The theory of 3-manifolds has been revolutionised in the last few years by work of Thurston [66-70]. He has shown that geometry has an important role to play in the theory in addition to the use of purely topological methods. The basic aim of this article is to discuss the various geometries which arise and explain their significance for the theory of 3-manifolds. The idea is that many 3-manifolds admit 'nice' metrics which give one new insight into properties of the manifolds. For the purposes of this article, the nicest metrics are those of constant curvature. An observer in a manifold with a constant curvature metric will see the same picture wherever he stands and in whichever direction he looks. Such manifolds have special topological properties. However, we will also need to consider nice metrics which are not of constant curvature. In this article, I will explain what is meant by a 'nice' metric and describe their classification in dimension three which is due to Thurston. Then I will discuss some of the 3-manifolds which admit these nice metrics and the relationship between their geometric and topological properties. In this introduction all manifolds and metrics will be assumed to be smooth so that the objects of interest are all Riemannian manifolds. It has been known since the nineteenth century that in dimension two there is a very close relationship between geometry and topology. I will start by describing some basic facts about closed surfaces. I will discuss these in more detail in §1. Each

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The geometries of 3-manifolds

Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifolds Infinite dimensional differential geometry Manifolds of mappings Further applications References Index.

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The Convenient Setting of Global Analysis

This article was widely circulated as a preprint, about 12 years ago. At that time the Bulletin did not accept research announcements, and after a couple of attempts to publish it, I gave up, and the preprint did not find a home. I very soon saw that there were many ramifications of this theory, and I talked extensively about it in a number of places. One year I devoted my graduate course to this theory, and notes of Bill Floyd and Michael Handel from that course were circulated for a while. The participants in a seminar at Orsay in 1976-1977 went over this material, and wrote a volume [FLP] including some original material as well. Another good general reference, from a somewhat different point of view, is a set of notes of lectures by A. Casson, taken by S. Bleiler [CasBlei]. There are by now several alternative ways to develop the classification of diffeomorphisms of surfaces described here. At the time I originally discovered the classification of diffeomorphism of surfaces, I was unfamiliar with two bodies of mathematics which were quite relevant: first, Riemann surfaces, quasiconformal maps and Teichmiiller's theory; and second, Nielsen's theory of the dynamical behavior of surface at infinity, and his near-understanding of geodesic laminations. After hearing about the classification of surface automorphisms from the point of view of the space of measured foliations, Lipman Bers [Bersl] developed a proof of the classification of surface automorphisms from the point of view of Teichmüller theory, generalizing Teichmiiller's theorem by allowing the Riemann surface to vary as well as the map. Dennis Sullivan first told me of some neglected articles by Nielsen which might be relevant. This point of view has been discussed by R. Miller, J. Gilman, M. Handel and me. The analogous theory, of measured laminations and 2-dimensional train tracks in three dimensions, has been considerable development. This has been applied to reinterpret some of Haken's work, to classify incompressible surfaces in particular classes of 3-manifolds in papers by me, Hatcher, Floyd, Oertel and others in various combinations. Shalen, Morgan, Culler and others have developed the related theory of groups acting on trees, and its relation to measured laminations, to define and analyze compactifications of representation spaces of groups in SL(2, C) and SO(n, 1); this has many interesting applications, including the theory of incompressible surfaces in 3-manifolds.

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On the geometry and dynamics of diffeomorphisms of surfaces

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the complex is finite-dimensional, but locally infinite. It was introduced by Harvey as an analogy, in the context of Teichmuller space, for Tits buildings for symmetric spaces, and has been studied by Harer and Ivanov as a tool for understanding mapping class groups of surfaces. 
In this paper we prove that, endowed with a natural metric, the complex is hyperbolic in the sense of Gromov. In a certain sense this hyperbolicity is an explanation of why the Teichmuller space has some negative-curvature properties in spite of not being itself hyperbolic: Hyperbolicity in the Teichmuller space fails most obviously in the regions corresponding to surfaces where some curve is extremely short. The complex of curves exactly encodes the intersection patterns of this family of regions (it is the "nerve" of the family), and we show that its hyperbolicity means that the Teichmuller space is "relatively hyperbolic" with respect to this family. A similar relative hyperbolicity result is proved for the mapping class group of a surface. 
We also show that the action of pseudo-Anosov mapping classes on the complex is hyperbolic, with a uniform bound on translation distance.

