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.