Cup and Cap Product: Poincaré Duality

Related lectures (31)

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

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

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

Covers cohomology in real projective spaces, focusing on associative properties and algebraic structures.

Covers the concept of naturality in algebraic topology and the generation of homology groups through induction.

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

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

Introduces reduced homology groups and explains their properties and applications in topology.

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

Covers algebraic topology, Betti numbers, and shape representation methods for efficient data shape measurement and analysis.