Über die Autorin bzw. den Autor

Benson Farb is professor of mathematics at the University of Chicago. He is the editor of Problems on Mapping Class Groups and Related Topics and the coauthor of Noncommutative Algebra. Dan Margalit is assistant professor of mathematics at Georgia Institute of Technology.

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A Primer on Mapping Class Groups

PRINCETON UNIVERSITY PRESS

Contents

Preface....................................................................xiAcknowledgments............................................................xiiiOverview...................................................................1PART 1. MAPPING CLASS GROUPS...............................................151. Curves, Surfaces, and Hyperbolic Geometry...............................172. Mapping Class Group Basics..............................................443. Dehn Twists.............................................................644. Generating the Mapping Class Group......................................895. Presentations and Low-dimensional Homology..............................1166. The Symplectic Representation and the Torelli Group.....................1627. Torsion.................................................................2008. The Dehn–Nielsen–Baer Theorem...............................2199. Braid Groups............................................................239PART 2. TEICHMÜLLER SPACE AND MODULI SPACE............................26110. Teichmüller Space.................................................26311. Teichmüller Geometry..............................................29412. Moduli Space...........................................................342PART 3. THE CLASSIFICATION AND PSEUDO-ANOSOV THEORY........................36513. The Nielsen–Thurston Classification..............................36714. Pseudo-Anosov Theory...................................................39015. Thurston's Proof.......................................................424Bibliography...............................................................447Index......................................................................465

Chapter One

A linear transformation of a vector space is determined by, and is best understood by, its action on vectors. In analogy with this, we shall see that an element of the mapping class group of a surface S is determined by, and is best understood by, its action on homotopy classes of simple closed curves in S. We therefore begin our study of the mapping class group by obtaining a good understanding of simple closed curves on surfaces.

Simple closed curves can most easily be studied via their geodesic representatives, and so we begin with the fact that every surface may be endowed with a constant-curvature Riemannian metric, and we study the relation between curves, the fundamental group, and geodesics. We then introduce the geometric intersection number, which we think of as an "inner product" for simple closed curves. A second fundamental tool is the change of coordinates principle, which is analogous to understanding change of basis in a vector space. After explaining these tools, we conclude this chapter with a discussion of some foundational technical issues in the theory of surface topology, such as homeomorphism versus diffeomorphism, and homotopy versus isotopy.

1.1 SURFACES AND HYPERBOLIC GEOMETRY

We begin by recalling some basic results about surfaces and hyperbolic geometry that we will use throughout the book. This is meant to be a brief review; see [208] or [119] for a more thorough discussion.

1.1.1 SURFACES

A surface is a 2-dimensional manifold. The following fundamental result about surfaces, often attributed to Möbius, was known in the mid-nineteenth century in the case of surfaces that admit a triangulation. Radò later proved, however, that every compact surface admits a triangulation. For proofs of both theorems, see, e.g., [204].

THEOREM 1.1 (Classification of surfaces) Any closed, connected, orientable surface is homeomorphic to the connect sum of a 2-dimensional sphere with g = 0 tori. Any compact, connected, orientable surface is obtained from a closed surface by removing b = 0 open disks with disjoint closures. The set of homeomorphism types of compact surfaces is in bijective correspondence with the set {(g, b) : g, b = 0}.

The g in Theorem 1.1 is the genus of the surface; the b is the number of boundary components. One way to obtain a noncompact surface from a compact surface S is to remove n points from the interior of S; in this case, we say that the resulting surface has n punctures.

Unless otherwise specified, when we say "surface" in this book, we will mean a compact, connected, oriented surface that is possibly punctured (of course, after we puncture a compact surface, it ceases to be compact). We can therefore specify our surfaces by the triple (g, b, n). We will denote by S_g,n a surface of genus g with n punctures and empty boundary; such a surface is homeomorphic to the interior of a compact surface with n boundary components. Also, for a closed surface of genus g, we will abbreviate S_g,0 as S_g. We will denote by ?S the (possibly disconnected) boundary of S.

Recall that the Euler characteristic of a surface S is

X(S) = 2 - 2g - (b + n).

It is a fact that X(S) is also equal to the alternating sum of the Betti numbers of S. Since X(S) is an invariant of the homeomorphism class of S, it follows that a surface S is determined up to homeomorphism by any three of the four numbers g, b, n, and X(S).

Occasionally, it will be convenient for us to think of punctures as marked points. That is, instead of deleting the points, we can make them distinguished. Marked points and punctures carry the same topological information, so we can go back and forth between punctures and marked points as is convenient. On the other hand, all surfaces will be assumed to be without marked points unless explicitly stated otherwise.

If X(S) = 0] and [partial derivative]S = [empty set], then the universal cover [??] is homeomorphic to R² (see, e.g., [199, Section 1.4]). We will see that, when x(S) < 0, we can take advantage of a hyperbolic structure on [??].

1.1.2 THE HYPERBOLIC PLANE

Let H² denote the hyperbolic plane. One model for H² is the upper half-plane model, namely, the subset of C with positive imaginary part (y > 0), endowed with the Riemannian metric

ds² = [dx² + dy²]/y²,

where dx² + dy² denotes the Euclidean metric on C. In this model the geodesics are semicircles and half-lines perpendicular to the real axis.

It is a fact from Riemannian geometry that any complete, simply connected Riemannian 2-manifold with constant sectional curvature -1 is isometric to H².

For the Poincaré disk model of H², we take the open unit disk in C with the Riemannian metric

ds² = 4 dx² + dy²/(1 - r²)².

In this model the geodesics are circles and lines perpendicular to the unit circle in C (intersected with the open unit disk).

Any Möbius transformation from the upper half-plane to the unit disk is an isometry between the upper half-plane model for H² and the Poincaré disk model of H². The...