By Benson Farb

The learn of the mapping category staff Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and team conception. This publication explains as many very important theorems, examples, and methods as attainable, fast and at once, whereas while giving complete info and protecting the textual content approximately self-contained. The booklet is acceptable for graduate students.A Primer on Mapping category teams starts via explaining the most group-theoretical houses of Mod(S), from finite new release by way of Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the way in which, valuable gadgets and instruments are brought, akin to the Birman special series, the complicated of curves, the braid team, the symplectic illustration, and the Torelli team. The ebook then introduces Teichmller area and its geometry, and makes use of the motion of Mod(S) on it to end up the Nielsen-Thurston type of floor homeomorphisms. issues comprise the topology of the moduli area of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov idea, and Thurston's method of the type.

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**Example text**

We call this technique the change of coordinates principle. One example of this principle is that, in order to prove a topological statement about an arbitrary nonseparating simple closed curve, we can prove it for any speciﬁc simple closed curve. We will see below that this idea applies to any conﬁguration of simple closed curves that is given by topological data. 1 C LASSIFICATION OF S IMPLE C LOSED C URVES As a prelude to our explanation of the change of coordinates principle, we present a classiﬁcation of simple closed curves in a surface.

We therefore want a method for converting homotopies to isotopies whenever possible. We already know i(a, b) is realized by geodesic representatives of a and b. Thus, in order to apply the above results on geometric intersection numbers to isotopy classes of curves, it sufﬁces to prove the following fact originally due to Baer. 10 Let α and β be two essential simple closed curves in a surface S. Then α is isotopic to β if and only if α is homotopic to β. Proof. One direction is vacuous since an isotopy is a homotopy.

The genus of a bounding pair in a closed surface is deﬁned similarly to the genus of a separating simple closed curve. 8 A genus 1 bounding pair. 3. Pairs (or k-tuples) of disjoint simple closed curves whose union does not separate. 4. Pairs of simple closed curves {α, β} with i(α, β) = |α ∩ β| = 2 and ˆi(α, β) = 0 and whose union does not separate. 5. Nonseparating simple proper arcs in a surface S that meet the same number of components of ∂S. 6. Chains of simple closed curves. A chain of simple closed curves in a surface S is a sequence α1 , .