Every mapping class group is generated by 6 involutions

Let Modg,b denote the mapping class group of a surface of genus g with b punctures. Luo asked in [Torsion elements in the mapping class group of a surface, math.GT/0004048, v1 8Apr2000] if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate Modg,b. We answer Luoʹs question by proving that 3 torsion elements suffice to generate Modg,0. We also prove the more delicate result that there is an upper bound, independent of genus, not only for the number of torsion elements needed to generate Modg,b but also for the order of those elements. In particular, our main result is that 6 involutions (i.e., orientation-preserving diffeomorphisms of order two) suffice to generate Modg,b for every genus g 3, b=0 and g 4, b=1.