Geometric realization: Simplicial sets

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Knot Theory: The Quadratic Linking Degree

Covers the quadratic linking degree in knot theory, exploring its definitions, properties, and significance in algebraic geometry.

Simplicial Homology: Structure and Complexes

Covers the structure of topological spaces with A-complexes and chain complexes.

Topology: Fundamental Groups and Applications

Provides an overview of fundamental groups in topology and their applications, focusing on the Seifert-van Kampen theorem and its implications for computing fundamental groups.

Bar Construction: Homology Groups and Classifying Space

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

Transformations and Inversions: Laplace and Fourier

Discusses Laplace and Fourier transformations, focusing on their inversion formulas and applications in solving differential equations.

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.

Homology Theorem

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

Homotopy Extension Property

Demonstrates how to obtain homotopy equivalences between different spaces using the homotopy extension property.

Homology and Homotopy

Explores the comparison of long exact sequences for vibrations and the relationship between homotopy and homology groups.

Homotopy Theory in Care Complexes

Explores the construction of cylinder objects in chain complexes over a field, focusing on left homotopy and interval chain complexes.