Eilenberg-Steenrod Axioms

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Related lectures (32)

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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 groups: Quotients

Covers homology groups of quotients, homotopy invariance, and exact sequences.

Homotopy Invariance: Homology Groups

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

Singular Homology: First Properties

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

Group Cohomology

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

Zig Zag Lemma

Covers the Zig Zag Lemma and the long exact sequence of relative homology.

Homology with coefficients

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

Relative Homology: Exact Sequence

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

Bar Construction: Homology Groups and Classifying Space

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

Homology and Homotopy

Explores the comparison of long exact sequences for vibrations and the relationship between homotopy and homology groups.