Homology and Homotopy

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Related lectures (31)

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Bar Construction: Homology Groups and Classifying Space

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

The Topological Künneth Theorem

Explores the topological Künneth Theorem, emphasizing commutativity and homotopy equivalence in 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.

Algebraic Kunneth Theorem

Covers the Algebraic Kunneth Theorem, explaining chain complexes and cohomology computations.

Cellular Homology: Applications

Delves into applying cellular homology to compute homology groups and Euler characteristic, showcasing its practical implications.

CW Approximation Theorem

Explores the CW Approximation Theorem, constructing CW complexes from spaces to ensure isomorphism on homology groups.

Long Exact Sequence in Homotopy

Explores the long exact sequence in homotopy, emphasizing the importance of sets and groups in the sequence.

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.

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.