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A Primer on Mapping Class Groups

Benson Farb and Dan Margalit
Princeton University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Scott A. Taylor
, on

A Primer on Mapping Class Groups (henceforth, the Primer) was circulated in preprint form by its authors for several years before its publication. By the time it was actually in print, it had already become the standard reference for the basic facts and techniques of mapping class groups. This fame is well deserved: the book is one of the best-written mathematics books I have read. Before beginning a proof, for example, the authors first give an outline, an overview of the important ideas, or an easier version. They often include multiple proofs of the same result to demonstrate the efficacy of certain techniques or to prepare the reader for more sophisticated ideas to come. The book is mostly self-contained, but includes references to recent literature guiding the reader to topics of current research. This book has become essential for anyone interested in low-dimensional topology, low-dimensional geometry, or geometric group theory. Different parts of the book make use of different prerequisites. At a minimum, the reader needs a solid foundation in topology (fundamental groups, covering spaces, homology theory), some familiarity with complex analysis, and some prior exposure to 2-dimensional hyperbolic geometry.

In order to describe some of the content of the book, and simply because it’s a beautiful area of mathematics, let me indulge in some definitions. For any topological space \(X\), the set of homeomorphisms \(\operatorname{Homeo}(X)\) from \(X\) to itself forms a group under composition. For interesting spaces (such as manifolds) this group is typically much too large to be useful. To get a more tractable group, but one which is still related in interesting ways to the topology of \(X\), we consider the quotient group obtained by considering two homeomorphisms \(f: X \to X\) and \(g: X \to X\) to be equivalent if they are homotopic. (Informally, this means that the map \(f\) can be continuously deformed to the map \(g\).) If \(X\) is an oriented surface (connected real 2-dimensional manifold) then the mapping class group of \(X\) is the group \(\operatorname{Mod}{X}\) of orientation-preserving homeomorphisms modulo equivalence up to homotopy.

The most basic, and most important, example of a homeomorphism representing a non-trivial mapping class is the Dehn twist. There is a Dehn twist around each non-contractible simple closed curve \(c\) in \(X\). To perform a Dehn twist, cut the surface open along \(c\), rotate one of the sides of \(c\) by \(2\pi\) and then reglue, as illustrated below. It turns out that Dehn twists generate the mapping class group. In fact, for a surface of genus \(g\), it is possible to choose \(2g + 1\) simple closed curves such that Dehn twists about those curves generate the mapping class group. Although each Dehn twist is itself of infinite order in \(\operatorname{Mod}(X)\), there are finite order mapping classes (can you find one?). This rather elementary fact points to some of the complexity inherent in the study of mapping class groups.

The first few chapters of the Primer are devoted to basic facts about mapping class groups and to finding presentations for \(\operatorname{Mod}(X)\). An important tool in this endeavor is the action of \(\operatorname{Mod}(X)\) on the first homology group (with integer coefficients) of \(X\). The kernel of the action is called the Torelli group of \(X\) and is a subject of active research. The action of \(\operatorname{Mod}(X)\) on the first homology of \(X\) preserves a symplectic form on the homology group, so the study of mapping class groups is also connected to symplectic geometry. The Primer gives a short, but very nice, introduction to these connections.

Later chapters of the Primer develop connections between mapping class groups and other important areas of mathematics such as hyperbolic geometry, Teichmüller theory, and 3-manifold theory. With all these connections to other areas of mathematics, I’m sure there was the temptation to wander far afield. Yet the authors do a remarkable job of staying focussed on mapping class groups while introducing just enough from these other areas to whet the appetite for further explorations. Chapter 8 of the Primer is a wonderful example of this. In chapter 8, the authors prove the “Dehn-Nielsen-Baer” theorem which relates the topology of \(X\) to the algebra of its fundamental group. More precisely, \(\operatorname{Mod}(X)\) is isomorphic to an index 2 subgroup of the the outer isomorphisms of the fundamental group \(\pi_1(X)\) of \(X\). The authors provide three (!) different proofs of the Dehn-Nielsen-Baer theorem. The first uses facts about the “linking at infinity” of axes of isometries of the hyperbolic plane. The second uses “pants decompositions” of surfaces, decompositions which are familiar to 3-manifold topologists. The third is a one-paragraph proof which uses harmonic maps and appeals to deep theorems.

Also of interest in the Primer is the account, following Bers, of the Nielsen-Thurston classification of self-homeomorphisms of surfaces. This is one of the most readable accounts I have seen; although other approaches are also interesting and useful.

In conclusion, I highly recommend this book — it should be on the shelf of everyone interested in the interplay between topology, geometry, and algebra in low-dimensions.

Scott Taylor is a 3-manifold topologist and knot theorist at Colby College.