Homology: Introduction and Applications

This lecture introduces the concept of homology as a tool to distinguish spaces in all dimensions, contrasting it with the limitations of fundamental groups. The instructor explains the motivation behind homology, its relation to higher homotopy groups, and the advantages it offers in proving theorems across various mathematical areas. The lecture covers the construction of simplicial homology for intuitive understanding and the transition to singular homology for theoretical handling. The use of delta complexes is presented as a combinatorial approach to defining homology, providing a more restrictive boundary condition. The instructor encourages exploring informal explanations of homology to gain a better grasp of the concept and hints at upcoming topics on cellular homology and applications of homology.