Relative Homology: Homotopy Invariance

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Delves into the universal coefficient theorems in homological algebra, showcasing their practical application in computing homology and cohomology groups.

Singular homology

Introduces singular homology, defining singular simplices and explaining the chain complex construction.

Naturality: Chain Complexes and Homology Groups

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

Hurewicz Theorem

Explores the proof of the Hurewicz Theorem and its applications to spheres and homotopy groups.

Homotopy Theory of Chain Complexes

Explores the homotopy theory of chain complexes over a field, focusing on closure properties and decomposition.

Cohomology Groups: Hopf Formula

Explores the Hopf formula in cohomology groups, emphasizing the 4-term exact sequence and its implications.

Transformations and Inversions: Laplace and Fourier

Discusses Laplace and Fourier transformations, focusing on their inversion formulas and applications in solving differential equations.

Acyclic Models: Cup Product and Cohomology

Covers the cup product on cohomology, acyclic models, and the universal coefficient theorem.

Simplicial Homology Revisited

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

EML Spaces and Cohomology

Covers spaces, homology, chain groups, and abelianization in CW complexes and maps.