Mapping class groups and generators

In this study, surface, surface homeomorphisms and isotopy classses of surface homeomorphisms are examined. These isotopy classes constitute a group under the name of the "mapping class group". The main purpose of this thesis is to focus on the generators of the mapping class group. Dehn twists, the most basic generators of the mapping class group, are examined. Dehn twists play an important role in generating the mapping class group of the surfaces. Different generator sets from Dehn twists are also investigated. On this purpose, the torsion (finite order) elements are added to the generating sets of the mapping class group of the surfaces. Then finding the minimum order generators are studied. On this, the generator set consisting of order two generators, namely involutions, are examined. "How many involutions can generate the mapping class group" is considered and the most recent studies to this problem are investigated.