@article{article_1559701, title={Equivalent Crossed Homomorphisms on The Mapping Class Group}, journal={Necmettin Erbakan Üniversitesi Fen ve Mühendislik Bilimleri Dergisi}, volume={7}, pages={206–213}, year={2025}, author={Ünlü Eroğlu, Hatice}, keywords={Çapraz homomorfizm, Dolanım sayısı, Gönderim sınıfı grubu}, abstract={The mapping class group of a surface, which describes the isotopy classes of orientation-preserving self-diffeomorphisms, plays an important role in many areas of mathematics, particularly in topology, algebra and geometry. In topology, mapping class groups are essential for studying 3-manifolds and fiber bundles, while in algebra and geometry, they are closely related to the theory of automorphisms, moduli spaces, and complex structures on surfaces. An interesting perspective on mapping class groups involves the study of their cohomology classes. Cohomology classes of the mapping class groups of orientable surfaces can be considered as characteristic classes of surface bundles. There are several constructions of the cohomology class of the mapping class groups of orientable surfaces given by Earle, Morita, Furuta, and Trapp. These constructions seem very different. Therefore, various authors have made efforts to better understand the relationships between these constructions by comparing them. The crossed homomorphisms which yield the cohomology classes of the mapping class groups, as proposed by Furuta and presented by Trapp, are related to winding numbers. In this study, we show the relation between these two different constructions.}, number={2}, publisher={Necmettin Erbakan Üniversitesi}