Lecture

Naturality: An Example

Related lectures (31)

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

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

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.

Cup and Cap Product: Poincaré Duality

Explains the cup and cap product in algebraic topology and their relation to homology groups.

Naturality: Chain Complexes and Homology Groups

Explores naturality in chain complexes, homology groups, and abelian groups, emphasizing the commutativity of squares and the Five-Lemma.

Chain Maps: Homotopy Invariance

Covers chain maps, homotopy invariance, homology groups, and induced homomorphisms between cycles and boundaries.

Homology groups: Quotients

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

Homology Groups: Basics

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

Shape of Data: Algebraic Topology and Shape Representation

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

Bar Construction: Homology Groups and Classifying Space

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

Induced Homomorphisms

Covers induced homomorphisms, homotopy invariance, and homology groups of quotients.

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.

Simplicial and Singular Homology Equivalence

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

The Topological Künneth Theorem

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

Homology Theorem

Covers the proof of Theorem A, discussing homology, quotients, and isomorphisms.

Cohomology: Cross Product

Explores cohomology and the cross product, demonstrating its application in group actions like conjugation.

Relative Homology: Homotopy Invariance

Explains relative homology, n-cycles, n-boundaries, and exact sequences of chain complexes.

Topology: Classification of Surfaces and Fundamental Groups

Discusses the classification of surfaces and their fundamental groups using the Seifert-van Kampen theorem and polygonal presentations.

Homotopy Theory of Chain Complexes

Explores the homotopy theory of chain complexes, including path object construction and fibrations.