Topology: Fundamental Groups and Applications

This lecture covers the fundamental concepts of topology, focusing on the theorem of Seifert-van Kampen and its applications. The instructor begins by discussing the corollaries of the theorem, emphasizing its importance in understanding the fundamental group of topological spaces. The lecture illustrates how to compute the fundamental group of a space obtained by attaching a 2-cell, using specific examples to clarify the process. The instructor explains the significance of choosing appropriate base points and how this choice affects the computation of fundamental groups. The discussion includes the relationship between homotopy and fundamental groups, highlighting how different choices of base points can lead to isomorphic groups. The lecture concludes with exercises that reinforce the concepts presented, encouraging students to explore the implications of the Seifert-van Kampen theorem in various topological contexts. Overall, the lecture provides a comprehensive overview of fundamental groups in topology, equipping students with the necessary tools to analyze and understand complex topological structures.