Homology with coefficients

This lecture introduces the concept of homology with coefficients, where homology groups are defined with respect to an arbitrary abelian group instead of just integers. The instructor explains how to construct chain complexes with coefficients in a given group, define boundary operators, and obtain homology groups. The lecture covers the augmented chain complex, relative chain complexes, homotopy invariance, long exact sequences, excision, and the universal coefficient theorem for homology. The instructor also demonstrates how to apply the concept to compute homology groups of projective spaces with different coefficient groups, emphasizing the impact of the coefficient group's characteristics on the resulting homology groups.