Lectures related to Euler Characteristic: Surfaces and Homotopy

Explores chain homotopy, projective complexes, and homotopy equivalences in chain complexes.

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

Model Categories: Properties and Structures

Covers the properties and structures of model categories, focusing on factorizations, model structures, and homotopy of continuous maps.

The Whitehead Lemma: Homotopy Equivalence in Model Categories

Explores the Whitehead Lemma, showing when a morphism is a weak equivalence.

Serre model structure on Top

Explores the Serre model structure on Top, focusing on right and left homotopy.

Serre model structure: Left and right homotopy

Explores the Serre model structure, focusing on left and right homotopy equivalences.

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.

The Topological Künneth Theorem

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

Fundamental Groups

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Cell Attachment and Homotopy

Explores cell attachment, homotopy, function existence, construction using universal property, and continuity.

Elementary Properties of Model Categories

Covers the elementary properties of model categories, emphasizing the duality between fibrations and cofibrations.

Cell Attachment and Homotopy

Covers cell attachment, homotopy, mappings, and universal properties in topology.

Homotopy Invariance

Explores homotopy invariance, emphasizing the preservation of properties under continuous functions and their relationship with topological spaces.

Universal Covering Construction

Introduces the concept of a universal covering construction with examples like Hawaiian rings.

Unifying left and right homotopy: The homotopy relation

Explores the equivalence of left and right homotopy for morphisms between bifibrant objects.

Quasi-Categories: Active Learning Session

Covers fibrant objects, lift of horns, and the adjunction between quasi-categories and Kan complexes, as well as the generalization of categories and Kan complexes.

Induced Homomorphisms on Relative Homology Groups

Covers induced homomorphisms on relative homology groups and their properties.

Homotopy Classes and Group Structures

Explores wedges in pointed spaces, group structures, and the Echman-Hilton argument.

Homotopy Theory: Cylinders and Path Objects

Covers cylinders, path objects, and homotopy in model categories.

Base B for the covering

Explores constructing a base B for a topology using homotopy classes and paths.