By Farb B., Margalit D.
The research of the mapping category staff Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and workforce thought. This e-book explains as many very important theorems, examples, and methods as attainable, speedy and at once, whereas even as giving complete info and conserving the textual content approximately self-contained. The ebook is appropriate for graduate students.A Primer on Mapping classification teams starts via explaining the most group-theoretical homes of Mod(S), from finite iteration via Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the way in which, vital gadgets and instruments are brought, comparable to the Birman unique series, the advanced 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 end up the Nielsen-Thurston type 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 class.
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Additional resources for A primer on mapping class groups
So if α ∈ π1 (S) is centralized by β, it follows that α and β have the same fixed points in ∂H2 . By the discreteness of the action of π1 (S), we would then have that the centralizer of α in π1 (S) is infinite cyclic. If π1 (S) had nontrivial center, it would then follow that π1 (S) ≈ Z. But then S would necessarily have infinite volume, a contradiction. 2 SIMPLE CLOSED CURVES Our study of simple closed curves in a surface S begins with the study of all closed curves in S, and the usefulness of geometry in understanding them.
If χ(S) < 0 then S admits a hyperbolic metric. If χ(S) = 0 then S 22 CHAPTER 1 admits a Euclidean metric. A surface endowed with a fixed hyperbolic metric will be called a hyperbolic surface. A surface with a Euclidean metric will be called a Euclidean surface or flat surface. 2 is consistent with the Gauss–Bonnet theorem which, in the case of a compact surface S with totally geodesic boundary, states that the integral of the curvature over S is equal to 2πχ(S). One way to get a hyperbolic metric on a closed surface Sg is to construct a free, properly discontinuous isometric action of π1 (Sg ) on H2 (as above, this requires g ≥ 2).
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.