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 « (#,-#/) # (# fc y fe 3 ) # (#/ («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.
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