a foliation by
S1
fibers.
