By Farb B., Margalit D.
The research 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 team thought. This e-book explains as many very important theorems, examples, and methods as attainable, fast and without delay, whereas while giving complete info and preserving the textual content approximately self-contained. The e-book is acceptable for graduate students.A Primer on Mapping classification teams starts off via explaining the most group-theoretical houses of Mod(S), from finite iteration by way of Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the best way, relevant items and instruments are brought, reminiscent of the Birman specified series, the complicated of curves, the braid crew, the symplectic illustration, and the Torelli crew. 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 type of floor homeomorphisms. themes comprise the topology of the moduli area of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov idea, and Thurston's method of the class.
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Additional resources for A primer on mapping class groups
Let us look at two of the simplest examples: the closed disk D and the closed annulus A. On D, any orientation-reversing homeomorphism f induces a degree −1 map on S 1 = ∂D, and from this follows that f is not isotopic to the identity. However, the straight-line homotopy gives a homotopy between f and the identity. On A = S 1 × I, the orientation-reversing map that fixes the S 1 factor and reflects the I factor is homotopic but not isotopic to the identity. It turns out that these two examples are the only examples of homotopic homeomorphisms that are not isotopic.
Proof. Assume χ(S) ≤ 0, so the universal cover S is homeomorphic to R2 (the case of χ(S) > 0 is an exercise). Let p : S → S be the covering map. Suppose the lifts α and β of α and β intersect in at least two points. It follows that there is an embedded disk D0 in S bounded by one subarc of α and one subarc of β. By compactness and transversality, the intersection (p−1 (α)∪ p−1 (β))∩ D0 is a finite graph, if we think of the intersection points as vertices. 3). Denote the two “vertices” of D by v1 and v2 , and the two “edges” of D by α1 and β1 .
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 . Up to homotopy, we can assume that γ passes through the image in T 2 of (0, 0) in R2 . Any path lifting of γ to R2 based at the origin terminates at some integral point (p, q). There is then a homotopy from γ to the standard straight-line representative of (p, q) ∈ π1 (T 2 ); indeed, the straight-line homotopy from the lift of γ to the straight line through (0, 0) and (p, q) is equivariant with respect to the group of deck transformations and thus descends to the desired homotopy.
A primer on mapping class groups by Farb B., Margalit D.