Excision: An Example

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Homology groups: Quotients

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

Homotopy Theory of Chain Complexes

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

Cellular Homology: Applications

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

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.

Chain Maps: Homotopy Invariance

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

Singular homology

Introduces singular homology, defining singular simplices and explaining the chain complex construction.

Shape of Data: Algebraic Topology and Shape Representation

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

A Lemma: Introduction to Homology Groups

Introduces the concept of homology groups and focuses on a lemma about free abelian groups.

Active Learning: Functors and Geometric Realization

Covers the computation of nerves and geometric realization in simplicial sets, along with functors into and out of the category of simplicial sets.

Higher-order Interactions in Brain Networks

Explores higher-order interactions in brain networks using simplicial complexes and information theory, analyzing data from fMRI, financial time-series, and infectious diseases.