Topology: Classification of Surfaces and Fundamental Groups

This lecture covers the classification of surfaces and their fundamental groups, focusing on the application of the Seifert-van Kampen theorem. The instructor discusses the standard polygonal presentation of surfaces and how to construct surfaces like tori and projective planes. The classification theorem is explained, emphasizing that every surface can be represented in a polygonal form. The lecture also highlights the importance of the fundamental group in distinguishing surfaces, including the role of the abelianization of these groups. The instructor illustrates how to calculate the fundamental group of a torus with holes and discusses the significance of the Euler characteristic in surface classification. The concept of covering spaces is introduced, along with the principle of uniqueness in lifting paths. The lecture concludes with exercises on calculating the fundamental groups of various surfaces, reinforcing the theoretical concepts presented throughout the session.