Foundations of Algebraic Topology: Homology and Cohomology

How do we mathematically distinguish a donut from a sphere, or classify complex geometric shapes? Algebraic topology provides the powerful framework to solve these geometric puzzles by converting topological spaces into algebraic structures. This text-based course guides you through the core concepts of singular homology and cohomology, giving you the theoretical tools to analyze spaces rigorously. You will build a strong mathematical intuition for how algebraic invariants reveal the fundamental properties of shapes. What you'll learn: - Understand the foundational definitions of topological spaces, chain complexes, and functors - Calculate singular homology groups for standard geometric shapes and CW complexes - Apply homological algebra techniques, including exact sequences and diagram chasing - Explore the structure of cohomology groups and the cup product structure - Analyze the duality of manifolds using Poincaré duality theorems - Discover modern perspectives on computational topology and persistent homology concepts The course begins with essential definitions of categories, functors, and chain complexes before progressing to singular homology, CW approximations, and advanced cohomology theories. You will practice through detailed written proofs, step-by-step derivations, and conceptual exercises. This course is designed for mathematics students and self-learners with a basic background in point-set topology and abstract algebra. No prior knowledge of algebraic topology is required. Start reading today to unlock the algebraic language of geometric spaces.