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Eilenberg-Steenrod Axioms
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Related lectures (32)
Group Cohomology
Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.
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.
Topology Seminar: Tower Sequences and Homomorphisms
Explores tower sequences, homomorphisms, and their applications in topology, including the computation of homology and the construction of telescopes.
Cohomology Groups: Hopf Formula
Explores the Hopf formula in cohomology groups, emphasizing the 4-term exact sequence and its implications.
Homology of Riemann Surfaces
Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.
CW Approximation Theorem
Explores the CW Approximation Theorem, constructing CW complexes from spaces to ensure isomorphism on homology groups.
EML Spaces and Cohomology
Covers spaces, homology, chain groups, and abelianization in CW complexes and maps.
Homotopy Theory of Chain Complexes
Explores the homotopy theory of chain complexes, including path object construction and fibrations.
Cohomology: Cross Product
Explores cohomology and the cross product, demonstrating its application in group actions like conjugation.
Introduction to Topology
Covers fundamental concepts of space, topology, groups, and homology theory.
The Topological Künneth Theorem
Explores the topological Künneth Theorem, emphasizing commutativity and homotopy equivalence in chain complexes.
Simplicial and Singular Homology Equivalence
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Demonstrates the equivalence between simplicial and singular homology, proving isomorphisms for finite s-complexes and discussing long exact sequences.
Homology groups: Quotients
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Covers homology groups of quotients, homotopy invariance, and exact sequences.
Homotopy Invariance: Homology Groups
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Explores homotopy invariance and its application to homology groups of quotients, showcasing isomorphism and chain homotopy.
Singular Homology: First Properties
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Covers the first properties of singular homology and the preservation of decomposition and path-connected components in topological spaces.
Zig Zag Lemma
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Covers the Zig Zag Lemma and the long exact sequence of relative homology.
Homology with coefficients
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Covers homology with coefficients, introducing the concept of defining homology groups with respect to arbitrary abelian groups.
Relative Homology: Exact Sequence
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Covers the long exact sequence of relative homology groups and chain complexes.
Chain Maps: Homotopy Invariance
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Covers chain maps, homotopy invariance, homology groups, and induced homomorphisms between cycles and boundaries.
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