Explore the concept of homology groups in algebraic topology through an in-depth examination of simple examples like the circle and disk. Learn how to define homology groups H_n(X) for a space X, understand the abstract process of considering complexes of abelian groups, and grasp the significance of boundary maps. Discover how different dimensions of homology groups measure various aspects of a space, such as connectivity, non-trivial loops, and 2-dimensional holes. Gain insights into the mathematical foundations of topology and develop a deeper understanding of how these abstract concepts relate to geometric structures.