Simplicial and Singular Homology Equivalence

This lecture covers the equivalence between simplicial and singular homology, demonstrating how every simplicial n-chain of a topological space X can be viewed as a singular n-chain. The inclusion of chain complexes induces isomorphisms between the homology groups, proven for finite s-complexes. The lecture also discusses the long exact sequences and the commutative diagrams of chain maps, emphasizing the isomorphisms between the homology groups. The proof is completed by assuming X to be finite-dimensional. The lecture concludes by showing that the induced maps are vertical chain maps, and the diagram commutes due to naturality.