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A primer on mapping class groups by Farb B., Margalit D.

By Farb B., Margalit D.

The learn of the mapping classification team Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and team thought. This e-book explains as many vital theorems, examples, and methods as attainable, quick and without delay, whereas while giving complete info and holding the textual content approximately self-contained. The publication is appropriate for graduate students.A Primer on Mapping type teams starts by way of explaining the most group-theoretical houses of Mod(S), from finite new release via Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the way in which, valuable gadgets and instruments are brought, similar to the Birman specific series, the advanced of curves, the braid workforce, the symplectic illustration, and the Torelli workforce. The ebook then introduces Teichmller area and its geometry, and makes use of the motion of Mod(S) on it to turn out the Nielsen-Thurston class of floor homeomorphisms. issues contain the topology of the moduli area of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov idea, and Thurston's method of the category.

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A primer on mapping class groups

The examine of the mapping type workforce Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and team thought. This booklet explains as many very important theorems, examples, and strategies as attainable, speedy and without delay, whereas whilst giving complete information and conserving the textual content approximately self-contained.

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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. 36 CHAPTER 1 Proof. One direction is vacuous, since an isotopy is a homotopy. So suppose that α is homotopic to β. We immediately have that i(α, β) = 0. By performing an isotopy of α, we may assume that α is transverse to β. If α and β are not disjoint then by the bigon criterion they form a bigon. A bigon prescribes an isotopy that reduces intersection. Thus, we may remove bigons one by one by isotopy until α and β are disjoint.

The minimum of {k, g − k} is called the genus of the separating simple closed curve. By the above, the genus of a curve determines and is determined by its topological type. Note that there are ⌊ 2g ⌋ topological types of essential separating simple closed curves in a closed surface. The uninitiated may have trouble visualizing separating simple closed curves that are not the “obvious” ones. 5, and we encourage the reader to draw even more complicated separating simple closed curves. 5 Some nonobvious separating simple closed curves.

A multicurve in S is the union of a finite collection of disjoint simple closed curves in S. The notion of intersection number extends directly to multicurves. 7) gives a version of the bigon criterion for multicurves: two multicurves are in minimal position if and only if no two component curves form a bigon. g. the geodesic) so that each pair of curves is in minimal position. 5 H OMOTOPY VERSUS ISOTOPY FOR SIMPLE CLOSED CURVES Two simple closed curves α and β are isotopic if there is a homotopy H : S 1 × [0, 1] → S from α to β, with the property that the closed curve H(S 1 × {t}) is simple for each t ∈ [0, 1].

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