By Benson Farb
The examine of the mapping category crew Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and workforce concept. This booklet explains as many vital theorems, examples, and strategies as attainable, fast and without delay, whereas while giving complete information and conserving the textual content approximately self-contained. The ebook is acceptable for graduate students.The publication starts through explaining the most group-theoretical homes of Mod(S), from finite iteration by way of Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the way in which, critical items and instruments are brought, reminiscent of the Birman distinctive series, the advanced of curves, the braid workforce, the symplectic illustration, and the Torelli team. The publication then introduces Teichmüller house and its geometry, and makes use of the motion of Mod(S) on it to end up the Nielsen-Thurston class of floor homeomorphisms. subject matters contain the topology of the moduli area of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov conception, and Thurston's method of the category.
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Extra resources for A Primer on Mapping Class Groups (Princeton Mathematical)
In higher dimensions, the situation is worse: it is not known if Diff(D 4 , ∂D4 ) is connected, and for infinitely many n we have that Diff(D n , ∂Dn ) is not connected. 1 also holds with D 2 replaced by a once-punctured disk (take the puncture/marked point to lie at the origin) and hence we also have the following: The mapping class group of a once-punctured disk is trivial. The sphere and the once-punctured sphere. There are two other mapping class groups Mod(Sg,n ) that are trivial, namely Mod(S0,1 ) and Mod(S 2 ).
We can classify the set of homotopy 28 CHAPTER 1 classes of simple closed curves in the torus T 2 as follows. Let R2 → T 2 be the usual covering map, where the deck transformation group is generated by the translations by (1, 0) and (0, 1). We know that π1 (T 2 ) ≈ Z2 , and, if we base π1 (T 2 ) at the image of the origin, one way to get a representative for (p, q) as a loop in T 2 is to take the straight line from (0, 0) to (p, q) in R2 and project it to T 2 . Let γ be any oriented simple closed curve in T 2 .
For hyperbolic surfaces, geodesics are the natural representatives of each free homotopy class, in the following sense. 6 Let S be a hyperbolic surface. Let α be a closed curve in S, not homotopic into a neighborhood of a puncture. 3. If α is simple then γ is simple. CURVES AND SURFACES 29 Proof. We begin by applying the following fact. A closed curve β in a hyperbolic surface S is simple if and only if the following properties hold: 1. Each lift of β to H2 is simple. 2. No two lifts of β intersect.