By Farb B., Margalit D.
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 crew thought. This ebook explains as many very important theorems, examples, and methods as attainable, fast and at once, whereas while giving complete info and preserving the textual content approximately self-contained. The booklet is acceptable for graduate students.A Primer on Mapping type teams starts off by way of explaining the most group-theoretical houses of Mod(S), from finite new release through Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the best way, relevant gadgets and instruments are brought, akin to the Birman unique series, the complicated of curves, the braid crew, the symplectic illustration, and the Torelli crew. The ebook then introduces Teichmller house and its geometry, and makes use of the motion of Mod(S) on it to turn out the Nielsen-Thurston class of floor homeomorphisms. themes contain the topology of the moduli area of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov conception, and Thurston's method of the class.
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Extra info for A primer on mapping class groups
By considering the number of fixed points of f in H2 , we obtain a classification of isometries of H2 , as follows. Elliptic. If f fixes a point p ∈ H2 then f is called elliptic, and it is a rotation about p. Elliptic isometries have no fixed points on ∂H2 . They correspond to elements of PSL(2, R) whose trace has absolute value less than 2. Parabolic. If f has exactly one fixed point in ∂H2 , then f is called parabolic, In the upper half-plane model, f is conjugate in Isom+ (H2 ) to z → z ± 1.
1 H OMOTOPY VERSUS ISOTOPY FOR HOMEOMORPHISMS When are two homotopic homeomorphisms isotopic? 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.
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. 10 (see [8, 9], and also ). 12 Let S be any compact surface, and let f and g be homotopic homeomorphisms of S. Then f and g are isotopic unless they are one of the two examples described above (on S = D and S = A). In particular, if f and g are orientation-preserving then they are isotopic.
A primer on mapping class groups by Farb B., Margalit D.