By Benson Farb
The research 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 staff concept. This publication explains as many very important theorems, examples, and strategies as attainable, speedy and without delay, whereas even as giving complete information and conserving the textual content approximately self-contained. The publication is acceptable for graduate students.The publication starts off by way of explaining the most group-theoretical homes of Mod(S), from finite new release through Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the way in which, relevant gadgets and instruments are brought, resembling the Birman certain series, the advanced of curves, the braid team, the symplectic illustration, and the Torelli workforce. The ebook then introduces Teichmüller house and its geometry, and makes use of the motion of Mod(S) on it to turn out the Nielsen-Thurston type of floor homeomorphisms. subject matters contain the topology of the moduli house of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov concept, and Thurston's method of the type.
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Additional info for A Primer on Mapping Class Groups (Princeton Mathematical)
In the literature, there are various other notations for the mapping class group, for instance: MCG(S), Map(S), M(S), and Γg,n . As a general rule, “mapping class group” refers to the group of homotopy classes of homeomorphisms of a surface, but there are plenty of variations: one can consider homeomorphisms that do not necessarily preserve the orientation of the surface, or that do not act as the identity on the boundary, or that fix each puncture individually, etc. Punctures versus marked points.
We give a different proof that two curves not in minimal position must form a bigon. 11 below). We may assume without loss of generality that α and β are transverse and that H is transverse to β (in particular, all maps are assumed to be smooth). Thus, the preimage H −1 (β) in the annulus S 1 × [0, 1] is a 1–submanifold. There are various possibilities for a connected component of H −1 (β): it could be a closed curve, an arc connecting distinct boundary components, or an arc connecting one boundary component to itself.
Minimal position. In practice, one computes the geometric intersection number between two homotopy classes a and b by finding representatives α 31 CURVES AND SURFACES and β that realize the minimal intersection in their homotopy classes, so that i(a, b) = |α ∩ β|. When this is the case we say that α and β are in minimal position. Two basic questions now arise. 1. Given two simple closed curves α and β, how can we tell if they are in minimal position? 2. Given two simple closed curves α and β, how do we find homotopic simple closed curves that are in minimal position?
A Primer on Mapping Class Groups (Princeton Mathematical) by Benson Farb