Yönlendirilemeyen Yüzeylerin Gönderim Sınıf Gruplarının Cebirsel Yapısı

Mapping class groups of surfaces play a central role in the theory of low dimensional topology. Any information about algebraic structure of these groups might be useful in the solution of some topological problem of low dimensional manifolds. That is one reason why understanding algebraic sturucture of mapping class groups is so important. Consider a nonorientable connected genus g surface with k marked points. The main task of this project was to obtain information on the algebraic structure of the mapping class gorups of these surfaces. Through the project we have obtained results about automorphism groups of curve complexes of nonorientable surfaces and we gave some immediate applications of these results. In particular, we have shown that any isomorphism between two finite index subgroups of the mapping class group is given by the restriction of an inner automorphism of the mapping class group. Moreover, we have shown that the outer automorphism group of a nonorientable punctured surface of genus at least five is tirivial. Finally, we have observed that the Torelli subgroup of a nonorientable surface contains the generators analogous to those of orientable surfaces.