By Benson Farb
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 staff concept. 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.The publication starts off 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, relevant gadgets and instruments are brought, reminiscent of the Birman particular series, the advanced of curves, the braid workforce, the symplectic illustration, and the Torelli staff. The ebook then introduces Teichmüller area and its geometry, and makes use of the motion of Mod(S) on it to end up the Nielsen-Thurston class of floor homeomorphisms. issues comprise the topology of the moduli house of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov conception, and Thurston's method of the class.
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Extra resources for A Primer on Mapping Class Groups (Princeton Mathematical)
In the statement, Homeo0 (S) is the connected component of the identity in the space of homeomorphisms of a surface S. 14 Let S be a compact surface, possibly minus a finite number of points from the interior. Assume that S is not homeomorphic to S 2 , R2 , D 2 , T 2 , the closed annulus, the once-punctured disk, or the oncepunctured plane. Then the space Homeo0 (S) is contractible. 14. 2, in the proof of the Birman exact sequence. 14; see  or . Chapter Two Mapping class group basics In this chapter we begin our study of the mapping class group of a surface.
Indeed, if α = β k , then β α = α while βαβ −1 = α. Note that the above correspondence does not hold for the torus T 2 . This is CURVES AND SURFACES 25 so because each closed curve has infinitely many lifts, while each element of π1 (T 2 ) ≈ Z2 is its own conjugacy class. Of course π1 (T 2 ) is its own center, and so the centralizer of each element is the whole group. Geodesic representatives. A priori the combinatorial topology of closed curves on surfaces has nothing to do with geometry. It was already realized in the nineteenth century, however, that the mere existence of constant curvature Riemannian metrics on surfaces has strong implications for the topology of the surface and of simple closed curves in it.
Let φ : D 2 → D 2 be a homeomorphism with φ|∂D2 equal to the identity. We define F (x, t) = (1 − t)φ x x 1−t 0 ≤ |x| < 1 − t 1 − t ≤ |x| ≤ 1. for 0 ≤ t < 1, and we define F (x, 1) to be the identity map of D 2 . The result is an isotopy F from φ to the identity. 4. The individual F (⋆, t) homeomorphisms appear at horizontal slices. 4 The picture for the Alexander trick. The isotopy given by the proof can be thought of as follows: at time t, do the original map φ on the disk of radius 1 − t, and apply the identity map outside of this disk.
A Primer on Mapping Class Groups (Princeton Mathematical) by Benson Farb