Geometry of the complex of curves I: Hyperbolicity

A presentation for the mapping class group of a closed orientable surface

To each rational number p/q, with q odd, there is associated the 2-bridge knot Kp/q shown in Fig. 1. QI bl Fig. 1. The 2-bridge knot Kp/q In (a), the central grid consists of lines of slope +p/q, which one can imagine as being drawn on a square \"pillowcase\". In (b) this \"pillowcase\" is punctured and flattened out onto a plane, making the two \"bridges\" more evident. The knot drawn is K3/5, which happens to be the figure eight knot. (We assume q odd in order to get a knot rather than a two-component link.) The double cover of S 3 branched along Kp/q is the lens space Lq,p. With this observation, attributed in [16] to Seifert, the isotopy classification of 2-bridge knots follows easily from the classification [14] of oriented lens spaces: K~/q =gp,/q, if and only if q'=q and p,_p+_l (modq). Basic references for 2-bridge knots are [2, 16, 17]. We shall derive in this paper the isotopy classification of the incompressible surfaces, orientable or not, in S3-Kp/q. As an application, we obtain some information about the manifold resulting from Dehn surgery on Kp/q: Excluding the cases when Kp/q is a torus knot (Dehn surgery on torus knots was completely analyzed in [10]), every Dehn surgery on K m yields an irreducible manifold, and all but finitely many Dehn surgeries yield non-Haken (i.e., not sufficiently large) non-Seifert-fibered manifolds. The case of the figure eight

Incompressible surfaces in 2-bridge knot complements

The Smale Conjecture [9] is the assertion that the inclusion of the orthogonal group 0(4) into Diff(S3), the diffeomorphism group of the 3-sphere with the Cw topology, is a homotopy equivalence. There are many equivalent forms of this conjecture, some of which are listed in the appendix to this paper. We shall prove

https://www.jstor.org/stable/pdfplus/2007035.pdf

A proof of the Smale Conjecture, Diff(S3) 0(4)

This is a Tex version, made in 2004, of a paper that appeared in Pac. J. Math. 99 (1982), 373-377, with some revisions in the exposition. Let M be an compact orientable 3 manifold whose boundary @M consists of a single torus. If a meridian and longitude in this torus are chosen, then isotopy classes of smoothly embedded circles in @M that do not bound disks correspond bijectively with elements of QP 1 E Q[f 1= 0g, regarded as slopes of these curves. We show in this paper that the set of slopes coming from boundary curves of incompressible, @ incompressible surfaces in M is finite. There is also a generalization to the case that @M consists of n tori T1;;Tn . Given a curve system in @M consisting of finitely many disjoint smoothly embedded circles that do not bound disks, then by choosing parallel orientations for the circles in each component of @M, we get an element of H 1 N@MO. Ignoring orientations amounts to factoring out multiplication by1 in each component, yielding a quotient of H 1 N@MO which can be identified with the set CSN@MO of isotopy classes of curve systems in@M. Each factorH1NTiO ofH1N@MO is the integer lattice inH1NTi;RO R 2 , and H 1 NT i ;RO= 1 is a cone, so CSN@MO can be viewed as the integer lattice in a product of cones, the space H1N@M;RO=Z n where Z n acts by inversions in the factors H 1 NT i ;RO.

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Allen Hatcher

Papers

A presentation for the mapping class group of a closed orientable surface

Incompressible surfaces in 2-bridge knot complements

A proof of the Smale Conjecture, Diff(S3) 0(4)

On the boundary curves of incompressible surfaces.

On triangulations of surfaces