Homogeneous Spaces and Quotients

This lecture starts with a brief mention of Sofia Kovalevskaia, the first woman to obtain a PhD and a professorship in mathematics. The main topic is the completion of the first chapter on group actions, focusing on homogeneous actions of subgroups on groups. Examples include orthogonal and special orthogonal groups, homotopy types, and homomorphisms. The lecture also covers push-outs, gluing, and cell attachments. The instructor addresses questions on separating quotient spaces, actions of compact spaces, and the separation of elements in group quotients.