Introduction to Algebraic Topology

Explore the foundational concepts of algebraic topology through this comprehensive lecture that bridges abstract mathematical theory with geometric intuition. Begin with an overview of algebraic topology's core principles before diving into the fundamental group of the circle, a crucial example that illustrates how algebraic structures can capture topological properties. Master the van Kampen theorem, a powerful tool for computing fundamental groups of complex spaces by decomposing them into simpler components. Discover how the fundamental groupoid can be understood as a topological space itself, providing deeper insight into the relationship between algebra and topology. Investigate the rich theory of surfaces, learning to classify and understand their topological properties. Conclude with the complete classification theorem for path-connected, compact surfaces, which demonstrates how algebraic topology provides elegant solutions to complex geometric problems.