A Lemma: Introduction to Homology Groups

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Explores naturality in chain complexes, homology groups, and abelian groups, emphasizing the commutativity of squares and the Five-Lemma.

Introduces reduced homology groups and explains their properties and applications in topology.

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Explores the Mayer-Vietoris sequence, exact homomorphisms, embedded spheres, and path-connected spaces.

Covers the proof of Theorem A, discussing homology, quotients, and isomorphisms.

Covers chain maps, homotopy invariance, homology groups, and induced homomorphisms between cycles and boundaries.

Explores simplicial and singular homology, reduced homology groups, and their connection to the fundamental group.

Covers the concept of simplicial homology, focusing on finite complexes and induced maps.

Explains relative homology, n-cycles, n-boundaries, and exact sequences of chain complexes.

Covers the long exact sequence of relative homology groups and chain complexes.