A Lemma: Introduction to Homology Groups

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Singular homology

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

Algebra: Chain Complexes

Explores chain complexes, abelian groups, homomorphisms, homology groups, and free abelian groups.

Singular Homology: First Properties

Covers the first properties of singular homology and the preservation of decomposition and path-connected components in topological spaces.

Simplicial Homology: Structure and Complexes

Covers the structure of topological spaces with A-complexes and chain complexes.

Simplicial and Singular Homology Equivalence

Demonstrates the equivalence between simplicial and singular homology, proving isomorphisms for finite s-complexes and discussing long exact sequences.

Homology of Riemann Surfaces

Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.

Homotopy Invariance: Homology Groups

Explores homotopy invariance and its application to homology groups of quotients, showcasing isomorphism and chain homotopy.

Homology with coefficients

Covers homology with coefficients, introducing the concept of defining homology groups with respect to arbitrary abelian groups.

Bar Construction: Homology Groups and Classifying Space

Covers the bar construction method, homology groups, classifying space, and the Hopf formula.

Universal Coefficient Theorems

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