Let S be either a sphere with 5 punctures or a torus with 3 punctures. We prove that the automorphism group of the complex of curves of S is isomorphic to the extended mapping class group M S. As applications we prove that surfaces of genus 1 are determined by their complexes of curves, and any isomorphism between two subgroups of M S of nite index is the… (More)
We construct examples of Lefschetz fibrations with prescribed singular fibers. By taking differences of pairs of such fibrations with the same singular fibers, we obtain new examples of surface bundles over surfaces with non-zero signature. From these we derive new upper bounds for the minimal genus of a surface representing a given element in the second… (More)
In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer representation of the braid group B n , which was shown to be faithful by Bigelow and Krammer. We obtain a faithful representation of the mapping class group of… (More)
There exists a (relatively minimal) genus g Lef-schetz fibration with only one singular fiber over a closed (Rie-mann) surface of genus h iff g ≥ 3 and h ≥ 2. The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of two commutators and no Dehn twist… (More)
In this survey paper, we give a complete list of known results on the first and the second homology groups of surface mapping class groups. Some known results on higher (co)homology are also mentioned .
Let S be a compact, connected, orientable surface of positive genus. Let HT (S) be the Hatcher-Thurston complex of S. We prove that Aut (HT (S)) is isomorphic to the extended mapping class group of S modulo its center.
Wajnryb proved in [W2] that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist.
S. Bigelow proved that the braid groups are linear. That is, there is a faithful representation of the braid group into the general linear group of some field. Using this, we deduce from previously known results that the mapping class group of a sphere with punctures and hyperelliptic mapping class groups are linear. In particular, the mapping class group… (More)