Group Cohomology

This lecture covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings. It explains the construction of group cohomology rings and their properties, such as homotopy equivalences, quasi-isomorphisms, and the role of classifying spaces. The lecture also delves into the relationship between chain complexes and homology, emphasizing the importance of negatively and non-negatively graded chain complexes. Furthermore, it discusses the dualization process and the definition of homology, showcasing the connection between chain homotopy and cochain complexes. The lecture concludes with the application of group cohomology in defining homomorphisms and inducing isomorphisms on homology groups.