1. GEOMETRIZATION OF THREE-MANIFOLDS 3 REMARK 1.2. Seifert's original definition [116] of a fiber space required the existence of a fiber-preserving diffeomorphism of a tubular neighborhood of each fiber to a neighborhood of a fiber in some quotient of S1 x V2 by a cyclic group action. But Epstein showed later [39] that if a compact 3- manifold is foliated by Sl fibers, then each fiber must possess a tubular neighborhood with the prescribed property. Hence the simpler definition we have given above is equivalent to the original. One says M3 is Haken if it is prime and contains an incompressible surface other than S2. In [122], Thurston proved the important result that if M3 is Haken (in particular, if M3 admits a nontrivial torus decomposition) then M3 admits a canonical decomposition into finitely many pieces J\ff such that each possesses a unique geometric structure. For now, the reader should regard a geometric structure as a complete locally homogeneous Riemannian metric gi on Mf. A more thorough discussion of geometric structures will be found in Sections 2 and 3 below. Thurston's Geometrization Conjecture asserts that a geometric de- composition holds for non-Haken manifolds as well, namely that every closed 3-manifold can be canonically decomposed into pieces such that each admits a unique geometric structure. In light of the Torus Decomposition Theo- rem and more recent results from topology, one can summarize what is currently known about the Geometrization Conjecture as follows. Let A/*3 be an irreducible orientable closed 3-manifold. If its fundamental group 7Ti(A/r3) contains a subgroup isomorphic to the fundamental group Z © Z of the torus, then either ni(Af3) has a nontrivial center or else A/*3 contains an incompressible torus. (See [114] and also [124].) If 7Ti(A/'3) has a nontrivial center, then results of Casson-Jungreis [22] and Gabai [42] imply that A/"3 is a Seifert fiber space, all of which are known to be geometrizable. On the other hand, if A/*3 contains an incompressible torus, then it is Haken, hence geometrizable by Thurston's result. Thus only the following two cases of Thurston's Conjecture are open today: CONJECTURE 1.3 (Elliptization). Let A/"3 be an irreducible orientable closed 3-manifold of finite fundamental group ni (A/*3). ThenM3 is diffeo- morphic to a quotient S3/Y of the 3-sphere by a finite subgroup TofO (4). In particular, J\f3 admits a Riemannian metric of constant positive curvature. CONJECTURE 1.4 (Hyperbolization). LetM3 be an irreducible orientable closed 3-manifold of infinite fundamental group TT\ (AT3) such that TT^A/"3) contains no subgroup isomorphic to Z © Z. Then Af3 admits a complete hyperbolic metric of finite volume. REMARK 1.5. A complete proof of the Elliptization Conjecture would imply the Poincare Conjecture, which asserts that any homotopy 3-sphere is actually a topological 3-sphere. (See Chapter 6.) REMARK 1.6. Noteworthy progress toward the Hyperbolization Conjec- ture has come from topology. For example, Gabai has proved [43] that if
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