1547671870-The_Ricci_Flow__Chow

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^00 ). 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 M^3 is prime if M^3 is not diffeomorphic to the 3-sphere 53 and

if a connected sum decomposition M^3 = My#M~ is possible only if My

or M~ is itself diffeomorphic to 53. One says an orientable closed manifold

M^3 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 52 x 51.

A consequence of the Prime Decomposition Theorem [85, 97 ] is

that an orientable closed manifold M^3 can be decomposed into a finite

connected sum of prime factors

M^3 ~ (#jX}) # (#kY2) # (#c (5^2 x 51 )).

Each X} is irreducible with finite fundamental group and universal cover a

homotopy 3-sphere. Each Y2 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. From

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 I;^2 be a two-

sided compact properly embedded surface in a manifold-with-boundary N^3.

Assume that I;^2 has no components diffeomorphic to the 2-disc D^2 , and that

I;^2 either lies in 8N^3 or intersects 8N^3 only in 8I;^2. Under these conditions,

one says I;^2 is incompressible if for each D^2 c N^3 with D^2 n I;^2 = fJD,

there exists a disc D c I;^2 with fJD = fJD.

Let M^3 be irreducible. The Torus Decomposition Theorem [79, 80 ]

says that there exists a finite (possibly empty) collection of disjoint incom-

pressible 2-tori T/ such that each component N^3 of M^3 \ U ~^2 is either geo-

metrically atoroidal or a Seifert fiber space, and that a minimal such collec-

tion {'Ii} is unique up to homotopy. One says an irreducible manifold-with-

boundary N^3 is geometrically atoroidal if every incompressible torus

72 C N^3 is isotopic to a component of fJN. One says a compact manifold

N^3 is a Seifert fiber space if it admits a foliation by 51 fibers.